Linear Algebra Over a Ring

LINEAR ALGEBRA O VER A RING IV O HERZOG Abstract. Giv en an associative, not necessarily commutat ive, r i ng R with ident ity , a formal matrix calculus is int ro duced and developed for pairs of matrices ov er R. This calculus subsumes the theory of homogeneous s ystems of linear equations with co efficien ts in R. In the case when the ring R i s a field, every pair is equiv alen t to a homogeneous system. Using the f ormal matrix calculus, t wo alternate presentat ions are given for the Grothendiec k group K 0 ( R -mod , ⊕ ) of the category R -mod of finitely p resente d mo d- ules. One of these pr esen tations suggests a homological interpreta tion, and so a complex is i n tro duced whose 0-dimensional homology i s naturally isomorphic to K 0 ( R -mod , ⊕ ) . A computation shows that if R = k is a field, then the 1-dim ensional homology group is give n by ( k × ) ab / {± 1 } , where k × denotes the mu ltiplicitav e group of k , and ( k × ) ab its abelianization. The formal matrix calculus, which consists of three r ules of matrix op eration, is the syn tax of a deductiv e system whose completeness was pr o ve d by Prest. The three rules of inf er ence of this deductiv e system corresp ond to the three r ules of matrix operation, which app ear in the f ormal matrix calculus as the Rules of Divisi bility . Let R b e an a sso ciative, not neces sarily commutativ e, ring with identit y . F or ev ery natural num ber n, define L ′ n ( R ) to be the collection of pairs ( B | A ) where A and B are matrices with entries in R such that A has n columns, and B has the same num ber of rows as A. Define the rela tion ( B | A ) ≤ n ( B ′ | A ′ ) to hold in L ′ n ( R ) provided there exist ma trices U, V a nd G, of appr o priate size, such tha t U B = B ′ V a nd U A = A ′ + B ′ G. Separating o ut the individual r oles o f the three ma trices, one verifies easily (Theorem 2) that this relation is the least pre-order on L ′ n ( R ) satisfying the following three Rules of (Left) Divis ibility (RoD) for ma trices: (1) if U is a matrix with m columns, then ( B | A ) ≤ n ( U B | U A ) . (2) if V is a matrix with k rows, then ( B V | A ) ≤ n ( B | A ) . (3) if G is a k × n matrix, then ( B | A + B G ) ≤ n ( B | A ) In this article, we develop the formal matrix ca lc ulus that arises from these rules. Two pairs in L ′ n ( R ) are equiv alen t ( B | A ) ≈ n ( B ′ | A ′ ) provided b oth ( B | A ) ≤ n ( B ′ | A ′ ) and ( B ′ | A ′ ) ≤ n ( B | A ) hold; an n -ary matrix p air [ B | A ] is defined to b e an equiv alence cla ss of this rela tion. The collection of n -ary matrix pairs is denoted by L n ( R ); the partial order induced on L n ( R ) by ≤ n is denoted using t he same notatio n. This pa rtial order L n ( R ) o f n -a ry matrix pairs has a maximum element 1 n that sa tisfies Prop ositio n 9. [ B | A ] = 1 n if and only if there exists a matr ix W such that A = B W . 2000 Mathematics Subje ct Classific ation. 03B22, 06C05, 15A24, 16E20, 18F30, 19D55. Key wor ds a nd p hr ases. Grothendiec k group, h omology , finitely presented mo dule, mat rix divisibil it y , systems of linear equations. The author was partially supported by a gran t from Y onsei Universit y and NSF Gran t DMS-05-01207. 1 The element ( B | A ) in L ′ n ( R ) may th us b e interpreted as the prop os ition ” B divides A on the left” and t he n -ary matrix pair [ B | A ] as a p oint in the partial o rder L n ( R ) that measures the exten t to which B divides A o n the left. Seen in this light, RoD (1), for example, a sserts that the likeliho o d that U B divides U A on the left is a t least as hig h a s the likeliho o d that B divides A on the left. An element of L ′ n ( R ) is called a ( homo gene ous ) system ( of line ar e quations in n variables ) provided it is of the form ( 0 | A ) . The m atrix of c o efficients of this system is A a nd, for simplicity , we write ( A ) := ( 0 | A ) , and deno te the asso ciated n - ary matrix pair by [ A ] := [ 0 | A ] . It follows fro m the definition that if A and A ′ are matrices, both with n columns, then ( A ) ≈ n ( A ′ ) in L ′ n ( R ) if and only if there exist matrices U and U ′ such that U A = A ′ and U ′ A ′ = A. This implies that for every left R -mo dule R M , the t wo solution subgroups of ( R M ) n of the homogeneous systems with matrices of co efficients A a nd A ′ , resp ectively , are the same. Corollary 12. The ring R is von Neumann reg ula r if and only if for every n ≥ 1 (resp., n = 1 ) , every ( B | A ) ∈ L ′ n ( R ) is equiv a lent to a ho mo geneous system. A field k is certainly a von Neumann r egular ring, so the formal matrix ca lc ulus on L n ( k ) coincides with the study of homogeneous systems of linear equations in n v a riables, with co efficients in k . The other main features of this matrix calculus may b e s ummarized as follows. Corollary 1 5. The map [ B | A ] 7→ " B tr 0 A tr I n # is an anti-isomorphism b etw ee n L n ( R ) and L n ( R op ) . Theorem 16. The partial order L n ( R ) is a mo dular lattice with ma xim um and minimum elements. Theorem 18 . A mo r phism f : R → S of rings is a n epimorphis m if a nd only if fo r every n ≥ 1 (resp. n = 2), the induced morphism L n ( f ) : L n ( R ) → L n ( S ) is onto. The elements ( B | A ) of L ′ n ( R ) may be interpreted as the syntax of a deductive system, whose completeness was proved by Prest (Lemma 1.1.13 of [10]. The three matrices that app ear in the statement o f Pr e st’s r esult co rresp ond to the three rules of infer e nce (cf. the commentary pre ceding ibid. ) of the deductiv e sy s tem, a nd in the for mal matrix calc ulus find expression as the Rules of Divisibilit y (RoD) (1 )-(3). P recisely , let L ( R ) b e the langua g e for left R -mo dules, and deno te by T ( R ) the standard collection of axioms, express ible in L ( R ) , for a left R -mo dule. Associa te to the element ( B | A ) of L ′ n ( R ) the for mula ( B | A )( v ) := ∃ w ( B w . = A v ) , where v is a co lumn n - vector of v ariables v i and w a c olumn k -vector of v ariables w j . Then we may introduce a pr e-order or der on L ′ n ( R ) by defining ( B | A ) ⊢ n ( B ′ | A ′ ) to ho ld provided that T ( R ) ⊢ ∀ v ( B | A v → B ′ | A ′ v ) . It is re adily verified that this pre- order obe y s the three Rules of Divisibility , and since ≤ n is the le ast pre- order that ob eys these rules, ( B | A ) ≤ n ( B ′ | A ′ ) implies ( B | A ) ⊢ n ( B ′ | A ′ ) . Theorem 39 ( Lemm a Presta I. ) [1 0, Lemma 1 .1.13 and Cor. 1.1 .1 6] Given ( B | A ) a nd ( B ′ | A ′ ) in L ′ n ( R ) , ( B | A ) ≤ n ( B ′ | A ′ ) if and only if ( B | A ) ⊢ n ( B ′ | A ′ ) . 2 Lemma Pres ta is a completeness theorem, for it shows that a n y implication b etw een formulae of the form ( B | A )( v ) that is pr ov able rela tiv e to the ax ioms T ( R ) is pr ov able using the thre e Rules of Divisibility , co nstrued as rules of inferenc e . Once the relations hip b etw een these t wo partial order s on L ′ n ( R ) is established, the r esults cited a b ove (Coro lla ries 12 and 15 and Theorems 16 and 18) app e a r as familiar results fro m the mo del theo ry o f mo dules. Corollar y 12 is just elemination of quan tifiers [1 0, Thm. 2.3.24]; Co rollary 1 5 the ant i- isomorphism, discov ered b y Prest [10, § 1.3.1 ], and indep endently , b y Huisgen-Zimmerma nn and Z immermann [1 5], betw een the resp ective partial or ders of p ositive-primitive for m ulae ov er R and the o ppo site ring R op ; Theor e m 1 6 asserts nothing more than the fact tha t these pa rtial or ders ar e mo dular la ttices (cf. [1 0, § 1 .1.3]); and Theor em 18 is a v ariation o f Prest’s r e sult [1 0, Thm. 6.1 .8] that if f : R → S is a ring epimorphism, then the sub categ ory S-Mo d ⊆ R-Mo d is a xiomatizable in the la nguage L ( R ) . Let A b e a n m × n matrix; B and m × k matrix a nd R M a left R - mo dule. Let us co ns ider the nonhomo gene ous system of linear equations A v . = b , where v is a column n -vector of v ariables ( v i ) and b a column k -vector with entries from M . Denote by Sol M ( A v . = b ) := { a ∈ ( R M ) n : A a = b } the s ubg roup o f ( R M ) n of solutions in M to the nonho mogeneous sy s tem. Given an element ( B | A ) ∈ L ′ n ( R ) , define ( B | A )( R M ) := [ b ∈ B M k Sol M ( A v . = b ) . This is consistent with mo del-theoretic notation, b ecause ( B | A )( R M ) is the subg roup of ( R M ) n defined in R M by the for m ula ( B | A )( v ) . Let us introduce the relation ( B | A ) | = n ( B ′ | A ′ ) to hold in L ′ n ( R ) pr ovided that ( B | A )( R M ) ⊆ ( B ′ | A ′ )( R M ) for every left R - mo dule R M . Equiv ale ntly , T ( R ) | = ∀ v ( B | A v → B ′ | A ′ v ) . By G¨ odel’s Co mpleteness Theorem, the rela tions ⊢ n and | = n are the same on L ′ n ( R ) , so that o ne obtains a second version o f Lemma Presta. Prop ositio n 40 ( Lemma Presta I I. ) Given ( B | A ) and ( B ′ | A ′ ) in L ′ n ( R ) , ( B | A ) ≤ n ( B ′ | A ′ ) if and only if ( B | A ) | = n ( B ′ | A ′ ) . Seen in this light, the Rules of Divisibility declare the r elationships b etw een s pa ces of nonhomogeneo us systems o f linear equations . F or example, RoD (2) may b e seen a s the statement that for every le ft R - mo dule R M , [ b ∈ B V · M k ′ Sol M ( A v . = b ) ⊆ [ c ∈ B · M k Sol M ( A v . = c ) . Suppo se that A and A ′ are matrices, b oth with n columns, with the prop erty that for every left R -mo dule R M , the tw o solution subgroups of ( R M ) n of the tw o re s pective homog eneous systems of linear equations ar e equal. This is expressible in L ( R ) by T ( R ) | = ∀ v [( A v . = 0 ) ↔ ( A ′ v . = 0 )] . Equiv ale ntly , ( A ) | = n ( A ′ ) a nd ( A ′ ) | = n ( A ) . By Lemma Presta I I, the tw o systems ( A ) and ( A ′ ) ar e equiv alent in L ′ n ( R ) . The first section of the article is devoted to a general ex po sition of the formal matrix calculus; the se cond sectio n of the a rticle a pplies the forma l matrix calculus to obta in v a rious 3 presentations of the Grothendieck gr oup K 0 ( R -mo d , ⊕ ) o f the ca teg ory R -mo d of finitely presented left R -mo dules. One of these pre s en tations sugges ts a homolo gical interpretation, and so a complex C ∗ ( R ) is int ro duced in the third section who se 0 - dimensional homology H 0 ( R ) is naturally is omorphic (Theor em 33) to the Grothendieck group K 0 ( R -mo d , ⊕ ) . If R = k is a field, then the 1-dimensio nal homo lo gy H 1 ( k ) is isomorphic (Coro llary 38) to ( k × ) ab / {± 1 } , where ( k × ) ab denotes the ab e lianization of the mult iplicitav e group of k . The last sec tio n des crib es some repr esentations of the partial order s L n ( R ) . These representation offer co ncrete exa mples, mostly coming fr o m the mo del theory of mo dels, of isomorphic partial order s, which also serve as a historica l refere nce for the formal matrix calc ulus they inspire. Throughout the a rticle, R will denote an asso ciative r ing with identit y 1 . The n × n ident ity matr ix will b e denoted by I n . A t first (in most of section § 1 .1) the dimensio ns of zero matr ic es are sp ecified; the m × n zero matr ix is denoted by m 0 n , but then these cum b ersome subscripts ar e dropp ed to denote a n y zer o matrix b y 0 , except the 1 × 1 , which is denoted 0 . If A is a n m × n matrix, then i A will deno te the i -th row of A and A j the j -th column. When useful, the matrix A is expressed as a row of co lumn ma tr ices A = ( A 1 , A 2 , . . . , A n ) . If the matrix A is m × ( n + 1 ) , the columns are indexed so that A = ( A 0 , A 1 , . . . , A n +1 ) . A g reat debt is owing to P uninsky , w ho showed me how to manipulate matrice s using Lemma Presta. The for mal matrix ca lculus presented here was first introduced at a lecture in the Durham Symp osium, ”New Directions in the Mo del Theory of Fields,” July 200 9. I am gra teful to M. Mak k ai and Ph. Rothmaler for their co nstructive feedback. 1. The Formal Calculus of Ma trix P airs This section is devoted to a developmen t of the formal matrix calculus defined in the Intro- duction. Let us inspec t more closely the definition of the relatio n ≤ n on L ′ n ( R ) . Recall tha t the e lemen ts ( B | A ) of L ′ n ( R ) ar e given by matr ic e s A and B (with ent ries in R ) that hav e the same num ber of c olumns. The num b e r of columns in A is n, but neither dimensio n of B is sp ecified. T o b etter keep track of the computations, we will a s sign dimensions so that B is an m × k matrix. In that ca se, the dimensions of A are m × n . W e ar e given that ( B | A ) ≤ n ( B ′ | A ′ ) if and only if there exist matrices U, V and G, o f appropr iate size, such that (1) U B = B ′ V and U A = A ′ + B ′ G. Let B ′ be an m ′ × k ′ matrix. Then we see that U is an m ′ × m matrix, V a k ′ × k ma trix, and G a k ′ × n matr ix . 1.1. Prelimi nary O bserv ations. Let us b egin by verifying some o f the claims made in the Int ro duction. When it is clear from the context, we will dr op the subscr ipt in the no tation ≤ n . Prop ositio n 1. The r elation ≤ on L ′ n ( R ) is a pr e-or der (r eflexive and tr ansitive). Pr o of. The reflexive pro per t y ( B | A ) ≤ ( B | A ) is es ta blished by letting U = I m , V = I n , and G = k 0 n . T o verify tr a nsitivity , suppo se that ( B 1 | A 1 ) ≤ ( B 2 | A 2 ) a nd ( B 2 | A 2 ) ≤ ( B 3 | A 3 ) , with corres p onding matr ices U i , V i and G i , i = 1 , 2 . Th us we have U 1 B 1 = B 2 V 1 and U 2 B 2 = B 3 V 2 , which implies ( U 2 U 1 ) B 1 = U 2 B 2 V 1 = B 2 ( V 2 V 1 ) . Let U = U 2 U 1 and V = V 2 V 1 . Then U A 1 = U 2 U 1 A 1 = U 2 ( A 2 + B 2 G 1 ) = U 2 A 2 + U 2 B 2 G 1 = A 3 + B 3 G 2 + B 3 V 2 G 1 = A 3 + B 3 G, where G = G 2 + V 2 G 1 .  4 Let us isolate the rˆ oles played by each of the three pa rameters U, V and G in the definition of the pre-o rder on L ′ n ( R ) (cf. the comments pr eceding Lemma 1.1 .13 o f [10]). Theorem 2. L et ( B | A ) ∈ L ′ n ( R ) , wher e B is an m × k matrix. (1) If U is a matrix with m c olumns, then ( B | A ) ≤ n ( U B | U A ) . (2) If V is a matrix with k r ows, then ( B V | A ) ≤ n ( B | A ) . (3) If G is a k × n matrix, then ( B | A + B G ) ≤ n ( A | B ) . The r elation ≤ n is the le ast pr e-or der on L ′ n ( R ) satisfying (1), (2) and (3). Pr o of. F or (1 ), let V = I k and G = k 0 n ; for (2), let U = I m and G = k 0 n ; and fo r (3 ), let U = I m and V = I k . Suppos e that ≺ is a partial order on L ′ n ( R ) satisfying (1), (2) a nd (3 ). If ( B | A ) ≤ ( B ′ | A ′ ) and U, V , and G sa tisfy Equations 1, then ( B | A ) ≺ ( U B | U A ) = ( B ′ V | A ′ + B ′ G ) ≺ ( B ′ | A ′ + B ′ G ) ≺ ( B ′ | A ′ ) .  In the sequel, the three statements of Theor em 2 a r e referre d to as the Rules of Divisibility (RoD) (1)-(3). The par tial o rder ≤ imp o ses on L ′ n ( R ) the equiv alence relation ( B | A ) ≈ n ( B ′ | A ′ ) defined to hold provided ( B | A ) ≤ n ( B ′ | A ′ ) and ( B | A ) ≤ n ( B ′ | A ′ ) . The equiv alence class of ( B | A ) will b e denoted by [ B | A ] . Definition. An n -ary matrix p air is an eq uiv alence class [ B | A ] of some element ( B | A ) of L ′ n ( R ) mo dulo the rela tio n ≈ n . The set of n -ar y ma trix pa irs is denoted by L n ( R ) . The partial order ≤ n on L ′ n ( R ) induces a partial order (r eflexive, symmetric and transitive) on L n ( R ) , which will also b e denoted ≤ n , (or ≤ , whe n there is no danger o f confusio n.) The nex t co r ollary indicates tho s e situa tio ns in whic h the Rules o f Divis ibilit y yield equality . Corollary 3. L et [ B | A ] ∈ L n ( R ) , wher e B is an m × k matrix. (1) If P is an invertible m × m matrix, then [ B | A ] = [ P B | P A ] . (2) If Q is an invertible k × k matrix, then [ B | A ] = [ B Q | A ] . (3) If G is a k × n matrix, then [ B | A + B G ] = [ B | A ] . Pr o of. (1). By RoD (1), [ B | A ] ≤ [ P B | P A ] . Replacing P by P − 1 yields [ P B | P A ] ≤ [ B | A ] . (2). B y RoD (2), [ B Q | A ] ≤ [ B | A ] . Repla cing Q by Q − 1 yields [ B | A ] ≤ [ B Q | A ] . (3). By RoD (3), [ B | A + B G ] ≤ [ B | A ] . Replacing G by − G yields [ B | A ] ≤ [ B | A + B G ] .  If we take the matrices P or Q in Corolla ry 3 to b e p ermutation matrices , we see that per mu tating (simultaneously) the r ows of A and B , or the co lumns of B , do es not change the matr ix pa ir. Similar ly , if P or Q are taken to b e elementary matrice s , then Cor ollary 3 implies that we may per form elementary row o per ations (simult aneously ) o n A and B , or elementary co lumn op erations on B , without changing the matrix pair . The next pro po sition describ es the affect of adding rows to an n -ary matr ix pair. Lemma 4. Le t [ B | A ] and [ B ′ | A ′ ] b elong to L n ( R ) , with B an m × k matrix, and B ′ an m ′ × k matrix. Then  B A B ′ A ′  ≤ [ B | A ] . If A ′ and B ′ ar e zer o matric es, then e quality holds. Pr o of. F or the fir st pa rt, apply RoD (1) with U = ( I m , m 0 m ′ ) . This yields the inequality in L ′ n ( R ) , and therefor e in L n ( R ) . If A ′ and B ′ are zer o matr ices, then the inequa lit y in the opp osite directio n also follows using RoD (1 ), but with U =  I m m ′ 0 m  .  5 If a n n -ar y matrix pair ha s an extra row of zeros, then that r ow may be removed, without changing the matrix pair. F o r, we may p ermute the rows of the matrix pair to put the zero row at the b ottom, a nd then apply Lemma 4 to r emov e it. The next pr op osition describ es the situatio n when c olumns are added to the matrix B . Lemma 5. L et [ B | A ] ∈ L n , with B an m × k matrix. If B ′ is an m × k ′ matrix, then [ B | A ] ≤ [ B , B ′ | A ] . If B ′ is the zer o matrix, t hen e quality holds. Pr o of. The first a s sertion follows fro m RoD (2) with V =  I k k ′ 0 k  . If B ′ is a zero matrix, then the inequality in the opp osite direction also follows using RoD (2), but with V = ( I k , k 0 k ′ ) .  Lemma 5 implies that an ex tra column of zeros may b e removed from the left matrix B without changing the n -ary matrix pair. The next pro p os ition shows how to compute the infim um o f tw o n -a r y matrix pair s. Prop ositio n 6. The infimum of two element s [ B | A ] and [ B ′ | A ′ ] of L n ( R ) is given by [ B | A ] ∧ [ B ′ | A ′ ] :=  B 0 A 0 B ′ A ′  . Pr o of. By Lemmata 4 and 5, it is cle a r that [ B | A ] ∧ [ B ′ | A ′ ] ≤ [ B | A ] and [ B | A ] ∧ [ B ′ | A ′ ] ≤ [ B ′ | A ′ ] . So supp ose that ( B ′′ | A ′′ ) ≤ ( B | A ) and ( B ′′ | A ′′ ) ≤ ( B ′ | A ′ ) in L ′ n ( R ) . Ther e are U, V and G such that U B ′′ = B V and U A ′′ = A + B G ; and there are U ′ , V ′ and G ′ such that U ′ B ′′ = B ′ V ′ and U ′ A ′′ = A ′ + B ′ G ′ . T o see that ( B ′′ | A ′′ ) ≤ ( B | A ) ∧ ( B ′ | A ′ ) , just note that  U U ′  B ′′ =  B V B ′ V ′  =  B 0 0 B ′   V V ′  and  U U ′  A ′′ =  U A ′′ U ′ A ′′  =  A + B G A ′ + B ′ G ′  =  A A ′  +  B 0 0 B ′   G G ′  .  Example 7. If D is a diagonal matrix with diagonal entries d ii , then Pr op osition 6 implies that [ D | A ] = ^ i [ d ii | i A ] , wher e i A denotes the i -t h r ow of A. This is r elevant, for example, in the c ase when R is a c ommut ative PID. By The or em 3.8 of [3] , the m × k matrix B may b e diagona liz e d in the sense that ther e ar e invertible matric es P and Q such that P B Q = D is a diagonal matrix . Applying Ro D (1) and (2) with t he invertible matric es P and Q, r esp e ctively, we se e that for any n -ary matrix p air, [ B | A ] = [ P B | P A ] = [ P AQ | P A ] = [ D | P A ] is the infim u m of n -ary matrix p airs [ d ii | i ( P A )] , wher e the matrix on the left is a 1 × 1 sc alar matrix. The partial order L n ( R ) has a minimum element 0 n . F or if [ B | A ] ∈ L n with B an m × k matrix, then [ n 0 1 | I n ] ≤ [ m 0 1 | A ] = [ B · k 0 1 | A ] ≤ [ B | A ] . The fir st inequality follows by RoD (1) with U = A ; the s econd by RoD (2) with V = k 0 1 . Prop ositio n 8. A n element [ B | A ] ∈ L n ( R ) is the minimu m element 0 n if and only if ther e exists a matrix U s u ch that U B = n 0 k and U A = I n . 6 Pr o of. If [ B | A ] satisfies the co ndition, then [ B | A ] ≤ [ n 0 k | I n ] , by (1) of Theorem 2 with U as given. But [ n 0 k | I n ] = 0 n , by Lemma 5. F or the co n verse, supp ose that ( B | A ) ≤ ( n 0 1 | I n ) . Then there exist matrices U, V , and G such that U B = n 0 1 · V = n 0 k and U A = I n + n 0 1 · G = I n .  The partial order L n ( R ) also has a maximum element 1 n . F or if [ B | A ] ∈ L n with B a n m × k matrix, then [ B | A ] = [ I m · B | A ] ≤ [ I m | A ] . This matr ix pair [ I m | A ] will b e seen to b e the maximum element once it is put int o a form independent of the matrix A and dimensio n m. By Ro D (3) with G = − A, [ I m | A ] = [ I m | m 0 n ] = ^ [1 | 1 0 n ] = [1 | 1 0 n ] . The second equality follows fro m Pro po sition 6 and Lemma 5. The maximum ele ment is therefore g iven by 1 n = [1 | 1 0 n ] . There are other forms [ B | A ] that repre s ent the maximum element. F or e xample, if B is any m × k matrix, then 1 n = [ I k | k 0 n ] ≤ [ B | m 0 n ] , which is o btained using Ro D (1 ) with U = B . Mor e genera lly , we have the following. Prop ositio n 9. An element [ B | A ] ∈ L n ( R ) is the m ax imum element 1 n if and only if ther e exists a matrix W such that B W = A. Pr o of. Suppo se tha t A = B W for some k × n matrix W. Then [ B | A ] = [ B | B W ] = [ B | m 0 n ] = 1 n , by RoD (3) with G = − W. On the other ha nd, if [ B | A ] = 1 n , then (1 | 1 0 n ) ≤ ( B | A ) , so ther e exist ma tr ices U, V , and G such tha t U · 1 = B V and U · 1 0 n = A + B G. Then A = B ( − G ) , and so W = − G.  T o aid our intuition, we may co nstrue the sym b ol ( B | A ) as the prop osition ” B divides A on the left.” This prop osition is then assig ned a pos ition in the partial o r der L n ( R ) , g ov e r ned by the Rules of Divisibility in L ′ n ( R ) . These rules cor r esp ond to the ma trix op erations that preserve the r elation ” B divides A on the left.” Prop osition 9 a sserts that the pr op osition [ B | A ] is assigned the ma ximu m v alue 1 n if a nd only if, B | A is true, that is if B do es, in fact, divide A on the left: [ B | A ] = 1 n if and only if B | A. 1.2. Homog eneous Systems . An element of L ′ n ( R ) is ca lled a ( homo gene ous ) system ( of line ar e quations in n variables ) if it is of the for m ( 0 | A ); the matrix of c o efficients of this system is m × n matrix A. F or s implicit y , we will denote such a sy stem by ( A ) := ( 0 | A ) , and its n -ary matrix pa ir b y [ A ] := [ 0 | A ] . F or example, the maximum element 1 n = [ 1 0 n ] and minimum element 0 n = [ I n ] ar e b oth matr ix pairs of sy s tems. The basic pro per ties o f systems ar e given in the following prop osition. Prop ositio n 10. Le t A b e an m × n matrix. Then the fol lowing hold: (1) if A ′ is another matrix with n c olumn s , then ( A ) ≈ n ( A ′ ) in L ′ n ( R ) if and only if ther e exist matric es U and U ′ such that U A = A ′ and U ′ A ′ = A ; (2) ( A ) = ( B ′ | A ′ ) in L ′ n ( R ) if and only if ther e exist matric es U, W, and G su ch that U B ′ = 0 ; U A ′ = A ; and W A = A ′ + B ′ G ; (3) supp ose that A ′ is an m × k matrix, with ( A, A ′ ) ≤ n + k ( B | C ′ , C ′′ ) in L ′ n + k ( R ) , wher e C ′ and C ′′ have n (r esp., k ) c olumn s. Then ( A | A ′ ) ≤ k ( B , C ′ | C ′′ ) . 7 Pr o of. Both (1) and (2) follow from the definition. T o prove (3), supp ose that we hav e matrices U , V , and G = ( G ′ , G ′′ ) such that U · 0 = B V and U ( A, A ′ ) = ( C ′ , C ′′ ) + B ( G ′ , G ′′ ) . The tw o equations U A = C ′ + B G ′ and U A ′ = C ′′ + B G ′′ may b e rewritten as U A = ( B , C ′ )  G ′ I n  and U A ′ = C ′′ + ( B , C ′ )  G ′′ 0  .  If A a nd A ′ satisfy Co ndition (1) of Prop os ition 10, then it is immediate that for every left R - mo dule R M , the re spec tiv e solution subgroups of ( R M ) n of the corresp onding homogeneous systems of linear equations (in the c lassical s ense) are equal. A matrix B is called re gular if ther e exists a ma trix C such that B C B = B . Theorem 11. The fol lowing ar e e quivalent for an m × k matrix B : (1) it is r e gular; (2) for every m × n matrix A, ther e exists a matrix A ′ with n c olumns such t hat [ B | A ] = [ A ′ ]; (3) ther e exists a matrix A ′ with n c olumns such t hat [ B | I n ] = [ A ′ ] . Pr o of. (1) ⇒ (2). Suppose that B is regular. By RoD (2) with V = C B and V = C, resp ectively , Then [ B | A ] = [ B C B | A ] ≤ [ B C | A ] ≤ [ B | A ] . Then ( B C ) 2 = ( B C )( B C ) = ( B C B ) C = B C a nd we ma y r eplace B by the n × n idemp otent matrix E = B C in the n -a ry matrix pa ir. Then [ E | A ] ≤ [ 0 | ( I n − E ) A ] ≤ [ E | A − E A ] ≤ [ E | A ] . The fir st inequality follows from RoD (1 ) with U = I n − E ; the second fr o m RoD (2 ) with V = 0 ; and the third from RoD (3) with G = A. Now let A ′ = ( I n − E ) A. (3) ⇒ (1). W e ar e g iven that ( B | I n ) ≤ ( 0 | A ′ ) and ( 0 | A ′ ) ≤ ( B | I n ) . F rom the first inequality , w e obtain a matrix U such that U B = 0 and U = A. In short, AB = 0 . F ro m the second inequality , we obtain matric e s U ′ and G ′ such that U ′ A = I n + B G ′ . Multiplying on the rig ht by B yields the equa tion 0 = B + B G ′ B . Whence B = B C B with C = − G ′ .  The ring R is von Neumann r e gular if for every r ∈ R , ther e exists an s ∈ R s uc h that rsr = r. By Theore m 1.7 o f [4], every matr ix ov er a von Neumann regular r ing is r egular. Corollary 12. The fol lowi ng ar e e qu ivalent for the ring R : (1) it is von Neumann r e gular; (2) for every n ≥ 1 , every element ( B | A ) of L ′ n ( R ) is e quivalent to a system; (3) every element ( B | A ) of L ′ 1 ( R ) is e quivalent to a system; Pr o of. T o prov e (1) ⇒ (2) just note that B is a r egular matrix, and a pply Theor em 11. F or (3) ⇒ (1), let r ∈ R , and apply the hypothesis a nd Theorem 1 1 to ( r | 1) ∈ L ′ 1 ( R ) .  Example 13. Supp ose that R = k is a field. Evidently, it is von Neumann r e gular, so t hat every n -ary matrix p air is the class of some system ( A ) of line ar e quations. By Cor ol lary 3, we may cho ose the matrix of c o efficients A to b e a r ow r e duc e d e chelon matrix. By Pr op o- sition 10 .(1), this choic e of A is unique. Ther efor e, the n -ary matrix p airs in L n ( k ) ar e in bije ctive c orr esp ondenc e with r ow r e duc e d e chelon matric es with n c olumns. Example 1 3 to gether with Prop ositio n 10 indicates that the formal ma trix calculus pre- sented reduce s ov er k to the study of homog eneous systems o f linear equations. 8 1.3. Duality . The o ppo s ite ring of R is denoted R op . Its e le ments are those of R, as is the underlying ab elian gr oup s tr ucture, but the multiplication ∗ in R op is given by r ∗ s = sr, where the multiplication on the r ight is carr ied out in R . Mor e genera lly , mu ltiplication o f matrices over R op is denoted A ∗ B . It is related to multiplication of matrices ov er R by the equation ( A ∗ B ) tr = ( B tr )( A tr ) . Theorem 14. If ( B | A ) ≤ ( B ′ | A ′ ) in L ′ n ( R ) , t hen in L ′ n ( R op ) , ( B ′ ) tr 0 ( A ′ ) tr I n ! ≤ B tr 0 A tr I n ! . Pr o of. W e a re given ma trices U, V a nd G s uc h that U B = B ′ V and U A = A ′ + B ′ G. In short, ( B ′ , A ′ )  V G 0 I n  = U ( B , A ) . In L n ( R op ) , this yields V tr 0 G tr I n ! ∗  ( B ′ ) tr ( A ′ ) tr  =  B tr A tr  ∗ U tr . But als o note tha t V tr 0 G tr I n ! ∗  0 I n  =  0 I n  . Letting U ′ = V tr 0 G tr I n ! , V ′ = U tr , and G ′ = 0 establishes the a ssertion.  Theorem 14 implies tha t the rule g iven by [ B | A ] 7→ [ B | A ] ∗ := " B tr 0 A tr I n # . is a well-defined anti-morphism from the partia l orde r L n ( R ) to the partial order L n ( R op ) . There e x ists a similarly defined function in the o ppo site directio n, fro m L n ( R op ) to L n ( R ) . It is also denoted by [ B | A ] 7→ [ B | A ] ∗ with A a nd B matric e s ov er R op . Let us verify that these ma ps are mutual inv erses. Firs t note that [ B | A ] ∗∗ = " B tr 0 A tr I n # ∗ =  B A 0 0 I n I n  . Multiplying the b ottom ”r ow” by A on the left, and substra cting from the top ”r ow” yields the equa tion  B A 0 0 I n I n  =  B 0 − A 0 I n I n  . This is just the infimum [ I n | I n ] ∧ [ B | − A ] = [ B | − A ] . But [ B | − A ] = [ B | A ] as a result of multiplying b oth A and B by − I m on the left and then multiplying B by − I k on the rig ht . Corollary 1 5. The map [ B | A ] 7→ [ B | A ] ∗ is an anti-isomorphism b etwe en L n ( R ) and L n ( R op ) . Consider the n -ar y matrix pair [ A ] asso c iated to a system. According to this a n ti- isomorphism, its dual in L n ( R op ) is given b y [ A ] ∗ = [ 0 | A ] ∗ = [ A tr | I n ] . 9 This obser v ation le nds impor tance to the family of n -ar y ma trix pairs in L n ( R ) of the form [ B | I n ] fo r some n × k matrix B . It is the family dual, in the sense of this ant i- isomorphism, to the family of n -ary matrix pairs asso ciated to systems in L ′ n ( R ) . Co ndition (3) o f Theo rem 11 describ es those n -a ry matrix pairs tha t b elong to the intersection o f these t wo families. The supre mum op eratio n in L n ( R ) may then b e g iven in terms of the infim um ope ration in L n ( R op ) : [ B | A ] + [ B ′ | A ′ ] := ([ B | A ] ∗ ∧ [ B ′ | A ′ ] ∗ ) ∗ . It is readily computed a s [ B | A ] + [ B ′ | A ′ ] :=   B A 0 0 0 0 0 B ′ A ′ 0 0 I n 0 I n I n   . With the infim um and supre m um o per ations now b oth defined, the partial order L n ( R ) acquires the str uc tur e o f a lattic e with minimum and maximu m elements. Recall that a lattice is mo dular if a ≤ b implies that ( a + c ) ∧ b = a + ( b ∧ c ) . Theorem 16. The p artial or der L n ( R ) is a mo dular lattic e with m aximum and minimum elements. Pr o of. Let us verify the equation for mo dularity with a = [ B | A ] , b = [ B ′ | A ′ ] , and c = [ B ′′ | A ′′ ] . W e are given that a ≤ b so there ex ist ma tr ices U, V , and G such that U B = B ′ V and U A = A ′ + B ′ G. Le t us apply some op erations (explained b elow) to the n -ary matr ix pair ( a + c ) ∧ b :     B A 0 0 0 0 0 0 B ′′ A ′′ 0 0 0 I n 0 I n 0 I n 0 0 0 0 B ′ A ′     =     B A 0 0 0 0 0 0 B ′′ A ′′ 0 0 0 I n 0 I n 0 I n U B U A 0 0 B ′ A ′     =     B A 0 0 0 0 0 0 B ′′ A ′′ 0 0 0 I n 0 I n 0 I n B ′ V A ′ + B ′ G 0 0 B ′ A ′     =     B A 0 0 0 0 0 0 B ′′ A ′′ 0 0 0 I n 0 I n 0 I n 0 A ′ 0 0 B ′ A ′     . In the first equa lit y , we multiplied the top ”row” o n the left by U and added it to the bo to m. In the last equality , we subtra c ted right multip les, b y V and G, res pectively , of the fifth column from the first and s e cond, resp ectively . Now w e mult iply the thir d ”row” by A ′ (on the left) and substract fro m the b ottom ”r ow” to g et     B A 0 0 0 0 0 0 B ′′ A ′′ 0 0 0 I n 0 I n 0 I n 0 0 0 − A ′ B ′ 0     =     B A 0 0 0 0 0 0 B ′′ A ′′ 0 0 0 I n 0 I n 0 I n 0 0 0 A ′ B ′ 0     . The equality fo llows by multiplying the b ottom ” row” and fifth c o lumn by − I . All that remains is to p ermute simultaneously some rows of bo th ma tr ices, a nd some co lumns of the left matrix to obtain     B A 0 0 0 0 0 0 B ′ 0 A ′ 0 0 0 0 B ′′ A ′′ 0 0 I n 0 0 I n I n     = a + ( b ∧ c ) .  10 1.4. Morphism s of Ring s . Let f : R → S b e a mor phism of rings. If A is a matr ix with entries in R , denote by f ( A ) the matrix over S, of the same dimens ions, o btained by applying f to the entries o f A. Then it is e asy to se e that if ( B | A ) ≤ ( B ′ | A ′ ) in L ′ n ( R ) , then ( f ( B ) | f ( A )) ≤ ( f ( B ′ ) | f ( A ′ )) in L ′ n ( S ) . T his is b ecause the three matrices U , V , a nd G that arise fr om the for mer ineq ua lit y are taken by f to matr ic e s f ( U ) , f ( V ) , a nd f ( G ) that esta blish the latter. This induces a morphism of pa rtial orders [ B | A ] 7→ [ f ( B ) | f ( A )] , which is denoted by L n ( f ) : L n ( R ) → L n ( S ) . Recall that a morphis m f : R → S is an epimorphism if w he never r ing mor phisms g , h : S → T are giv en such that g f = hf , then g = h. Silv er [1 3] and Mazet [9] prov ed that a morphis m f : R → S of r ings is a n epimorphism if and only if e very element s ∈ S, considered as a 1 × 1 matrix, may b e factored as X P Y = s, where X , P, a nd Y are matrices of appropr iate size, such that the entries o f X P , P , and P Y all lie in the imag e of f . W e s ha ll require the fo llowing slight strengthening o f their result. Lemma 1 7. A morphism f : R → S of rings is an epimorphism if and only if every matrix A with entries in S has a factorization A = X P Y , such that X P, P and P Y have entries in the image of f . Pr o of. The cor resp onding morphism M n ( f ) : M n ( R ) → M n ( S ) of n × n matrix ring s is also a ring epimorphism, so if A is a squar e matrix, say n × n, then the theor em of Ma zet and Silver applied to M n ( f ) prov es the claim. Supp ose now that A is an m × n matr ix, wher e m > n. Let k = m − n, and apply the fo r egoing to the squar e matr ix ( A, m 0 k ) . W e obtain a Silver-Mazet factoriza tion ( A, m 0 k ) = X P Y = X P ( Y ′ , Y ′′ ) where Y = ( Y ′ , Y ′′ ) has b een decomp osed so that Y ′ has n columns and Y ′′ has k columns. Thu s X P, P , and P Y = P ( Y ′ , Y ′′ ) hav e e ntries in R. But then A = X P Y ′ and P Y ′ also has entries from R . The case when m < n is handled similarly .  Theorem 18. The fol lowing ar e e quivalent for a morphism f : R → S of rings: (1) it is an epimorphism; (2) for every matrix A, t he syst em [ A ] is in the image of L n ( f ) , wher e n is the numb er of c olumn s of A ; (3) for every n ≥ 1 , the induc e d morphism L n ( f ) : L n ( R ) → L n ( S ) is onto; (4) the induc e d morphism L 2 ( f ) : L 2 ( R ) → L 2 ( S ) is onto. Pr o of. (1) ⇒ (2). F actor A = X P Y , according to the lemma, so that X P, P, and P Y have ent ries in the image of f . Then  X P 0 − P P Y  =  0 X P Y − P P Y  = [ 0 | A ] ∧ [ − P | P Y ] = [ A ] . The first e quality is obtained by m ultiplying the b ottom ”r ow” on the left by X and adding it to the top; the seco nd follows fr o m Prop ositio n 6; and the last b ecaus e [ − P | P Y ] = 1 n (Prop osition 9). (2) ⇒ (3). L et [ B | A ] ∈ L n ( S ) , where B has k columns. By assumption [ B , A ] = [ B ′ | A ′ , A ′′ ] in L k + n ( S ) , wher e A ′ has k columns, A ′ has n co lumns, and the entries of A ′ , A ′′ , and B ′ all lie in the imag e of f . By Pro po s ition 10.(3), [ B | A ] = [ B ′ , A ′ | A ′′ ] . 11 (4) ⇒ (1). Let g , h : S → T b e morphisms such that g f = hf . Pick s ∈ S ; we must show that g ( s ) = h ( s ) . Consider the 1 × 2 matrix A = (1 , s ) . By hypothesis, there are matrices A ′ and B ′ with entries from the imag e of f such tha t [(1 , s )] = [ B ′ | A ′ ] . By a ssumption, the morphisms g and h ag ree on the entries of A ′ and B ′ : g ( A ′ ) = h ( A ′ ) and g ( B ′ ) = h ( B ′ ) . By Pr op osition 10.(2), there are ma tr ices U, W , a nd G such tha t U B ′ = 0 ; U A ′ = (1 , s ); and W (1 , s ) = A ′ + B ′ G. Now A ′ and G are also matrices with 2 co lumns so we may expr ess them as A ′ = ( A ′ 1 , A ′ 2 ) and G = ( G 1 , G 2 ) . The equations then b ecome U B ′ = 0 ; U ( A ′ 1 , A ′ 2 ) = (1 , s ); and W (1 , s ) = ( A ′ 1 , A ′ 2 ) + B ′ ( G 1 , G 2 ) . Eliminating W from the third equation yields A ′ 2 + B ′ G 2 = W s = A ′ 1 s + B ′ G 1 s. Apply p = g − h, a function that is not a ring morphism, to b oth s ides of the equa tio n, to obtain g ( B ′ ) p ( G 2 ) = g ( A ′ 1 ) p ( s ) + g ( B ′ ) p ( G 1 s ) . Then multiply on the le ft b y g ( U ) a nd use the fac t that g ( U ) g ( B ′ ) = g ( U B ′ ) = 0 , and g ( U ) g ( A ′ 1 ) = g ( U A ′ 1 ) = g (1) = 1 . Whence p ( s ) = 0 .  There ar e morphism f : R → S that ar e no t epimorphisms , but hav e the prop erty that L 1 ( f ) : L 1 ( R ) → L 1 ( S ) is onto. F or exa mple, if f : k → k ′ is an extensio n of c omm utative fields with nontrivial Galois group, Gal( k ′ /k ) 6 = 1 . Then f is not an epimorphis m, but bo th L 1 ( k ) and L 1 ( k ′ ) contain nothing mo re tha n the resp ective maximum and minimum elements, which L 1 ( f ) res pects. 2. The Grothendieck Group of Finitel y Presented Mod ules In this section, we apply the for mal moartix calculus to give several prese n tations of the Gr othendieck gr o up K 0 ( R -mo d , ⊕ ) of finitely presented left R -mo dules. One of these presentations is used in the next section a s the basis for a ho mology theory . A left R -mo dule R M is finitely pr esent e d if there is an e x act sequence, a fr e e pr esentation of R M , of the for m R R m ϕ ✲ R R n ✲ M ✲ 0 . The morphism ϕ : R m → R n is given by m ultiplication on the rig ht by an m × n ma trix A, ϕ = − × A. One says that R M is pr esente d by the matrix A, a nd writes R M = M A . Two matrices A and B ar e e quivalent, deno ted A ∼ B , if they present isomorphic mo dules, M A ∼ = M B . The equiv alence class of a matrix A is denoted by { A } . Let R -mo d denote the c ate gory of finitely pr esented mo dules and define K 0 ( R -mo d , ⊕ ) to b e the free gr oup on the symbols { M } , M ∈ R -mo d , mo dulo the r elations { M ⊕ N } = { M } + { N } . It may a lso b e defined as the fr ee group on the equiv a lence clas ses { A } of matrices , mo dulo the re lations  A 0 0 B  = { A } + { B } . This group K 0 ( R -mo d , ⊕ ) is isomo rphic to the Gr othendie ck gr oup K 0 (Ab( R )) of the fr e e ab elian c ate gory Ab( R ) ov er R . This follows fro m the fac t that the sub category of pro jective ob jects of Ab( R ) is dual to R - mo d [1]. 12 Theorem 19. [7, Theorem 6 .1] Two matric es A and A ′ with entries in the ring R ar e e quivalent A ∼ A ′ if and only if A ′ may b e obtaine d fr om A by a se quenc e of (invertible) op er ations of the fol lowing form: (1) addition or deletion of an extr a ro w of zer os; (2) r eplac ement of a m atr ix C by  C 0 0 1  , or the r everse; (3) p ermu tation of r ows or c olumn s ; (4) addition of a left (re sp., right) sc alar multiple of a ro w (r esp., c olumn) to another r ow (r esp., c olumn). Theorem 19.(2) implies that for any n ≥ 1 , the symbo l asso ciated to the identit y matrix { I n } = n { 1 } = 0; any iden tit y matrix I n corres p onds to the finitely presented module 0 . On the other hand, the v alue of the m × k zero ma trix m 0 k may b e computed using Theorem 1 9.(1), { m 0 k } = { k 0 k } = k { 0 } ; the 1 × 1 ma trix { 0 } corresp ondes to the finitely presented mo dule R R. The theorem implies that the asso c iation R 7→ K 0 ( R -mo d , ⊕ ) is functorial, for if f : R → S, is g iven, then o ne may eas ily verify that whenever tw o matrices A and A ′ , with entries from R , are equiv alent, then so are the matr ic es f ( A ) and f ( A ′ ) with entries from S. The rule { A } 7→ { f ( A ) } fr om K 0 ( R -mo d , ⊕ ) to K 0 ( S -mod , ⊕ ) is therefor e well-defined o n the generator s, and extends linearly to a morphism K 0 ( f ) : K 0 ( R -mo d , ⊕ ) → K 0 ( S -mod , ⊕ ) of a belia n groups . 2.1. The Goursat Group. The Goursat gr oup, denoted by G ( R ) , is the free g roup on the elements of ∪ n ≥ 1 L n ( R ) , mo dulo the rela tio ns: (1) for every three matrice s A, A ′ and B with the same n umber o f rows, [ B , A | A ′ ] − [ B | A ′ ] = [ B , A ′ | A ] − [ B | A ]; (2) for [ B | A ] ∈ L m and [ B ′ | A ′ ] ∈ L n ,  B 0 A 0 0 B ′ 0 A ′  = [ B | A ] + [ B ′ | A ′ ]; (3) for every n ≥ 1 , 0 n = 0 . The re la tions of the Goursa t group repres ent in a formal way the group isomo r phism that app ears in a celebrated theore m of Goursat [5]. The definition of the Gours a t gro up o f R is also functorial: if f : R → S is a morphism of r ings, then the induced morphisms L n ( f ) : L n ( R ) → L n ( S ) , n ≥ 1 , given by [ B | A ] 7→ [ f ( B ) | f ( A )] induce a well-defined function from the generato rs of G ( R ) to G ( S ) , which extends linearly to a morphism G ( f ) : G ( R ) → G ( S ) of Go ursat gr oups. The 0 -Dimensional Goursat Gr oup, denoted b y G 0 ( R ) , is the free g roup o n unary matrix pairs [ B | A ] , the elements of L 1 ( R ) , mo dulo the relations: (1) if A and A ′ are column matrices , and all thr e e matrices A, A ′ , and B hav e the same nu mber o f rows, then [ B , A | A ′ ] − [ B | A ′ ] = [ B , A ′ | A ] − [ B | A ]; (2) 0 1 = 0 . Like the definitions o f K 0 ( R -mo d , ⊕ ) and G ( R ) , the definition of G 0 ( R ) is also functor ial. It is clear that the rule [ B | A ] 7→ [ B | A ] , defined on the gener ators of G 0 ( R ) with v a lues in G ( R ) is well-defined and r esp ects the r elations of G 0 ( R ) . It therefore extends linearly to a morphism ι R : G 0 ( R ) → G ( R ) . If f : R → S is a morphism of r ings, then the comm utativity of the diagr am 13 G 0 ( R ) ι R ✲ G ( R ) G 0 ( f ) ❄ ❄ G ( f ) G 0 ( S ) ι S ✲ G ( S ) is ea s ily established. It shows that the class of morphisms ι R : G 0 ( R ) → G ( R ) co nstitutes a natural transfor mation ι : G 0 → G of functors fr om the ca tegory Ring of asso ciative rings with identit y to the ca tegory Ab of ab elian gr oups. 2.2. The Natural T ransformation γ : G → K 0 . Let the function γ ′ : ∪ n ≥ 1 L ′ n ( R ) → K 0 ( R -mo d , ⊕ ) be defined by the rule ( B | A ) 7→ { B , A } − { B } . In order to show that this function induces as a well-defined function on the generato rs o f G ( R ) , we need the following le mma . Lemma 20 . If ( B | A ) ≤ ( B ′ | A ′ ) in L ′ n ( R ) , t hen  B A 0 0 0 B ′  ∼  B A 0 0 A ′ B ′  . Pr o of. W e are given U, V and G such that U B = B ′ V and U A = A ′ + B ′ G. One o btains the following sequence (justified b elow) of e quiv alences of ma trices:  B A 0 0 0 B ′  ∼  B A 0 U B U A B ′  =  B A 0 B ′ V U A B ′  ∼  B A 0 0 U A B ′  =  B A 0 0 A ′ + B ′ G B ′  ∼  B A 0 0 A ′ B ′  . The first equiv alence is obtained by mu ltiplying the top ”r ow” on the left by U and adding it to the seco nd. The other e q uiv alences are o btained by multiplying the third ”column” on the right by V and G, r esp ectively , then subtra cting it from the first, resp ectively , second.  The arg umen t in the pr o o f of Lemma 20 was alr e ady used in the pr o of of Theorem 1 6. Theorem 21. Th e rule γ ′ : ∪ n ≥ 1 L ′ n ( R ) → K 0 ( R -mo d , ⊕ ) induc es a natu ra l morphi sm γ R : G ( R ) → K 0 ( R -mo d , ⊕ ) of ab elian gr oups, whose values on the gener ators of G ( R ) ar e given by γ R : [ B | A ] 7→ { B , A } − { B } . Pr o of. T o b egin, let us s how that γ R is well-defined on the gener ators o f G ( R ) . If [ B | A ] = [ B ′ | A ′ ] in L n ( R ) , for so me n ≥ 1 , then ( B | A ) ≤ ( B ′ | A ′ ) and ( B ′ | A ′ ) ≤ ( B | A ) in L ′ n ( R ) . One obser ves the following seq uenc e of equiv alences ,  B A 0 0 0 B ′  ∼  B A 0 0 A ′ B ′  ∼  B ′ A ′ 0 0 A B  ∼  B ′ A ′ 0 0 0 B  , where the lemma is us e d to obtain the fir st and third equiv alences, and Theore m 1 9.(3) to obtain the second. Thus { B , A } + { B ′ } = { B ′ , A ′ } + { B } in the Gro thendiec k gro up K 0 ( R -mo d , ⊕ ) . Whence γ R ([ B | A ]) = { B , A } − { B } = { B ′ , A ′ } − { B ′ } = γ R ([ B ′ | A ′ ]) . 14 Let us now verify that the relatio ns of G ( R ) are als o re s pected by γ R . If γ R is applied to the firs t family of r elations [ B , A 1 | A 2 ] − [ B | A 2 ] = [ B , A 2 | A 1 ] − [ B | A 1 ] , then { B , A 1 , A 2 } − { B , A 1 } − ( { B , A 2 } − { B } ) = { B , A 2 , A 1 } − { B , A 2 } − ( { B , A 1 } − { B } ) , which holds in K 0 ( R -mo d , ⊕ ) , by Theorem 1 9.(3). The second family o f relations that ho ld in G ( R ) ar e of the form  B 0 A 0 0 B ′ 0 A ′  = [ B | A ] + [ B ′ | A ′ ] . If γ R is a pplied, then  B 0 A 0 0 B ′ 0 A ′  −  B 0 0 B ′  = { B , A } − { B } + { B ′ , A ′ } − { B ′ } . But this clear ly holds in K 0 ( R -mo d , ⊕ ) , b e c ause  B 0 A 0 0 B ′ 0 A ′  = { B , A } + { B ′ , A ′ } and  B 0 0 B ′  = { B } + { B ′ } . Finally , to see that γ R : 0 n 7→ 0 , r ecall that 0 n = [ n 0 1 | I n ] . Thus γ R (0 n ) = { n 0 1 , I n } − { n 0 1 } . Theorem 19.(2) implies that { n 0 1 , I n } = { 0 , 1 } , while Theo rem 1 9.(1) shows that { n 0 1 } = { 0 } . But { 0 , 1 } =  0 1 0 0  =  0 0 0 1  = { 0 } , by (1), (3), and (2), resp ectively , of Theorem 19.  Let us observe that if A is an m × n matrix, then γ R ([ A ]) = { A } . F or , γ R ([ A ]) = γ ([ m 0 1 | A ]) = { m 0 1 , A } − { m 0 1 } . But { m 0 1 , A } =  0 0 m 0 1 A  = { 0 } + { A } = { m 0 1 } + { A } , by tw o applica tio ns of Theo r em 19.(1). 2.3. The Natural T ransformation κ : K 0 → G 0 . Define the function κ ′ on the collection of a ll matrices with entries from R to G 0 ( R ) b y the rule A = ( A 1 , A 2 , . . . , A n ) 7→ n X i =1 [ A 1 , . . . , A i − 1 | A i ] = [ A 1 ] + [ A 1 | A 2 ] + · · · + [ A 1 , . . . , A n − 2 | A n − 1 ] + [ A 1 , . . . , A n − 1 | A n ] . In o r der to show that this function induces as a w ell-defined rule on the generators o f K 0 ( R -mo d , Ab) , we ne e d to verify tha t it is inv ar iant under the four o per ations cited in Theorem 1 9. W e shall use the observ ation that κ ′ R ( A ) = κ ′ R ( A 1 , A 2 , . . . , A n ) = κ ′ R ( A 1 , A 2 , . . . , A n − 1 ) + [ A 1 , A 2 , . . . , A n − 1 | A n ] . (1) Addition or del etion of an extra ro w of zeros . Supp ose tha t A ′ is o btained from A by adding an extr a row of zeros. Then, for each i, [ A ′ 1 , . . . , A ′ i − 1 | A ′ i ] =  A 1 , . . . , A i − 1 A i 0 , 0 , 0 0  = [ A 1 , . . . , A i − 1 | A i ] 15 in L 1 ( R ) , b y the commentary following Lemma 4. Consequently , κ ′ R ( A ) = n X i =1 [ A 1 , . . . , A i − 1 | A i ] = n X i =1 [ A ′ 1 , . . . , A ′ i − 1 | A ′ i ] = κ ′ R ( A ′ ) . (2) Repl acemen t of a matrix C b y  C 0 0 1  , or the rev erse. Simply note that κ ′ R  C 0 0 1  = κ ′ R  C 0  +  C 0 0 1  = κ ′ R ( C ) + ([ C | 0 ] ∧ [ 0 | 1]) = κ ′ R ( C ) + (1 1 ∧ 0 1 ) = κ ′ R ( C ) + 0 1 = κ ′ R ( C ) , which uses the relation 0 1 = 0 in G 0 ( R ) . (3) P erm utation of ro ws or colum ns. Suppo se that A ′ is obtained from A by p er- m utation of rows. Then A ′ = P A for some p ermutation, hence in vertible, matrix P , a nd so κ ′ R ( A ′ ) = n X i =1 [ P A 1 , . . . , P A i − 1 | P A i ] = n X i =1 [ A 1 , . . . , A i − 1 | A i ] = κ ′ R ( A ) , by RoD (1) with U = P. Suppo se, on the other hand, that A ′ is obta ined from A by p ermutation of columns. The symmetric gr oup S n on n elements is gene r ated by consecutive transp ositions, so it suffices to prov e the claim in ca se A ′ is obtained from A by p ermuting consec utive co lumns j − 1 and j, where 1 < j ≤ n. Let us co nsider a t ypical summand [ A 1 , . . . , A i − 1 | A i ] of κ ′ R ( A ) . If i > j, then a transp os itio n of the j -th a nd ( j − 1)-th columns will not affect the matrix pair, beca use it is the result of multiplying the left matrix by a p ermutation matrix Q on the rig h t. But this leav es the matrix pair inv ariant, by RoD (2) with V = Q. Also, it is immediate that if i < j − 1 , then trans po sing the j -th and ( j − 1)-th columns ha s no effect. The rema ining tw o cases ar e i = j − 1 a nd i = j, so we need to show that [ A 1 , . . . , A j − 2 | A j − 1 ] + [ A 1 , . . . , A j − 1 | A j ] = [ A 1 , . . . , A j − 2 | A j ] + [ A 1 , . . . , A j − 2 , A j | A j − 1 ] holds in G 0 ( R ) . But [ A 1 , . . . , A j − 1 | A j ] − [ A 1 , . . . , A j − 2 | A j ] = [ A 1 , . . . , A j − 2 , A j | A j − 1 ] − [ A 1 , . . . , A j − 2 | A j − 1 ] is an instance o f Relatio n (1) in the definition of G 0 ( R ) . (4) Addition of a left (resp. , righ t) scalar multiple of a ro w (resp. , column) to another row (resp., column). Suppo se that A ′ is obtained from A b y adding a left sc a lar mu ltiple of a row to another row. Then A ′ = P A for some e le mentary , hence inv er tible, ma trix P . Then κ ′ R ( A ′ ) = κ ′ R ( A ) , just as in the case when A ′ is obtained from A by p ermutation of rows. So s uppo s e that A ′ is obtained fro m A by adding a rig ht sca lar mult iple of a co lumn to another column. T o prove that κ ′ R ( A ′ ) = κ ′ R ( A ) , we ar e fre e, by (3) ab ove to p ermute the columns o f A ′ and as sume, that it is a rig h t scalar multip le A j r for j < n that is b eing added to A n , the last c olumn, to obtain κ ′ R ( A ′ ) = [ A 1 , . . . , A n − 1 | A n + A j r ] + X i 0 , then ∂ n is defined on an ( n + 1)-ary matr ix pair b y the r ule [ B | A ] 7→ n X i =0 ( − 1) i ( E i [ B | A ] − N i [ B | A ]) , and extended linearly; (2) if n = 0 , then ∂ 0 : Q 0 ( R ) → Z is the augmentation map ǫ, defined on a unary matrix pair by the r ule ǫ : [ B | A ] 7→ 1 , and extended linearly; and 22 (3) if n ≤ − 1 , then, of course, ∂ n = 0 . If the definitio n of the i -face op erato rs E i and N i is extended linea rly , to morphisms E i and N i from Q n ( R ) → Q n − 1 , then, for n > 0 , we may expres s the bo undary op erator as ∂ n = n X i =0 ( − 1) i ( E i − N i ) . It is routine (cf. p.13 of [8 ]) to verify that Q ∗ ( R ) is indeed a complex . Prop ositio n 30. F or every inte ger n, the c omp osition ∂ n ◦ ∂ n +1 = 0 . The complex Q ∗ ( R ) is ca lled the c omplex of matrix p airs o f R. If f : R → S is a morphism of rings, then for every n ≥ 0 , the morphism L n +1 ( f ) : L n +1 ( R ) → L n +1 ( S ) of partially ordered sets may b e extended linear ly , to obtain a morphism Q n ( f ) : Q n ( R ) → Q n ( S ) of ab elian gr o ups. If the morphism Q − 1 ( f ) : Z → Z is defined to the identit y , and Q m ( f ) = 0 for m < − 1 , then we obtain the following, which requires a routine verification (cf. § II.3 of [8]). Prop ositio n 31. F or every m orphism f : R → S of rings, the family Q ∗ ( f ) : Q ∗ ( R ) → Q ∗ ( S ) is a morphism of c omplexes. The anti-isomorphism [ B | A ] 7→ [ B | A ] ∗ from L n +1 ( R ) to L n +1 ( R op ) may b e ex tended linearly , to obtain a n isomorphism p n : Q n ( R ) → Q n ( R op ) of ab elian g r oups, for n ≥ 0 . Prop osition 2 8 implies that these morphisms p n satisfy the equa tions p n − 1 N i = E ∗ i p n and p n − 1 E i = N ∗ i p n . Define P n : Q n ( R ) → Q n ( R op ) to be the map P n :=    ( − 1) n +1 p n if n ≥ 0; − 1 Z if n = − 1; and 0 if n < − 1 . It is routine to verify that P ∗ : Q ∗ ( R ) → Q ∗ ( R op ) is a morphism of co mplexes. The inv e r se of P ∗ is defined similarly , and yie lds the following. Prop ositio n 32. The morphism P ∗ : Q ∗ ( R ) → Q ∗ ( R op ) of c omplexes is an isomorphism. 3.3. 0 -Dim ensional Homology. The 0-dimensiona l homolo g y of R is defined to b e the homology o f the co mplex Q ∗ ( R ) at Q 0 ( R ) , · · · ∂ 2 ✲ Q 1 ( R ) ∂ 1 ✲ Q 0 ( R ) ǫ ✲ Z ✲ 0 . The kernel o f ǫ is the subgroup B 0 ( R ) of 0 - b oundaries; the image of ∂ 1 the s ubgroup Z 0 ( R ) of 0 -cycles. Because ǫ ◦ ∂ 1 = 0 , the inclus io n Z 0 ( R ) ⊆ B 0 ( R ) holds, and the quotient group H 0 ( R ) := B 0 ( R ) / Z 0 ( R ) is the 0 -dimensional homolo gy of R. All of these notions are functorial in R. Theorem 33. The morphism λ R : G 0 ( R ) → H 0 ( R ) induc e d by the rule λ ′ R : [ B | A ] 7→ [ B | A ] − 0 1 mo d B 0 ( R ) is a n atur al isomorphism. Pr o of. Let us verify that the r ule λ ′ R so defined r esp ects the relations that define G 0 ( R ) . The family o f re la tions given by [ B , A | A ′ ] − [ B | A ′ ] − ([ B , A ′ | A ] − [ B | A ]) is ma pp ed b y λ ′ R to ([ B , A | A ′ ] − 0 1 ) − ([ B | A ′ ] − 0 1 − ([ B , A ′ | A ] − 0 1 ) + ([ B | A ] − 0 1 ) = [ B , A | A ′ ] − [ B | A ′ ] − ([ B , A ′ | A ] − [ B | A ]) = E 0 [ B | A, A ′ ] − N 0 [ B | A, A ′ ] − ( E 1 [ B | A, A ′ ] − N 1 [ B | A, A ′ ]) , 23 which is the bo undary ∂ 1 [ B | A, A ′ ] . Also , the ele ment 0 1 maps to 0 1 − 0 1 = 0 . Let us define a map δ R : H 0 ( R ) → G 0 ( R ) in the other direction, which will be the inv er se of λ R . T o do so, note first that the group Z 0 ( R ) of 0-cy cles is free on the generato r s [ B | A ] − 0 1 , whe r e [ B | A ] ∈ L 1 ( R ) is no t the minimum element. This is seen b y co nsidering the splitting of the shor t exact se q uence 0 ✲ Z 0 ( R ) ✲ Q 0 ( R ) ǫ ✲ Z ✲ 0 that corr esp onds to the section s : Z → Q 0 ( R ) determined by 1 7→ 0 1 . Define δ ′ R : Z 0 ( R ) → G 0 ( R ) b y the rule δ ′ R : [ B | A ] − 0 1 7→ [ B | A ] . T o see that δ ′ R induces a morphism δ R : H 0 ( R ) → G 0 ( R ) , it suffices to chec k that it annihilates every 0-b oundar y . But if [ B | A, A ′ ] is a binary matrix pair, then δ ′ R ◦ ∂ 1 [ B | A, A ′ ] = [ B , A | A ′ ] − [ B | A ′ ] − ([ B , A ′ | A ] − [ B | A ]) is an instance o f Relatio n (1) o f G 0 ( R ) . The tw o morphisms λ R : G 0 ( R ) → H 0 ( R ) a nd δ R : H 0 ( R ) → G 0 ( R ) are clearly mutual inv er ses, for if we c o mpo se them in either or der, the g enerators remain fixed.  The isomorphisms of Theorem 3 3 and Prop osition 3 2 may be comp osed to yield an isomorphism G 0 ( R ) λ R ✲ H 0 ( R ) P 0 ✲ H 0 ( R op ) λ − 1 R op ✲ G 0 ( R op ) of 0-dimensio nal Goursat gro ups. Comp osing further with the isomo rphism κ of Theo - rem 24 yields an isomorphism b et ween the K 0 ( R -mo d , ⊕ ) and the Gr othendieck group K 0 (mo d- R, ⊕ ) of the ca teg ory mo d- R o f finitely presented right R -mo dules. O ne may chec k that if A is a matrix with n columns, then this isomor phism sends the c lass { A } in K 0 ( R -mo d , ⊕ ) to the element n { 0 } − { A tr } in K 0 (mo d- R, ⊕ ) . 3.4. Degeneracy. An ( n + 1 )-ary matrix pair [ B | A ] , n > 0 , is de gener ate if for some i, 0 ≤ i ≤ n + 1 , E i [ B | A ] = N i [ B | A ] . In order to show that the degenera te ma trix pairs generate a sub complex that does not change the 0-dimensiona l homolog y , we sha ll need the following lemma. Lemma 3 4. Supp ose t hat A is an m × n matrix, and A ′ an m × n ′ matrix. If ( B , A | A ′ ) ≤ ( B | A ′ ) in L ′ n ′ ( R ) , t hen ( B , A ′ | A ) ≤ ( B | A ) in L ′ n ( R ) . Pr o of. W e are given U, V = ( V ′ , V ′′ ) a nd G s uch that U ( B , A ) = B ( V ′ , V ′′ ) a nd U A ′ = A ′ + B G. F rom that, we g e t ( I m − U )( B , A ′ ) = ( B − U B , A ′ − U A ′ ) = ( B − B V ′ , − B G ) = B ( I − V ′ , − G ) and ( I m − U ) A = A − U A = A − B V ′′ = A + B ( − V ′′ ) .  Prop ositio n 35. If [ B | A, A ′ ] ∈ L 2 ( R ) , t hen E 0 [ B | A, A ′ ] = N 0 [ B | A, A ′ ] if and only if E 1 [ B | A, A ′ ] = N 1 [ B | A, A ′ ] . Pr o of. Suppo se tha t E 0 [ B | A, A ′ ] = [ B , A | A ′ ] = [ B | A ′ ] = N 0 [ B | A, A ′ ] . Then ( B , A | A ′ ) ≤ ( B | A ′ ) , implies , by the lemma, that ( B , A ′ | A ) ≤ ( B | A ) . On the other hand, Lemma 5 implies that ( B | A ) ≤ ( B , A ′ | A ) . Thus E 1 [ B | A, A ′ ] = [ B , A ′ | A ] = [ B | A ] = N 1 [ B | A, A ′ ] . These dir ection of the equiv alence holds, in pa rticular, in L 2 ( R op ) . The con verse now follows by an applica tio n of Pr op osition 28.  24 The prop ositio n implies that if [ B | A, A ′ ] ∈ L 2 ( R ) is deg enerate, then ∂ 1 [ B | A, A ′ ] = 0 . F or n > 0 , let M n ( R ) ⊆ Q n ( R ) b e the subgro up generated by degenerate matrix pairs. If n ≤ 0 , let M n ( R ) ⊆ Q n ( R ) be the zero subgroup. Prop ositio n 36. The family of sub gr oups M n ( R ) ⊆ Q n ( R ) , n ∈ Z , c onsitu tes a sub c omplex M ∗ ( R ) ⊆ Q ∗ ( R ) . Pr o of. Suppo se that n > 1 and [ B | A ] ∈ L n +1 is degenera te. Then E j [ B | A ] = N j [ B | A ] for so me j, 0 ≤ j ≤ n + 1 , and it is ea s y to chec k tha t ∂ n [ B | A ] = X i 6 = j ( − 1) i ( E i [ B | A ] − N i [ B | A ]) is a linear com bination of degenerate ( n − 1)-ary matrix pa irs. Thus ∂ n ( M n ( R )) ⊆ M n − 1 ( R ) for all n > 1 . F or n = 1 , ∂ 1 ( M 1 ( R )) = 0 = M 0 ( R ) is a c o nsequence of Prop ositio n 3 5.  The co mplex in which w e are interested is given by C ∗ ( R ) := Q ∗ ( R ) / M ∗ ( R ) . It is called the c omplex of nonde gener ate matrix p airs o f R . If f : R → S is a morphism of r ing s, then for every n, Q n ( f ) : M n ( R ) ⊆ M n ( S ) , which induces a morphism of co mplexes C ∗ ( f ) : C ∗ ( R ) → C ∗ ( S ) . If f : R → S is an epimorphism, then Q ∗ ( f ) is an epimor phis m of complexes, a s is the induced C ∗ ( f ) . F or every n, denote by B n ( R ) ⊆ C n ( R ) the subgr oup of n -b oundaries, and by Z n ( R ) the subgroup of n -cycles. Note that these definitions g eneralize the ab ov e definitions of B 0 ( R ) and Z 0 ( R ) , b ecause C 0 ( R ) = Q 0 ( R ) and ∂ 1 ( M 1 ( R )) = 0 . In pa rticular, if we define the n -dimensional homolo gy of R to b e the homology Z n ( R ) /B n ( R ) o f C ∗ ( R ) at C n ( R ) , then this co incides with the ea r lier definition of H 0 ( R ) for n = 0 . If f : R → S is an epimor phis m of rings, then the morphis m Q ∗ ( f ) : Q ∗ ( R ) → Q ∗ ( S ) is a n epimor phis m of complexes, so the induced morphism C ∗ ( f ) : C ∗ ( R ) → C ∗ ( S ) is also an epimorphism. Eac h of the ab elian g roups Q n ( S ) / M n ( S ) is free, so that a long exact sequence of homolo gy arises. 3.5. 1 -Dim ensional Homo logy of a Field. Let k b e a , not neces s arily c omm utative, field. Corollar y 12 implies that every ma trix pair over k is equiv a len t to a system. The mo dular lattice L 1 ( k ) is tr iv ial; it co ns ists of ex actly tw o unar y matrix pair s [0] = 1 1 and [1] = 0 1 , the maximum and minimum elements, resp ectively . It follows that Q 0 ( k ) = C 0 ( k ) = Z 0 1 ⊕ Z 1 1 , and that the g r oup o f 0- cycles is given by Z 0 ( k ) = Z (1 1 − 0 1 ) . The g roup o f 0 -bo undaries is trivial, B 0 ( k ) = 0 , be cause the nondegenerate elements of L 2 ( k ) hav e the form [1 , r ] , with r 6 = 0 , and ∂ 1 ([1 , r ]) = ([1 | r ] − [ r ]) − ([ r | 1] − [1]) = (1 1 − 0 1 ) − (1 1 − 0 1 ) = 0 . Since ∂ 1 = 0 every 1-chain is a 1 -cycle, C 1 ( k ) = Z 1 ( k ) , so that Z 1 ( k ) is the free ab elian group on the elements [1 | r ] , where r ∈ k × , the multiplicativ e group o f nonzero elements of k . There ar e t wo families o f nondegenera te g e ne r ators of C 2 ( k ) , given by [1 , r , s ] and  1 0 r 0 1 s  . The b oundar y map o f the first family may b e computed as ∂ 2 [1 , s, r ] = ([1 | s, r ] − [ s, r ]) − ([ s | 1 , r ] − [1 , r ]) + ([ r | 1 , s ] − [1 , s ]) = (1 2 − [1 , s − 1 r ]) − (1 2 − [1 , r ]) + (1 2 − [1 , s ]) = [1 , r ] − [1 , s ] − [1 , s − 1 r ] , 25 bec ause 1 2 is a de g enerate ter tiary matrix pair. Similar ly , ∂ 2  1 0 r 0 1 s  = (  1 0 r 0 1 s  −  0 r 1 s  ) − (  0 1 r 1 0 s  −  1 r 0 s  ) + (  r 1 0 s 0 1  −  1 0 0 1  ) . All three systems that app ear have a ma trix of co efficie nts row equiv alent to I 2 , and ar e therefore equiv alent to the de g enerate binar y matrix pair [ I 2 ] = 0 2 . T o simplify the other terms, just note that  1 0 r 0 1 s  = [1 | 0 , r ] ∧ [1 , s ] = 1 2 ∧ [1 , s ] = [1 , s ];  0 1 r 1 0 s  = [1 , r ] ∧ [1 | 0 , s ] = [1 , r ] ∧ 1 2 = [1 , r ]; and, the most in teresting case,  r 1 0 s 0 1  =  r 1 0 0 − sr − 1 1  = [ r | 1 , 0] ∧ [ − sr − 1 , 1] = 1 2 ∧ [1 , − rs − 1 ] = [1 , − rs − 1 ] . The g roup B 1 ( k ) ⊆ Z 1 ( k ) of 1-b ounda r ies is therefor e genera ted by the tw o families (2) [1 , r ] − [1 , s ] − [1 , s − 1 r ] and (3) [1 , s ] − [1 , r ] + [1 , − rs − 1 ] , as r and s v ar y ov er the m ultiplicative group k × . Theorem 37. If k is a field, then H 1 ( k ) is the ab elian gr oup gener ate d by the symb ols [1 , r ] , r ∈ k × mo dulo the r elations [1 , r ] − [1 , s ] − [1 , rs − 1 ] and [1 , − 1] . Pr o of. First, we s how that these tw o relations hold in H 1 ( k ) . Letting r = s = 1 in b oth relations yields [1 , 1] = [1 , − 1] in H 1 ( k ); then, le tting r = 1 in b oth relations y ie lds [1 , ± s − 1 ] = − [1 , s ] in H 1 ( k ); and, finally , letting s = t − 1 in the first r elation yields [1 , r ] + [1 , t ] = [1 , tr ] . Thu s [1 , t r ] = [1 , rt ] , w hich a pplied to the firs t relation, yields [1 , r ] − [1 , s ] = [1 , s − 1 r ] = [1 , rs − 1 ] . Suppo se, on the other hand, that we ar e given the rela tions [1 , r ] − [1 , s ] − [1 , rs − 1 ] and [1 , − 1] . Le tting r = s yields [1 , 1] = 0; then, letting r = 1 yields [1 , s − 1 ] = − [1 , s ]; and finally , letting s = t − 1 yields [1 , r ] + [1 , t ] = [1 , r ] − [1 , t − 1 ] = [1 , rt ] . Whence [1 , r ] − [1 , s ] = [1 , rs − 1 ] = [1 , s − 1 r ] and [1 , s ] − [1 , r ] = [1 , rs − 1 ] = [1 , rs − 1 ] − [1 , − 1] = [1 , − rs − 1 ] .  Recall [11, § 2.2 ] that the firs t K - group of a field k is the ab elianiza tio n o f the m ultiplicative group, K 1 ( k ) = ( k × ) ab . A standard group-theore tic arg umen t shows that if Γ is a g roup, then its ab elianization Γ ab is isomo rphic to the free ab elian group on the symbols x g , g ∈ Γ , mo dulo the family of relations x g − x h − x gh − 1 , as g and h v ary over Γ . The o rem 37 thus implies the following. 26 Corollary 38. H 1 ( k ) ∼ = ( k × ) ab / {± 1 } . 4. The Model Theor y of Modules In this final section, we consider several characteriza tions of the pr e-order ≤ n on L ′ n ( R ) . These v arious , but equiv alent, formulations may seem more co ncrete, a nd s erve as a historical reference fo r the origin of the pre-order ≤ n . 4.1. A Comple te ness Theorem. The language (cf. [10, § 1.1]) for left R -mo dules is L ( R ) = (+ , − , 0 , r ) r ∈ R . I t co n tains the la nguage (+ , − , 0) for ab elian groups together with unary function symbols r, one for every element r ∈ R, and denoted the s ame wa y . The symbols from the a be lia n gro up la nguage ar e in tended to in terpret the underlying ab elian group structure of a left R -mo dule R M , while the unary function symbo l r is intended to interpret the ac tio n of the cor resp onding ring element r ∈ R on M . The atomic formulae in L ( R ) are o f the form (4) b 1 w 1 ± b 2 w 2 ± · · · ± b m w m . = a 1 v 1 ± a 2 v 2 ± · · · ± a n v n , where the v i and w j are v ar iables, and a i and b j elements from R acting as sca la rs on the left. These formulae a re nothing more than linear equations. The standard axioms for a left R -mo dule ar e ex pressible in L ( R ) . In the language L ( R ) , it is no t po ssible to qua ntify ov er the ring R, so that this collection of axioms, denoted by T ( R ) , is usua lly infinite. F or exa mple, T ( R ) contains the ax io m schema: for every r ∈ R , ( ∀ v , w ) r ( v + w ) . = rv + r w . Relative to the a xioms T ( R ) , w e may rewrite the a to mic formula (4) as b 1 w 1 + b 2 w 2 + · · · + b m w m . = a 1 v 1 + a 2 v 2 + · · · + a n v n . A system of linea r equatio ns is expr essible as a finite conjunction of linear equations , B w . = A v , where A is an m × n matrix, B an m × k matrix a nd v = ( v i ) n i =1 and w = ( w j ) k j =1 c olumn ve ctors of n, (resp ectively , k ) v ar iables. A p ositive-primitive formula is a n existentially quantified system of linear equations: ∃ w ( B w . = A v ) . There ar e tw o extreme c ases of a p ositive-primitive formula; (1) if B = 0 , then the p o sitive-primitiv e formula is equiv a lent , r elative to T ( R ) , to a quantifier-fr e e formula, the homogeneous sy stem of linear eq ua tions A v . = 0 ; (2) if A = I n , the n × n ident ity matrix, then the p ositive-primitive fo rmula is equiv alent, relative to T ( R ) , to the divisibi lity c ondition B | v := ∃ w ( B w . = v ) . Using this no ta tion, we may express a g e neral pp-formula in the free v ariables v as ( B | A )( v ) := B | A v . The Rules of Divisibilit y corre s po nd to three prop erties of p ositive-primitiv e formulae: (1) T ( R ) ⊢ ∀ v [( B | A )( v ) → ( U B | U A )( v )]; (2) T ( R ) ⊢ ∀ v [( B V | A )( v ) → ( B | A )( v )]; and (3) T ( R ) ⊢ ∀ v [( B | A )( v ) → ( B | A + B G )( v )] , and may ther efore b e in terpreted a s rules o f inference. If the r elation ( B | A ) ⊢ n ( B ′ | A ′ ) is defined to hold in L ′ n ( R ) whenever T ( R ) ⊢ ∀ v ( B | A )( v ) → ( B ′ | A ′ )( v ) , then we see, in view of P rop osition 1, tha t ( B | A ) ≤ n ( B ′ | A ′ ) implies ( B | A ) ⊢ n ( B ′ | A ′ ) . The following completeness theorem is due to Pr est. 27 Theorem 39. (Lem m a Presta I) [10, L e mma 1.1.13 and Cor. 1.1.1 6] Given ( B | A ) and ( B ′ | A ′ ) in L ′ n ( R ) , ( B | A ) ≤ n ( B ′ | A ′ ) if and only if ( B | A ) ⊢ n ( B ′ | A ′ ) . Theorem 39 is a co mpleteness theorem, because it implies that any implication of the form T ( R ) ⊢ ∀ v ( B | A )( v ) → ( B ′ | A ′ )( v ) ha s a pro o f using exclusively the r ules o f inference RoD (1)-(3) 4.2. Nonhomo geneous Systems. Let A b e a n m × n matrix; B and m × k matrix and R M a left R -mo dule. Let us cons ider the nonhomo gene ous system of linea r equations A v . = b , where v is a column n -vector of v ariables ( v i ) and b a column k -vector with entries from M . Denote by Sol M ( A v . = b ) := { a ∈ ( R M ) n : A a = b } the subgro up o f ( R M ) n of solutio ns in M to the nonhomoge ne o us sys tem. As in the classica l case ov e r a field, if Sol M ( A v . = b ) is nonempty , then it is a coset of the subg roup of solutions Sol M ( A v . = 0 ) of the corr esp onding homogeneous system. Given an elemen t ( B | A ) ∈ L ′ n ( R ) , define ( B | A )( M ) := [ b ∈ B M k Sol M ( A v . = b ) . This is consistent with mo del-theor etic notatio n, b ecause ( B | A )( M ) is the subset o f ( R M ) n defined in R M b y the formula ( B | A )( v ) , ( B | A )( R M ) = { a ∈ ( R M ) n : ∃ c ∈ ( R M ) k ( B c = A a ) } = { a ∈ ( R M ) n : M | = ( B | A )( a ) } . It is an exercise to show that ( B | A )( M ) is a subgr oup of ( M n ) Z , functorial in R M . Subgroups of the for m a re called ( B | A )( M ) ar e known by several names : pp-definable sub gr oups [10] , finite matrix sub gr oups [1 4] , sub gr oups of fin ite definition [6] , etc. Let Latt n ( M Z ) denote the lattice of subgro ups of ( M n ) Z ; a map Ev ′ M : L ′ n ( R ) → Latt n ( M Z ) is obtained by the rule ( B | A ) 7→ ( B | A )( R M ) . By Lemma P resta I, this a morphism of pre- orders, a nd, b ecause Latt n ( M Z ) is a partial or der, the morphism factors through L n ( R ) to yield a mor phism Ev M : L n ( R ) → La tt n ( M Z ) , [ B | A ] 7→ ( B | A )( M ) satisfying the following prop erties: (1) if [ B | A ] ≤ n [ B ′ | A ′ ] holds in L n ( R ) , then ( B | A )( R M ) ≤ n ( B ′ | A ′ )( R M ) holds in La tt n ( M Z ); (2) [ B | A ] ∧ [ B ′ | A ′ ] 7→ ( B | A )( M ) ∩ ( B ′ | A ′ )( M ); (3) [ B | A ] + [ B ′ | A ′ ] 7→ ( B | A )( M ) + ( B ′ | A ′ )( M ); and (4) 1 n ( M ) = M n and 0 n ( M ) = 0 . W e may thus define the relatio n ( B | A ) | = n ( B ′ | A ′ ) to hold in L ′ n ( R ) pr ovided that ( B | A )( M ) ⊆ ( B ′ | A ′ )( M ) for every left R -mo dule R M . Eq uiv alently , T ( R ) | = ∀ v ( B | A v → B ′ | A ′ v ) . By G¨ odel’s Completeness Theor em, the rela tions ⊢ n and | = n on L ′ n ( R ) a re equal, so that one obtains a sec ond version of Lemma Presta . Prop ositio n 40. (Lem ma Presta I I.) Given ( B | A ) and ( B ′ | A ′ ) in L ′ n ( R ) , ( B | A ) ≤ n ( B ′ | A ′ ) if and only if ( B | A ) | = n ( B ′ | A ′ ) . 28 4.3. The T ensor Pro duct and Dualit y. If ( B | A ) ∈ L ′ n ( R ) , then, in a right R -mo dule K R , its dual de fines the subgro up B tr 0 A tr I n ! ( K R ) = { a ∈ K n : K | = ∃ w ( w A . = a ∧ w B . = 0 ) } , where the action of R op is wr itten on the right. Define the relation ( B | A ) ⊳ n ( B ′ | A ′ ) to hold in L ′ n ( R ) pr ovided that for every right R -mo dule K R and left R -mo dule R M , the subgroup ( B ′ ) tr 0 ( A ′ ) tr I n ! ( K ) ⊗ ( B | A )( M ) = 0 in the tensor pro duct K ⊗ R M . Let us verify that if ( B | A ) ≤ n ( B ′ | A ′ ) ho lds in L ′ n ( R ) , then ( B | A ) ⊳ n ( B ′ | A ′ ) . W e are given U, V a nd G such that U B = B ′ V a nd U A = A ′ + B ′ G. Le t a ∈ ( B ′ ) tr 0 ( A ′ ) tr I n ! ( K ) and b ∈ ( B | A )( M ) . Then there exist c ∈ K m ′ such that c A ′ = a and c B ′ = 0 ; and d ∈ M k such that B d = A b . Then the tensor a ⊗ b = n X i =1 a i ⊗ b i simplifies in K ⊗ R M as a ⊗ b = c A ′ ⊗ b = c ( U A − B ′ G ) ⊗ b = c U A ⊗ b = c U ⊗ A b = c U ⊗ B d = c U B ⊗ d = c B ′ V ⊗ d = 0 ⊗ d = 0 . Note how the prop erty that c B ′ = 0 has b een used twice in this simplification. By Cor o llary 1.3.8 of [10], the co nverse also holds. Prop ositio n 41. Given ( B | A ) and ( B ′ | A ′ ) in L ′ n ( R ) , ( B | A ) ≤ n ( B ′ | A ′ ) if and only if ( B | A ) ⊳ n ( B ′ | A ′ ) . References [1] Adelman, M., Ab el i an categories ov er additive ones, Journal of P ur e and Applied Algebra 3 (1973), 103-117. [2] Cr a wley-Bo ev ey , W.W., R epr esent ations of A lgebr as and R elate d T opics, T achik aw a, H. and Brenner, S., eds. London Mathematical So ciety Lecture Note Series 1 68 , 127-184. [3] Jacobson, N. , Basic Algebr a I, 2nd ed., W.H. F ri eman and Co., 1985. [4] Go o dearl, K., V on Neumann R e gular Rings, Krieger Publishing Co., 1991. [5] Goursat, M. 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[13] Silver, L., Noncommuta tive lo calization and applications, J. of Algebra 7 (1967), 44-76. 29 [14] Zimmermann, W., Rein injektive direkte Summen von M oduln, Communications in Algebra 5 (1977) , 1083-1117. [15] Zimmermann-Huisgen, B. and Zimmermann, W., On the sparsity of represent ations of rings of pur e global dimension zero, T ransactions of the AMS 320 (1990), 695-713. Yonsei University, Sinchon-dong, Seodaemun-gu , Seoul 120 -749, South Korea E-mail addr e ss : herzog@yonsei. ac.kr The Ohio St a te University at Lim a, Lima, OH 4580 4 USA E-mail addr e ss : herzog.23@osu. edu 30