Stability of rings

Stabilit y of rings ∗ V. M. P etec h uk † Abstract The conditions for stabilit y of the elements of linear groups ov er the associative rings with id entit y and th eir conn ection with the stabilit y of rings are analyzed in the article. The stability of rings whic h are com- mutativ e, satisfy the conditions of stabilit y of rank ≥ 2, von N eu mann regular, integer-algebraic, nearly local rings introduced by the author, is examined. The most important clas sical results of H. Bass, L. V aserstein, S.H. Kh lebutin, A.A. Suslin, J.S. Wilson, I.Z.Golub chik, are considered from the unified standp oint. The study of stability of rings takes ro ot from the w ell-known commutator formula S L ( n, R ) = [ S L ( n, R ) , S L ( n, R )] and Jor dan-Dixon’s theorem (1870) ab out the simplicit y of P S L ( n, R ) ov er the field R when n ≥ 3. Later, due to the works of Dieudonn´ e in the 5 0s, it turned out that these statements ar e v a lid ov er skew fie lds as well. The problem of the stabilit y o f local rings was for m ulated and solved in the 60s by Klingenberg . As it turned out, the gene r alization of the comm u- tator formula ov er lo cal rings with n ≥ 3 are the formulas [ C ( n, I ) , E ( n, R )] = E ( n, I ) ⊳ GL ( n, R ) whic h hold for all idea ls I of the ring R ; and the general- ization o f the J o rdan-Dixon fo r mula are the inclusio ns E ( n, I 0 ) ⊂ G ⊂ C ( n, I 0 ) for in v aria nt with r esp ect to E ( n, R ) subgro ups G of the gro up GL ( n, R ) and their resp ective ideals I 0 of the ring R . Asso ciative rings with identit y tha t satisfy the aforementioned generaliza- tions shall b e c alled st able . The work o f H. Ba ss in 196 4 was the fundamental bre akthrough in the pro of of stability of wider c lasses of rings. Therein, he proved the stability of ring s that satisfy the co ndition of stability o f r ank ≥ 2. F or instance, se milo cal rings that satisfy the rank 1 stability condition are sta ble as well. In 1 972-1 977 Wilso n, I.Z . Golub chik, A.A. Suslin proved that commutativ e rings with identit y are stable. The stabilit y o f rings that ar e finitely generated ov er their ce nt ers is the g e neralization of this res ult. This theo r em was o btained independently in the w orks of A.A. Suslin and L. V aser stein. Moreov er, in 19 8 1 L. V asers tein ha s prov ed the stability of a sso ciative rings with identit y if their lo calizations ov er all ma ximal ideals of the ring’s cen ter satisfy the co nditio n o f stability of r ank ≥ 2. In 20 01, V.M. Petec huk pr ov e d the s tability of asso cia tive ∗ The Ukr ainian version of this article was published in [ 23]. † Email: v asil-petec huk@rambler .ru 1 rings with ident ity sho uld their lo ca lizations over all maximal ideals of the ring’s center b e stable. In 1986 S.H. Khlebutin and L. V as e r stein, independently prov ed the sta bilit y of ring s reg ular in von Neumann sense. In 20 0 1, V.M. Petech uk gene r alized this result to rings that are in teger a lgebraic over ar bitrary subr ings o f own c e nt ers. In 1 995, the principally new appr oach to establishing the stability of weakly No etherian rings that co nt ain infinite fields in own centers was pr op osed b y I.Z. Golubchik. In 199 7 he announced ab out the sta bility o f the blo ck integer- algebraic ring s . Stabilit y of rings is considered in the Chev alley groups as well. Stabilit y of commutativ e rings in the Chev a lley gr oups is pro ved in [2, 3, 26, 31] a nd Lie-type rings in [15]. In the current pap er, the stability of ass o ciative rings with identit y is pr ov ed from a wider than earlier notion of the s ta bilit y of e lement s of the genera l linear group. The conditions of stability of elemen ts of linear groups ov e r the asso c ia tive rings with identit y ar e considered in the pap er as well. The stability of ring s that satisfy the condition of stability of ra nk ≥ 2, comm utative, r egular in von Neumann sense, blo ck in teger-a lgebraic and the nearly loca l rings, introduced by the a uthor, follow from these conditio ns . The s ta tement s ab out the s ta bilit y o f some rings ar e mo st systematica lly laid out in [6 , 1 7]. Rema rks in the third par agra ph o n p.1 22 o f [1 7] do not corres p o nd to reality . Stabilit y of rings plays an imp orta nt role in a pplications [4, 7, 1 8, 24, 25, 28, 32], particularly in the description of the homomor phisms of linea r groups ov e r asso ciative rings [1, 11, 12, 1 6, 17, 20] Let R be an asso ciative ring with identit y , R ∗ - group o f inv ertible e lement s of the ring R , J ( R ) - Jaco bson’s radical of the r ing R , ξ R and ξ R ∗ - centers of R and R ∗ resp ectively . Let R n = M n ( R ) denote the ring of all n × n matrices ov er R and GL ( n, R ) = R ∗ n - resp ectively the genera l line a r gro up o f inv ertible matrices. Under e ij ∈ M n ( R ) w e shall understand the matrix with iden tity at the place ( i, j ), and zero s at the rest. Such matrices we shall call standar d identity matric es . The elemen t t ij ( r ) = 1 + r e ij , where 1 is the identit y matrix, i 6 = j, r ∈ R shall be ca lled a tr ans ve ction . Sometimes the identit y matr ix sha ll b e denoted by E . If X is a subset of the ring R then under t ij ( X ) w e s ha ll under- stand the set { t ij ( r ) | r ∈ X } for fixed i 6 = j , 1 ≤ i, j ≤ n , E X = h t ij ( X ) i - subgroup of the group GL ( n, R ), gene r ated b y the set of all t ij ( X ), 1 ≤ i, j ≤ n . In the par ticula r ca se when X = R we shall also use a no tation E ( n, R ) = E R . Non-identit y tr ansvections from t ij ( R ) and t j i ( R ) shall b e ca lled opp osite . Let I b e an arbitra ry t wo-sided ideal o f the r ing R . Define by Λ I : R → R / I , Λ I : M n ( R ) → M n  R / I  , Λ I : GL ( n, R ) → GL  n, R / I  the natural homomo rphisms of the rings R , M n ( R ) and the group GL ( n, R ). Let us define a subgr oup C I = ker Λ I that we shall call the main c ongruen c e- 2 sub gr oup of level I in the group GL ( n, R ). The full preimage of the c e nter of the group GL  n, R / I  shall be denoted b y C ( n, I ) = Λ − 1 I ξ GL  n, R / I  and under E ( n, I ) we s ha ll understand the no rmal closure of the gr oup E I in E ( n, R ). It is easy to se e that E ( n, I ) ⊆ C I ⊆ C ( n, I ) . Let N and G b e s ubgroups of the group GL ( n, R ) that are inv a riant with resp ect to the g roup E ( n, R ) and N do es not co ntain non-identit y transvections. Under I 0 we shall understand the lar gest ideal of ring R such that E ( n, I 0 ) ⊆ G . F or elements o f an arbitrary group w e shall use the notations a b = ba b − 1 , [ a, b ] = aba − 1 b − 1 and [ a 1 , . . . , a l ] = [[ a 1 , . . . , a l − 1 ] , a l ] a nd the commutator for- m ulas [ ab, c ] = [ b , c ] a · [ a, c ], [ a, bc ] = [ a, b ] · [ a, c ] b , and P . Hall’s identit y  a − 1 , b, c  a ·  c − 1 , a, b  c ·  b − 1 , c, a  b = 1 . F urther we sha ll a ssume that n ≥ 3. F or tw o non-opp o s ite trans vections t ik ( x ) and t lj ( y ), ( l , j ) 6 = ( k , i ) the fol- lowing matrix co mm utator for mulas hold: [ t ik ( x ) , t lj ( y )] =  t ij ( δ kl xy ) , i 6 = j t lk ( − δ ij y x ) , l 6 = k , where δ ij , δ kl are Kronecker’s deltas. F ro m this formula in particular , it follo ws that the commutator o f tw o non- opp osite transvections commutes with ea ch o ne of them. The following result, first obta ine d b y L. V aser stein, a shorter pro of of whic h we shall pr e sent, holds. Lemma 1 L et I b e an ide al of t he ring R with identity. Then E ( n, I ) = D t ij ( I ) t ji ( R ) | 1 ≤ i 6 = j ≤ n E . Pr o of . L et T = D t ij ( I ) t ji ( R ) | 1 ≤ i 6 = j ≤ n E . It is clear that E I ⊆ T ⊆ E ( n, I ) and t ij ( I ) t kl ( R ) ⊆ E I when ( k , l ) 6 = ( j, i ) . Therefore, for any trans vection τ the fo llowing inclus ion is v alid E τ I ⊆ T and, as a consequence , E τ τ 1 I ⊆ T , where τ 1 is an a r bitrary transvection co mm uting with τ . If the transvection τ / ∈ t ij ( R ) then the commutators [ t j i ( R ) , τ ] are transvec- tions, comm uting with tra ns vections τ and t j i ( R ), and it then follows from the matrix commutator form ulas that  t ij ( I ) t ji ( R )  τ ⊆ ( t ij ( I ) τ ) t ji ( R )[ t ji ( R ) ,τ ] ⊆ E t ji ( R )[ t ji ( R ) ,τ ] I ⊆ T . 3 When τ ∈ t ij ( R ), by c ho osing s 6 = i , j we ha ve  t ij ( I ) t ji ( R )  τ ⊆  [ t is ( I ) , t sj ( R )] t ji ( R )  τ ⊆ [ t j s ( I ) t is ( I ) , t si ( R ) t sj ( R )] τ ⊆ ⊆ [ E I , t si ( R ) t sj ( R )] ⊆ T . Therefore, T τ ⊆ T , E ( n, I ) ⊆ T and E ( n, I ) = T and so Lemma 1 is prov ed. Corollary 1 L et I, J ide als of the ring R with identity. Then E ( n, I J ) ⊆ [ E I , E J ] . In p articular, E  n, I 2  ⊆ E I . Indeed, for an y tw o pa irwise distinct num b er s 1 ≤ i , j, s ≤ n the following inclusions hold t ij ( I J ) t ji ( R ) ⊆ [ t is ( I ) , t sj ( J )] t ji ( R ) ⊆ [ t j s ( I ) t is ( I ) , t si ( J ) t sj ( J )] ⊆ [ E I , E J ] . In view of Lemma 1, E ( n, I J ) = D t ij ( I J ) t ji ( R ) | 1 ≤ i 6 = j ≤ n, E ⊆ [ E I , E J ] . By taking I = J, w e hav e E  n, I 2  ⊆ E I . The description o f the norma l structure of linea r groups ov er some ring s, as usual, consists of tw o co mp o nents. F rom one side, it is proved that for the group G , whic h is no rmalized by the g roup E ( n, R ), there exists an ideal I of the ring R suc h that E ( n, I ) ⊆ G ⊆ C ( n, I ) , and from the o ther, the v alidity of ide ntit y [ C ( n, I ) , E ( n, R )] = E ( n, I ) ⊳ GL ( n, R ) for a n arbitrary idea l I of the ring R is prov ed. It should b e noted that b oth comp onents o f the normal structur e o f linea r gr oups are not alwa ys v alid at the s ame time. Moreo ver, there exist some r ings over which no ne of them is v alid. How ever, the class of ring s for which b oth comp onents o f the descr iption of the normal str ucture of linear gr oups ar e v alid is quite wide. In par ticula r, it contains comm utative rings, rings finitely generated ov er their centers, and others. The sea rch for conditions for rings, that would be b oth nece ssary and sufficient fo r the aforementioned compo nent s of the normal s tructure o f linear groups, is contin uing on. The present ar ticle represents o ne such attempt. Definition 1 The asso ciative ring R with identity is c al le d c ommutator if for al l ide als I of the ring R t he fol lowing identity holds [ C ( n, I ) , E ( n, R )] = E ( n, I ) and E ( n, I ) - normal sub gr oup of GL ( n, R ) . Definition 2 The asso ciative ring R with identity is c al le d we akly-c ommutator if ther e exists a p ositive inte ger k such that   C ( n, I ) , E ( n, R ) , . . . , E ( n, R ) | {z } k times   = E ( n, I ) ⊳ GL ( n, R ) 4 simultane ously for al l ide als I of the ring R . The num b er k is c al le d the length of the we akly-c ommutator ring R . It is not har d to notice that in an arbitrar y as s o ciative ring R with identit y one has the inclusion E ( n, I ) ⊆ [ C ( n, I ) , E ( n, R )], wher e I is a n ar bitrary ideal of R . Mor eov er, the as so ciative ring R with identit y is commutator iff [ C ( n, I ) , E ( n, J )] ⊆ E ( n, I ) \ E ( n, J ) for all ideals I and J of the ring R . Obviously , commutator ring s are weakly- commutator, and in the commutator r ings the subgroup E ( n, R ) is a normal subgroup of the group GL ( n, R ). Definition 3 The asso ciative ring R with identity is c al le d n ormal if for an arbitr ary s ub gr oup G, invaria nt with r esp e ct to the gr oup E ( n, R ) , t her e exist s an ide al I of the ring R such that E ( n, I ) ⊆ G ⊆ C ( n, I ) . Definition 4 The asso ciative ring R with identity is c al le d p artial ly normal if an arbitr ary sub gr oup N, invariant with r esp e ct to the gr oup E ( n, R ) and n ot c ontaining non-identity tr ansve ctions, is c ontaine d in ξ GL ( n, R ) . Obviously the quotient r ings of norma l rings are par tially normal. Definition 5 As s o ciative rings that ar e c ommutator and normal at the same time ar e c al le d s t able. It sho uld b e hig hlig hted that commutator proper ty , normality and, as a consequence, sta bility of ring s a re defined in the group GL ( n, R ) and, therefore, depe nd on n . Lemma 2 The we akly-c ommutator ring R, the quotient rings of which ar e p ar- tial ly n ormal, is stable. Pr o of. Let I 0 be the larges t ideal of R such that E ( n, I 0 ) ⊆ G. If Λ I 0 ( G ) contains transvections, then there exists a nonzero set J 0 = { r ∈ R | Λ I 0 ( t ij ( r ) ) ∈ Λ I 0 ( G ) for some i 6 = j } . It is easy to see that J 0 is a n ideal, containing I 0 . Since Λ I 0 E ( n, J 0 ) ⊆ Λ I 0 ( G ), we have E ( n, J 0 ) ⊆ GC I 0 . Therefor e, for r ∈ J 0 there exis ts g ∈ G such that t ij ( r ) g ∈ C I 0 ⊆ C ( n, I 0 ). Since R is a weakly-commutator ring of length k , then    C ( n, I 0 ) , E ( n, R ) , . . . , E ( n, R ) | {z } k times    = E ( n, I 0 ) ⊆ G. Therefore, 5 [ t ij ( r ) g , E ( n, R ) , . . . , E ( n, R )] ⊆ G and, as a consequenc e , t ij ( r ) ⊆ G. This means that E ( n, J 0 ) ⊆ G , which contradicts the definition of ideal I 0 . Therefore, Λ I 0 ( G ) do es not contain non-identit y tra ns vections. Since the ring R / I 0 is partially no r mal, w e have Λ I 0 ( G ) ⊂ ξ GL  n, R / I 0  , i.e. G ⊆ C ( n, I 0 ) . As a re s ult, E ( n, I 0 ) ⊆ G ⊆ C ( n, I 0 ) . This prov es that R is a normal ring. Let’s prove that R is a commutator ring. Let g ∈ GL ( n, R ) , H = E ( n, R ) g and H 0 = H E ( n,R ) be a nor mal clo sure of H with r esp ect to E ( n , R ). Naturally , H ⊆ H 0 and H 0 is an inv ariant subgroup with r esp ect to the gro up E ( n, R ). Since R is a norma l ring, then there exists an ideal I o f the r ing R such that E ( n, I ) ⊆ H 0 ⊆ C ( n, I ) . T ak ing into account the fact that C ( n, I ) is a normal subgro up of the gro up GL ( n, R ) a nd E ( n, R ) g ⊆ H 0 ⊆ C ( n, I ), we hav e I = R , E ( n, R ) ⊆ H 0 and [ E ( n, R ) , H ] ⊆ [ H 0 , H ]. It follo ws from E ( n, R ) = [ E ( n, R ) , E ( n, R )] that H = [ H , H ] ⊆ [ H 0 , H ]. This prov es that H 0 = H E ( n,R ) ⊆ [ E ( n, R ) , H ] H ⊆ [ H 0 , H ] . Since the r ing R is w eakly-c o mmut ator o f length k , it follows that   GL ( n, R ) , E ( n, R ) , . . . , E ( n, R ) | {z } k times   ⊆ E ( n, R ) . Therefore H 0 ⊆   H 0 , H, . . . , H | {z } k times   ⊆   GL ( n, R ) , H , . . . , H | {z } k times   ⊆ H . This means that H 0 = H , E ( n, R ) ⊆ H and E ( n, R ) g ⊆ E ( n, R ) for all g ∈ GL ( n, R ). Thus, we have pr ov ed that E ( n, R ) is a normal subgroup of the group GL ( n, R ). By taking into account the fact that R is a weakly-commutator ring of length k , i.e. E ( n, I ) =   C ( n, I ) , E ( n, R ) , . . . , E ( n, R ) | {z } k times   we obtain, as a conse quence, that E ( n, I ) is a normal subgro up of the group GL ( n, R ) for all idea ls I of the ring R . If k ≥ 2, we denote 6 C 1 =    C ( n, I ) , E ( n, R ) , . . . , E ( n, R ) | {z } k − 2 times    . Then [ C 1 , E ( n, R ) , E ( n, R )] = E ( n, I ) and [ E ( n, R ) , C 1 , E ( n, R )] = E ( n, I ) . F ro m P . Hall’s commutator identit y , by taking account that E ( n, I ) is a normal subgroup of the group GL ( n, R ), we r eceive [ E ( n, R ) , E ( n, R ) , C 1 ] ⊆ E ( n, I ). This means that [ C 1 , E ( n, R )] ⊆ E ( n, I ) . Therefore,    C ( n, I ) , E ( n, R ) , . . . , E ( n, R ) | {z } k − 1 times    = E ( n, I ) . Pro ceeding a nalogous ly we obtain [ C ( n, I ) , E ( n, R )] = E ( n, I ). So we prov ed tha t R is a comm utator ring and, a s a conseque nc e , R is a stable ring. F ro m the pro o f o f Lemma 2 w e receive Corollary 2 We akly-c ommut ator normal rings ar e stable. It should b e noted tha t in a comm uta to r r ing R for a subgroup L of the group GL ( n, R ) , that for some ide a l I 0 of the ring R satisfies the condition E ( n, I 0 ) ⊆ L ⊆ C ( n, I 0 ), one has the following inc lus ions E ( n, I 0 ) ⊆ [ L, E ( n , R )] ⊆ [ C ( n, I 0 ) , E ( n, R )] = E ( n, I 0 ) . Therefore, [ L, E ( n, R )] = E ( n, I 0 ) ⊆ L . This means that L is E ( n, R ) - normal subgr oup of the gr oup GL ( n, R ) and ideal I 0 is uniquely defined by the subgroup L . Let N denote an E ( n, R )-inv ariant subgroup of the group GL ( n, R ) , which do es not co ntain non-identit y transvections. If I is a tw o- sided ideal of the r ing R then the annihilator AnnI = { r ∈ R | r I = I r = 0 } of ideal I in R is a tw o-sided idea l as well. Lemma 3 L et R b e an asso ciative ring with identity, g = ( g ij ) ∈ N and ther e exists x ∈ R such that g ij x = 0 for some fixe d 1 ≤ i, j ≤ n . Then g ∈ C ( n, AnnRxR ) if i 6 = j and x = 0 otherwise. Pr o of. It is not hard to see that in the case k 6 = j the i th r ow of the matrix g 1 = [ g , t j k ( x )] ∈ N co incides with the i th row o f the matr ix t j k ( − x ). Suppo se that i 6 = j . Then the i th row of the matrix t j k ( − x ) and the ide ntit y matrix coincide. If g 1 6 = 1, then N c ontains transvections of the type [ g 1 , t li ( R )] for all l 6 = i , 1 ≤ l ≤ n . Since N do es no t contain non-identit y transvections, 7 we ha ve g 1 = 1 for all k 6 = j . It follows from the identit y [ g , t j k ( x )] = 1 that xg ks = 0 fo r all s 6 = k , g sj x = 0 for all s 6 = j and xg kk = g j j x. F ollowing analog ously we prove that g − 1 commutes with all matrices t mk ( x ), where m 6 = k , x ∈ R . Conseq uent ly , g commutes with a ll the trans vections from the group E x . This is equiv a le nt to the condition tha t g x = xg is a scalar matrix (how ever, not necessarily central). Since g ij xR = 0, we hav e that g xr = xrg is a scalar ma trix fo r all r ∈ R . T ak ing into a ccount i 6 = j w e hav e xRg ij = 0 and, co nsequently , R xRg ij = 0 . As a b ov e we prov e that r ′ xrg = g r ′ xr is a scalar matrix for all r , r ′ ∈ R. This means that elements from R xR a nnihilate from the left and right the elemen ts g pq , g pp − g pq of matrix g for all 1 ≤ p 6 = q ≤ n. Hence, it is pr ov ed that g ∈ C ( n, AnnRxR ) . In particular, when g has a zer o no n-diagona l element which, obviously , is annihilated by all the elemen ts of the r ing R, we obtain g ∈ C ( n, AnnR ) = ξ GL ( n, R ). Since g 1 = [ g , t j k ( x )] ∈ N and g 1 has a zer o non-diago nal element(as n ≥ 3), we hav e [ g , t j k ( x )] ∈ ξ GL ( n, R ) . Let us consider the case when i = j . As [ g , t ik ( x )] ∈ ξ GL ( n, R ) and the i th row o f ma trix g xe ik g − 1 is all zero , then for some element r ∈ ξ R T R ∗ , the i th row of matrix rt ik ( x ) − E = ( r − 1) E + rxe ik is all zero a s well. The r efore, r = 1 and x = 0. Remark 1 F ol lowing analo gously, the ar gument s in Le mma 3 r emain valid when inste ad of the e quality g ij x = 0 one c onsiders the e quality xg ij = 0 . It follows fr o m Lemma 3 that the diagona l elements of ma trices o f the gro up N do not have left or rig ht zero dividers. In particular, the diagona l elemen ts of the matrices g ∈ N cannot be equal to zero. Corollary 3 If g ∈ N and for some x ∈ R the c ommutator [ g , t ij ( x )] has a zer o element, then g ∈ C ( n, AnnR xR ) . Pr o of. Since N is an E ( n , R )-in v aria nt group, then [ g , t ij ( x )] ∈ N and, according to Lemma 3, the inclusion [ g , t ij ( x )] ∈ ξ GL ( n, R ) ho lds. The r efore, there exists r ∈ ξ R T R ∗ such that g t ij ( x ) g − 1 = rt ij ( x ). This means that g xe ij = ( r t ij ( x ) − E ) g = ( r − 1) g + rxe ij g . In such a case ( r − 1 ) g ll = 0 where l 6 = i , j . Since the diagonal elemen ts of matrix g do not have zer o diviso rs, then r = 1 a nd g xe ij = xe ij g . Thus, g si x = 0 for all s 6 = i . Acco rding to Lemma 3, the inclusion g ∈ C ( n, AnnRxR ) holds. Lemma 4 L et g ∈ N and x 1 , . . . , x n ∈ R such that g i 1 x 1 + · · · + g in x n = 0 and at le ast one of the elements x i is e qual to zer o. Then g ∈ C ( n, Ann ( Rx 1 R + . . . + Rx n R )) . Pr o of. Suppose that x j = 0 for some 1 ≤ j ≤ n. Then the i th r ow of the co mm utator g 1 = [ g , t 1 j ( x 1 ) · · · t nj ( x n )] ∈ N coincides with the i th row of the ma tr ix t ij ( − x i ). Since it co nt ains zero non-dia gonal elements, then, 8 according to Lemma 3, the c ommutator g 1 ∈ ξ GL ( n, R ). Reca lling that there is a n identit y on the ( i, i )th place in matrix t ij ( − x i ), we hav e g 1 = 1. Therefor e, for a n ar bitrary 1 ≤ l ≤ n there exists 1 ≤ s 6 = k ≤ n such that x l g sk = 0 . According to Le mma 3, g ∈ C ( n, AnnRx l R ) for all 1 ≤ l ≤ n . In this cas e, the elements g pq and g pp − g qq , where 1 ≤ p 6 = q ≤ n are annihilated fr o m left and right by elements of the ideals Rx l R for all 1 ≤ l ≤ n and, therefore, by elements of the sum Rx 1 R + · · · + Rx n R. This prov es the inclusion g ∈ C ( n, Ann ( Rx 1 R + · · · + R x n R )) . Remark 2 Analo gously, if g ∈ N and x 1 , . . . , x n ∈ R su ch that x 1 g 1 j + . . . + x n g nj = 0 and at le ast one of the elements x i is zer o, then g ∈ C ( n, Ann ( Rx 1 R + · · · + Rx n R )) . Corollary 4 If g ∈ N and at le ast one, not ne c essarily diagonal, element of the matrix g has a left or right inverse, then g ∈ ξ GL ( n, R ) . Pr o of. Supp os e that g − 1 ij is a left in verse of g ij . F or k 6 = i put x i = − g kj g − 1 ij , x k = 1 and x s = 0 for a ll s 6 = i , k . Then x 1 g 1 j + . . . + x n g nj = 0 and Rx 1 R + · · · + Rx n R = R . Accor ding to Remark 2 of Lemma 4, the inclusion g ∈ ξ GL ( n, R ) is v a lid. Lemma 5 L et g = g 1 g 2 ∈ N b e such that ( g 1 ) ki = δ ki and ( g 2 ) kj = δ kj for al l 1 ≤ k ≤ n and fixe d 1 ≤ i, j ≤ n . Then g ∈ ξ GL ( n, R ) . Pr o of. The condition o f Lemma 5 means that the i th column of the matrix g 1 − 1 and the j th column of ma trix g 2 − 1 are zero . If i = j , then the i th c o lumn of matrix g coincides with the i th column of the iden tity matrix. Ther efore, g has an invertible element and, in co rresp ondenc e w ith Co rollar y 4 of Lemma 4, g ∈ ξ GL ( n, R ). Suppo se that i 6 = j . Without loss of gener ality , up to the s imilarity b y a matrix from the group E ( n, R ), we ca n ass ume tha t i = 1 and j = n . Then g =  1 x 0 A   B 0 y 1  , where A, B ∈ GL ( n − 1 , R ) and x, y - rows of leng th n- 1 . Let X =  1 − xA − 1 0 E  , Y =  E 0 − y 1  , g 0 = X g Y where E is a n identit y ( n − 1) × ( n − 1) matrix. Then g 0 =  1 0 0 A   B 0 0 1  . Since X ∈ E ( n, R ) and Y co mmutes with t n 1 (1), then X [ g , t n 1 (1)] X − 1 ∈ N and [ g 0 , t n 1 (1)] ∈ N [ X , t n 1 (1)]. O n the other hand, the first row of matrices  1 0 0 A    B 0 0 1  , t n 1 (1)   1 0 0 A  − 1 ,  1 0 0 A  , t n 1 (1)  9 and, as a co nsequence, o f ma trix [ g 0 , t n 1 (1)] c oincides with the first row of the ident ity matr ix . Hence, [ g 0 , t n 1 (1) , t n 1 (1) , t n 1 (1)] = E . This means that h = [ X , t n 1 (1) , t n 1 (1) , t n 1 (1)] ∈ N . Since the seco nd row of ma trix h coincides with the sec o nd row of the identit y matrix, then, in accordance with the corollar y 4 o f Lemma 4, h = E . Let − xA − 1 = ( x 2 , . . . , x n ). Dir e c t co mputation shows that from the equa lity h = E o ne has x 4 n = 0. Th us , [ t n 1 (1) , X ] 11 = 1 − x n ∈ R ∗ . As a co r ollary we obtain that element of the ma trix [ g 0 , t n 1 (1)] · [ t n 1 (1) , X ] , which is at the p osition (1 , 1) , coincides with 1 − x n and, therefore, is invertible. T ak ing in to account the fact that Y co mm utes with t n 1 (1) , from the com- m utator for mu las one has a n e q uality [ g , t n 1 (1)] =  X − 1 g 0 , t n 1 (1)  = [ g 0 , t n 1 (1)] X − 1  X − 1 , t n 1 (1)  . Hence, its obvious coro lla ry ha s place [ g , t n 1 (1)] X = [ g 0 , t n 1 (1)] [ t n 1 (1) , X ]. Since g ∈ N , then [ g , t n 1 (1)] X ∈ N and, according to the Corollar y 4 o f Lemma 4, [ g , t n 1 (1)] X ∈ ξ GL ( n, R ) and, a s a co nsequence, [ g , t n 1 (1)] ∈ ξ GL ( n, R ) and g ∈ ξ GL ( n, R ) . W e sha ll ne e d some statements v alid for a rbitrar y rings. The fo llowing has place Lemma 6 L et a, b, c b e some elements of the asso ciative ring R with identity. Element 1 + ab ∈ R ∗ if and only if 1 + ba ∈ R ∗ . In p articular, if a 2 = b 2 = 0 and 1 + ab ∈ R ∗ then one has a de c omp osition 1 + ab = (1 + b (1 − γ )) [1 − b, 1 + a ] (1 + (1 − γ ) a ) (1 + ba ) , wher e γ = ( 1 + ab ) − 1 . Pr o of. The first half of Lemma 6 follows from the fact that the equality (1 + ab ) c = 1 draws the equa lit y (1 + ba ) (1 − bca ) = 1. If a 2 = b 2 = 0 a nd 1 + ab ∈ R ∗ , then 1 + a , 1 − b - invertible ele ments and γ (1 + ab ) = (1 + ab ) γ = 1 , where γ = (1 + ab ) − 1 . Thus, γ ab + γ − 1 = abγ + γ − 1 = 0 . As a co r ollary we obtain bγ aba + bγ a − ba = bγ 2 a − bγ a + bγ abγ a = bγ ab + bγ − b = abγ a + γ a − a = 0 and also a (1 − γ ) = ( 1 − γ ) b = 0 and aγ a = bγ b = 0 . In this ca se (1 − a ) (1 + (1 − γ ) a ) = 1 − γ a − a (1 − γ ) a = 1 − γ a and (1 + b (1 − γ )) (1 − b ) = 1 − bγ − b (1 − γ ) b = 1 − bγ . 10 By direc t calculations we establish that (1 − bγ ) (1 + a ) (1 + b ) (1 − γ a ) = (1 + a − bγ − bγ a ) (1 + b − γ a − bγ a ) = = 1 + ab − bγ a and 1 + ab = (1 + ab − b γ a ) ( 1 + ba ). This means that the following equality holds 1 + ab = (1 − bγ ) (1 + a ) (1 + b ) (1 − γ a ) (1 + ba ) , from which the formula of Lemma 6 follows. If in Lemma 6 one puts a = xe ik and b = − y e lj , where x, y ∈ R, then in the c a se i 6 = k and l 6 = j, the equalities a 2 = b 2 = 0 and ab = − δ kl xy e ij , ba = − δ ij y xe lk are v alid. If i 6 = j , then ba = 0 and γ = (1 + ab ) − 1 = 1 − ab . Thu s, 1 + b (1 − γ ) = 1 + (1 − γ ) a = 1. T aking into ac c o unt that 1 + a = t ik ( x ), 1 − b = t lj ( y ), 1 + ab = t ij ( − δ kl xy ) we r e ceive a formula t ij ( − δ kl xy ) = [ t lj ( y ) , t ik ( x )] from which one obtains well-known commutator formulas t ij ( δ kl xy ) = [ t ik ( x ) , t lj ( y )] a nd t lk ( δ ij y x ) = [ t lj ( y ) , t ik ( x )] . Let g ∈ GL ( n, R ) a nd U = g e ii , V = e ij g − 1 , r , r 0 ∈ R . O bviously , V U = δ ij e ii . F urther we sha ll ass ume tha t i 6 = j . Hence, V U = 0. Let x 1 , . . . , x n be some elements of R and V 0 = x 1 e i 1 + · · · + x n e in , α = x 1 g 1 i + · · · + x n g ni . Supp ose tha t x l = 0 fo r some 1 ≤ l ≤ n a nd x k = r g − 1 j k for some 1 ≤ k ≤ n . Put W = rV − V 0 . Obviously , V 0 U = αe ii = − W U and W ik = 0. Under these notations, t ij ( r 0 r ) g = 1 + g e ii r 0 re ij g − 1 = 1 + U r 0 rV . It is esay to chec k tha t t ij ( r 0 r ) g (1 − U r 0 W ) = (1 + U r 0 rV ) (1 − U r 0 W ) = = 1 + U r 0 ( rV − W ) = 1 + U r 0 V 0 . Suppo se that 1 + U r 0 V 0 ∈ GL ( n, R ). Then 1 − U r 0 W ∈ GL ( n, R ) and t ij ( r 0 r ) g = (1 + U r 0 V 0 ) ( 1 − U r 0 W ) − 1 and, in par ticular, t ij ( r ) g = (1 + U V 0 ) (1 − U W ) − 1 . The r epresentation o f the matrix t ij ( r ) g th us obtained is useful beca us e the i th r ows of eac h of the ma trix V 0 and W contain at least one zero elemen t. In the long run, this will allow to use Lemma 6 and decomp ose the re sp ective commutators into the pro duct of transvections and diagonal elements. F or g ′ ∈ GL ( n, R ) we ana lo gously define U ′ = g ′ e ii , V ′ = e ij ( g ′ ) − 1 , V ′ 0 = x ′ 1 e i 1 + . . . + x ′ n e in , α ′ = x ′ 1 g ′ 1 i + . . . + x ′ n g ′ ni , where x ′ 1 , . . . , x ′ n ∈ R , x ′ l = 0 and x ′ k = r ( g ′ ) − 1 j k for corr esp onding 1 ≤ i, j, l , k ≤ n and r 0 , r ∈ R. Let W ′ = r V ′ − V ′ 0 and 1 + U ′ r 0 V ′ 0 ∈ GL ( n, R ) . Then 11 t ij ( r 0 r ) g ′ = (1 + U ′ r 0 V ′ 0 ) (1 − U ′ r 0 W ′ ) − 1 . One has Lemma 7 (main) L et I and J b e ide als of R, c ∈ ξ R , r ∈ R , 1 + V 0 c 2 U ∈ GL ( n, R ) , 1 + ( V 0 − x k e ik ) c 2 U ∈ GL ( n, R ) , 1 + V ′ 0 c 2 U ′ ∈ GL ( n, R ) , 1 + ( V ′ 0 − x ′ k e ik ) c 2 U ′ ∈ GL ( n, R ) , h ∈ C I , g ′ = g h − 1 , x s − x ′ s ∈ I , 1 ≤ s ≤ n (for s = l , k it is automatic al ly satisfie d due to x l = x ′ l = 0 , x k = r g − 1 j k , x ′ k = r  g ′ j k  − 1 ). Then 1)  h, t ij  c 2 r  g ′ ⊆ E ( n, cI ) T E ( n, J ) , if r ∈ J ; 2) g ∈ C  n, AnnRc 2 rR  , if g ∈ N . Pr o of. Assume that u s = g si , d s = 1 + αc 2 e ss , where 1 ≤ s ≤ n . Since V 0 U = αe ii and by the condition 1 + αc 2 e ii = 1 + V 0 c 2 U ∈ GL ( n, R ), then 1 + αc 2 ∈ R ∗ and d s = diag    1 , . . . , 1 + αc 2 | {z } s-th place , 1 , . . . , 1    ∈ GL ( n, R ) . In particula r, d i = 1 + V 0 c 2 U. Analogously one can define u ′ s = g ′ si , d ′ s = 1 + α ′ c 2 e ss , 1 + α ′ c 2 ∈ R ∗ , d ′ s ∈ GL ( n, R ), d ′ i = 1 + V ′ 0 c 2 U ′ . Let us consider the matrices a = ( U e il − u l e ll ) c and b = e li V 0 c. Obviously , a =             0 u 1 . . . u l − 1 0 0 0 0 0 u l +1 . . . u n 0             c and b =   0 0 0 x 1 · · · x l − 1 0 x l +1 · · · x n 0 0 0   c It is not hard to see that a Ra = bRb = e ll Ra = bR e ll = 0. F rom the definition of matrice s a and b it follows that a + u l e ll c = U e il c , e il b = V 0 c , e ll b = b , be ll = 0, b d l = b . Moreov er, ab is a matrix with zero l -th row and l -th c o lumn, ba = αc 2 e ll , 1 + ba = 1 + αc 2 e ll = d l ∈ GL ( n, R ). In acco rdance with Lemma 6, 1 + ab ∈ GL ( n, R ) and 1 + ab = (1 + b (1 − γ )) [1 − b, 1 + a ] (1 + (1 − γ ) a ) (1 + ba ), where γ = (1 + a b ) − 1 . 12 Obviously , 1 − γ is a matrix with zero l -th row a nd l - th c o lumn and matr ices 1 ± a , 1 ± b , 1 + d l u l cb , 1 + (1 − γ ) a , 1 + b (1 − γ ) are pro ducts of transvections. Consider an eq ua lity 1 + U c 2 V 0 = 1 + U ce il b = 1 + ( a + u l e ll c ) b = (1 + ab ) (1 + u l cb ) and the equality (1 + ba ) (1 + u l cb ) = d l (1 + u l cb ) = (1 + d l u l cb ) d l . Let us denote T l ( V 0 ) = (1 + b (1 − γ )) [1 − b, 1 + a ] (1 + (1 − γ ) a ) (1 + d l u l cb ) . Clearly , T l ( V 0 ) is a pro duct of tra nsvections contained in the g roup E cR and 1 + U c 2 V 0 = (1 + ab ) (1 + u l cb ) = = (1 + b (1 − γ )) [1 − b, 1 + a ] (1 + (1 − γ ) a ) (1 + ba ) (1 + u l cb ) = T l ( V 0 ) d l . It should b e noted that the cons tr uction of decomp osition o f matrices 1 + U c 2 V 0 int o a pro duct of transvections and diag onal elements is v alid for an a rbitrar y column U = ( u 1 . . . u n ) T and arbitra ry row V 0 = ( x 1 . . . x n ) for which x l = 0 and 1 + V 0 U c 2 ∈ R ∗ . Its view is deter mined b y the formula in Lemma 6. Thu s 1 − U c 2 W = T k ( − W ) d k ∈ GL ( n, R ) . Since t ij ( rc 2 ) g = (1 + U c 2 V 0 )(1 − U c 2 W ) − 1 , then t ij ( rc 2 ) g = T ( g ) d l d − 1 k , where T ( g ) = T l ( V 0 )( T k ( − W ) − 1 ) d l d − 1 k . By taking into account ana logous arguments, we hav e t ij  rc 2  g ′ = T ( g ′ ) d ′ l ( d ′ k ) − 1 , where T ( g ′ ) is exactly the same pro duct of transvections modulo the ideal cI as T ( g ). Ther efore,  h, t ij  rc 2  g ′ = t ij  rc 2  g t ij  − rc 2  g ′ = T ( g ) d l d − 1 k d ′ k ( d ′ l ) − 1 T ( g ′ ) − 1 . It is not hard to see that in the case x ∈ R ∗ one has the formula  x 0 0 x − 1  =  1 x 0 1   1 0 − x − 1 1   1 x 0 1   1 − 1 0 1   1 0 1 1   1 − 1 0 1  In particula r, if I is an idea l of the ring R and x ∈ 1 + cI , then the matrix  x 0 0 x − 1    1 − 1 0 1   = 13 =  1 x − 1 0 1   1 0 1 − x − 1 1   1 1 − x 0 0    1 x − 1 0 1  ,  1 0 − 1 1  is contained in E (2 , cI ) . If 1 + α ′ c ∈ R ∗ and α − α ′ ∈ I then (1 + αc ) (1 + α ′ c ) − 1 ∈ 1 + cI . Hence,  1 + αc 0 0 (1 + αc ) − 1   1 + α ′ c 0 0 (1 + α ′ c ) − 1  − 1 = =  (1 + αc ) (1 + α ′ c ) − 1 0 0 (1 + αc ) − 1 (1 + α ′ c )  ∈ E (2 , cI ) . This means tha t d l d − 1 k d ′ k ( d ′ l ) − 1 ∈ E  n, c 2 I  . T ak ing in to account that for arbitr ary transvections τ 1 and τ 2 from E cR such that τ 1 ≡ τ 2 mo d E cI , one has the inclusions τ 1 E ( n, cI ) τ − 1 2 ⊆ E ( n, cI ) , we obtain T ( g ) E ( n, cI ) T ( g ′ ) − 1 ⊆ E ( n, cI ) . Thu s, it is prov ed that  h, t ij  rc 2  g ′ ⊆ T ( g ) E ( n, cI ) T ( g ′ ) − 1 ⊆ E ( n, cI ) . Let J b e an idea l of R , r ∈ J . Then V 0 + W = r V is a matrix over J and x k ∈ J . Let us prov e that t ij ( r ) g ∈ E ( n, J ). As ab ove, t ij  rc 2  g =  1 + U c 2 V 0   1 − U c 2 W  − 1 = T ( g ) d l d − 1 k is a pro duct of transvections and dia g onal elemen ts, which, as is known, nor- malize the gr oup E ( n, I ) . Consider V ∗ 0 = V 0 − x k e ik . Since x l = 0, then ( V ∗ 0 ) il = ( V 0 ) il − ( x k e ik ) il = x l − δ kl x k = 0. It is clear that ( V ∗ 0 ) ik = W ik = 0 and V ∗ 0 + W is also a matrix ov er J . It is given that 1 + V ∗ 0 c 2 U ∈ GL ( n, R ) . Note that in the case 1 + J ⊆ R ∗ it automatica lly follows from the equality V 0 U = V ∗ 0 U + x k e ll U and inclusion 1 + V 0 c 2 U ∈ GL ( n, R ) that 1 + V ∗ 0 c 2 U ∈ GL ( n, R ). As was shown a b ove, ma trices 1 + U c 2 V 0 and 1 + U c 2 V ∗ 0 , 1 + U c 2 V ∗ 0 and 1 − U c 2 W can b e expanded into re sp ectively iden tical construc tio ns of pro ducts of transvections and diago nal elemen ts. By taking into account the fact that V 0 ≡ V ∗ 0 mo d J a nd V ∗ 0 ≡ − W mo d J w e o btain the cong r uences 1 + U c 2 V 0 ≡  1 + U c 2 V ∗ 0  mo d E ( n, J ) , 1 + U c 2 V ∗ 0 ≡  1 − U c 2 W  mo d E ( n, J ) . Thu s, we have pr ov ed the following: 14 t ij  rc 2  g ∈ E ( n, J ), if 1 + V 0 c 2 U ∈ GL ( n, R ) and 1 + ( V 0 − x k e ik ) c 2 U ∈ GL ( n, R ) . Similarly one ca n prov e tha t t ij  rc 2  g ′ ∈ E ( n, J ), if 1 + V ′ 0 c 2 U ′ ∈ GL ( n, R ) a nd 1 + ( V ′ 0 − x ′ k e ik ) c 2 U ′ ∈ GL ( n, R ) . In the end, it is proved that  h, t ij  rc 2  g ′ = t ij  c 2 r  g t ij  − c 2 r  g ′ ⊆ E ( n, J ) , if 1 + V 0 c 2 U, 1 + ( V 0 − x k e ik ) c 2 U, 1 + ( V ′ 0 − x ′ k e ik ) c 2 U ′ , 1 + V ′ 0 c 2 U ′ are contained in GL ( n, R ) . This finishes the pro of of 1 ). Let’s note tha t if in 1 ) the element g ′ normalizes the g roups E ( n, cI ) and E ( n, J ) , then  h, t ij  c 2 r  ⊆ E ( n, cI ) \ E ( n, J ). Let’s prov e 2). Let us introduce the following notations. A = 1 + U c 2 V 0 , B = (1 − U c 2 W ) − 1 , C = t ij  − c 2 r  . Under these nota tions  g , t ij  c 2 r  = t ij  c 2 r  g t ij  − c 2 r  = A ( B C ) and  t ij  − c 2 r  , g  = t ij  − c 2 r  t ij  c 2 r  g = ( C A ) B . Obviously , the l -th column of matrix U c 2 V 0 and the k - th column o f matrix U c 2 W are all zero . Suppo se that the eq ua lity k = j = l is no t satisfied. Then j 6 = k or j 6 = l . If j 6 = k , then the k -th column of matrix B C − E is all zero. Hence, the commutator  g , t ij  c 2 r  is a pro duct of tw o matrices A and BC , which s a tisfy the co nditions of Lemma 5. In suc h a case  g , t ij  c 2 r  ∈ ξ GL ( n, R ) and g ∈ C  n, AnnRc 2 rR  . Similarly , if j 6 = l then the commutator  t ij ( − c 2 r ) , g  is a pro duct of tw o matrices CA and B and the l -th column of the matrix C A − E is a ll zero. According to Lemma 5,  t ij  − c 2 r  , g  ∈ ξ GL ( n, R ) and g ∈ C  n, AnnRc 2 rR  . Thu s, the only case le ft to co nsider is k = j = l . Then 0 = x k = r  g − 1  j j . Since, acco rding to Lemma 3, the diago nal elements of matrix g ∈ N a re not zero divisors, then r = 0 , AnnRc 2 rR = Ann 0 = R , g ∈ GL ( n, R ) ≡ C ( n, R ) ≡ C  n, AnnRc 2 rR  . In the end 2) is proved. Let’s note that the inclusions of Lemma 6 lo o k like f ( c ) = 0, where f is a po lynomial with co efficients from the ring R n and c ∈ ξ R . In par ticular, if the 15 inclusions of Lemma 6 ho ld for all c ∈ ξ R then the elemen ts of ξ R are the r o ots of po lynomial f. Let V 0 = x 1 e i 1 + · · · + x n e in , where x l = 0, x k = r  g − 1  j k , 1 + V 0 U ∈ GL ( n, R ), U = g e ii , g ∈ GL ( n, R ). Elemen ts of the type V 0 are not well defined by matrix g . They for m a whole cla ss of matrices and elements r ∈ R act in them a s left co efficients of  g − 1  j k e ik . Denote b y R ( g ) the additive subgr oup o f the ring R , ge nerated b y all e le- men ts r ∈ R , which appea r as le ft co efficients of the s ummands  g − 1  j k e ik of matrices V 0 = x 1 e i 1 + · · · + x n e in , where x l = 0, x k = r  g − 1  j k , 1 + V 0 U ∈ GL ( n, R ) , 1 + ( V 0 − x k e ik ) U ∈ GL ( n, R ) for some fixed matrix g ∈ GL ( n, R ) and all p os sible matric e s V 0 and 1 ≤ k , l ≤ n . Definition 6 F or fixe d 1 ≤ i 6 = j ≤ n , the element g ∈ GL ( n, R ) is c al le d ( R, i, j ) -stable if R ( g ) = R . In particula r, if among V 0 = x 1 e i 1 + · · · + x n e in there are so me that x l = 0 , x k = r  g − 1  j k , V 0 U ∈ J ( R ) e ii , x k g ki ∈ J ( R ) , then R V 0 U ∈ J ( R ) e ii , 1 + RV 0 U ⊆ 1 + J ( R ) e ii ⊆ GL ( n, R ) , 1 + R ( V 0 − x k e ik ) U ∈ GL ( n, R ) and, henceforth, R ( g ) contains the left ideal Rr . If at the same time r ∈ R ∗ , then R ( g ) = R a nd g is a ( R, i, j )-stable element. Therefore, the matrix g ∈ GL ( n, R ), for whic h the inclus ion  g − 1  j k g ki ∈ J ( R ) holds, is ( R, i, j )-stable. In order to show this, it is enough to choo se V 0 ∈ R  g − 1  j k e ik and use the inclusion 1 + V 0 U ⊆ GL ( n, R ) . Definition 7 Element g ∈ GL ( n, R ) which is ( R, i, j ) -stable for al l 1 ≤ i 6 = j ≤ n such that E ( n, R ) = h t ij ( R ) i is c al le d R-stable. The example of R -stable elemen t is the matrix g with g ij = g j i = 0 for all 1 ≤ i 6 = j ≤ n , where j is fixed. As in this case g is ( R , i, j )-stable and ( R, j, i )-stable element. Since E ( n, R ) = h t ij ( R ) , t j i ( R ) | 1 ≤ i 6 = j ≤ n, j − fix e d i , then g is R - stable element. It is no t hard to see that ( R, i, j )-stability and, a s a conseq ue nc e , R -stability of matrices is preserved when facto ring the r ing R . Let I b e an arbitrar y ideal o f the ring R, a nd N is a subg roup of GL ( n, R ) inv a r iant with resp ect to E ( n, R ), which do es not co nt ain no n- identit y transvec- tions. Definition 8 Element g ∈ GL ( n, R ) , for whi ch fr om g ∈ C I it fol lows that [ g , E ( n, J )] ⊆ E ( n, I ) T E ( n, J ) , and fr om g ∈ N , fol lows that g ∈ ξ GL ( n, R ) is c al le d stable. As shall b e shown b elow, for ins tance, elements of the g roup C ( n, J ( R )) a re stable. 16 Definition 9 We shal l say that the class of invertible matric es L g , which c on- tains an identity, up t o tr ansve ctions, appr oximates the element g ∈ GL ( n, R ) , if for g ∈ C I ther e exists an element g I ∈ L g such that g I ∈ E ( n , I ) g E ( n,R ) E ( n, I ) , and for g ∈ N t her e exists an element g N ∈ L g such that g N ∈ [ g , t ij ( R ∗ )] E ( n,R ) for some 1 ≤ i 6 = j ≤ n . Lemma 8 F r om the st ability of elements L g one has t he stability of element g. Pr o of. If g ∈ C I , then ther e exists a stable elemen t g I ∈ E ( n, I ) g E ( n,R ) E ( n, I ) ⊆ C I . F rom stability of element g I it follows that the following inclus io ns hold [ g I , E ( n, J )] ⊆ E ( n, I ) T E ( n, J ) and [ g , E ( n, J )] ⊆ E ( n, I ) T E ( n, J ) for all ideals I, J of the ring R . Similarly , if g ∈ N , then ther e exis ts a stable element g N such that g N ∈ [ g, t ij ( R ∗ )] E ( n,R ) ⊆ N . F rom stability o f element g N it follows that g N ∈ ξ GL ( n, R ). Hence, ther e exists an element r ∈ R ∗ such that [ g , t ij ( r ) ] ∈ ξ GL ( n, R ). According to Corollar y 3 we hav e g ∈ C ( n, AnnRr R ) = C ( n, AnnR ) = ξ GL ( n, R ) . Theorem 1 L et R b e an asso ciative ring with identity. Element s of the gr oup GL ( n, R ) , which, up to tra nsve ctions, ar e appr oximate d by t he class of R- stable matric es, ar e stable. If, up to tr ansve ctions, al l elements of the gr oup GL ( n, R ) ar e appr oximate d by classes of R-stable m atric es, then R is a stable ring. Pr o of. According to Lemma 8 it is s ufficient to prov e that R -stable elements are in fact stable. Let g ∈ GL ( n, R ) b e an R -sta ble ele ment. Then g is a ( R , i, j )-stable element for all pairs 1 ≤ i 6 = j ≤ n such tha t E ( n , R ) = h t ij ( R ) i . Let’s fix one s uch pair 1 ≤ i 6 = j ≤ n . Then the additive group R ( g ) = h r k 1 + V 0 U ∈ GL ( n, R ) , 1 + ( V 0 − x k e ik ) U ∈ GL ( n, R ) , V 0 = x 1 e i 1 + · · · + x n e in , x l = 0 , x k = r  g − 1  j k i Z = R. According to this, 1 + α ∈ R ∗ and 1 + α − x k g ki ∈ R ∗ . Let I, J b e idea ls of R , g ∈ C I , r ∈ J , U = g e ii , V = e ij g − 1 , αe ii = V 0 U . Obviously x i − α ∈ I . W e s hall use Le mma 7. In the case when the equality l = k = j fails, we put c = 1 , g ′ = 1 , x ′ i = (1 − δ ik ) (1 − δ il ) α, where x ′ k = r δ j k , x ′ s = x s for all 1 ≤ s 6 = i , k ≤ n. Then x ′ l = 0 . T his is obvious if l 6 = i, k or l = i. If l = k , then x ′ l = x ′ k = r δ j l = 0 w hen l = k 6 = j. Moreover, x ′ t − x t ∈ I for a ll 1 ≤ t 6 = i ≤ n, x ′ i − x i = α − x i − δ il α ∈ I . Since U ′ = e ii , we ha ve V ′ 0 U ′ = x ′ i e ii , ( V ′ 0 − x ′ k e ik ) U ′ = V ′ 0 U ′ − r δ j k e ik e ii = V ′ 0 U ′ − r δ j k δ ik e ii = V ′ 0 U ′ , and 1 + ( V ′ 0 − x ′ k e ik ) U ′ = 1 + V ′ 0 U ′ = 1 + x ′ i e ii ∈ R ∗ , as x ′ i = 0 o r x ′ i = α. In view of Lemma 7, 17 [ g , t ij ( r ) ] ⊆ E ( n, I ) T E ( n, J ) and, as a co nsequence, [ g , t ij ( R ( g ))] ⊆ E ( n, I ) T E ( n, J ) . If l = k = j , then 0 = x l = x k = r  g − 1  j k = r  g − 1  j j and, as a conse- quence, r ∈ I . It is not hard to see tha t V 0 = r V s atisfies all conditions, defined by the a dditive gr oup R ( g ) . Recall, that in Lemma 6 a = U e il − U l e ll , b = e li V 0 = r e li V . T ak ing into ac count that V U = 0 a nd αe ii = V 0 U w e obta in the equalities α = 0 , ba = − r e li V u l e ll = − r e j j V u l e j j = − r  g − 1  j j u l e j j = 0 . Henceforth, γ = (1 + ab ) − 1 = 1 − ab , (1 − γ ) a = b (1 − γ ) = 0 , d l = d k = 1 , t ij ( r ) g = T ( g ) = [1 − b , 1 + a ] ( 1 + u l b ). If r ∈ I T J then the matrices 1 − b , 1 + u l b ar e co nt ained in the gr oup E ( n, I ) T E ( n, J ) . Thu s, in the case l = k = j, we hav e the inclusion [ g , t ij ( r ) ] ⊆ E ( n, I ) T E ( n, J ) . as well. Therefore, it is prov e n that in all cas es [ g , t ij ( J T R ( g ))] ⊆ E ( n, I ) T E ( n, J ). The ( R, i, j )-stability of element g implies [ g , t ij ( J )] ⊆ E ( n, I ) T E ( n, J ) and [ g , t ij ( R )] ⊆ E ( n, I ). Hence, the R -stability of elemen t g implies the inclusion [ g , E ( n, R )] ⊆ E ( n, I ) . T ak ing into ac count the fa ct that E ( n, J ) = D t ij ( J ) E ( n,R ) E and [ E ( n, I ) , E ( n, J )] ⊆ E ( n, I ) T E ( n, J ) we obtain the necessary inclusion [ g , E ( n, J )] ⊆ E ( n, I ) \ E ( n, J ). Let N b e a group, inv a riant with r esp ect to E ( n, R ), not containing non-ide ntit y transvections. If g ∈ N , in a ccordance with Le mma 7, we have g ∈ C ( n, AnnRr R ) for all generator s r of the additive group R ( g ) = R. Hence, g ∈ C ( n, AnnR ) = ξ GL ( n, R ). Thus, it is prov ed that R -stability of elements of the gro up GL ( n, R ) implies their stabilit y . If, up to tr ansvections, all elements of the group GL ( n, R ) are approximated by a cla ss of stable matrices, then 18 [ C ( n, I ) , E ( n, R ) , E ( n, J )] ⊆ E ( n, I ) T E ( n, J ) and N ∈ ξ GL ( n, R ). If one puts J = R , then it follows that R is a weakly-commutator ring. Mo re- ov er , R is a partially normal ring. Since R -sta bilit y is preserved w hen taking quotients, then all quotient ring s of R a re partially normal a s w ell. According to Lemma 2, R is a sta ble ring. Theorem 1 implies that c ommutative rings with identity ar e stable [25, 3 2, 10, 9]. Indeed, let R b e a co mmutative ring with identit y , g ∈ GL ( n, R ). F or arbitrar y 1 ≤ i 6 = j ≤ n define V 0 =  g − 1  j k ( g ti e ik − g ki e it ) = g ti  g − 1  j k e ik − g ni  g − 1  j k e it , where 1 ≤ k 6 = t ≤ n . Then V 0 U = 0 and R g ti ⊆ R ( g ). By in terchanging k and t w e hav e R ( g ) = R . T his means that all elements of the g roup GL ( n, R ) ar e ( R, i, j )-stable and, as a consequence , R -stable. Ac- cording to Theo rem 1, R is a stable ring . Theorem 2 L et R b e an asso ciative ring with identity, g ∈ GL ( n, R ) and at le ast one element of the matrix g b elongs to t he r adic al J ( R ) . Then g is a stable element. Pr o of. Let g = ( g ij ) ∈ GL ( n, R ) and g ij ∈ J ( R ) . W e introduce the notation g 1 = e 1 eg e 2 , where e = t i 1   g − 1  j 1  · · · t in   g − 1  j n  , α = −  1 + g ij − ( g − 1 ) j i g ij  − 1 , e 1 = t 1 i  g 1 j α  · · · t ni  g nj α  , e 2 = t j 1  α  1 −  g − 1  j i  g i 1  · · · t j n  α  1 −  g − 1  j i  g in  . It is no t hard to see that ( g 1 ) il = ( g 1 ) sj = 0 fo r all 1 ≤ l 6 = j ≤ n , 1 ≤ s 6 = i ≤ n . Let I b e a n ideal o f R , g ∈ C I . If i = j then e, e 1 , e 2 are contained in E I , g 1 ∈ C I and g 1 satisfies the condition of R -stability . If i 6 = j , then there exists e 0 = t j i (1) t ij ( − 1) t j i (1) s uch that e 0 e 1 ee 2 ∈ E ( n, I ), e 0 g 1 ∈ C I and satisfies the condition of R - stability . Thu s, it is proved that g ∈ C I and is approximated, up to transvections, by R -stable matrices. Therefore, g satisfies the c o ndition of R - stability . If g ∈ N then 1+ g − 1 ki g ij ∈ R ∗ and, acco rding to Lemma 7, g ∈ C ( n, AnnR ) = ξ GL ( n, R ). Th us, it is proved that g is a stable element. In particula r the following inclusions hold h C ( n, J ( R )) \ C ( n, I ) , E ( n, J ) i ⊆ E ( n, I ) \ E ( n, J ) , N \ C ( n, J ( R )) ⊆ ξ GL ( n, R ) . If we put I = R , then [ C ( n, J ( R )) , E ( n, J )] ⊆ E ( n, J ). 19 Lemma 9 L et R b e an asso ciative ring with identity, J ( R ) - r adic al, R / J ( R ) - p artial ly normal ring. The n R is a p artial ly normal ring. Pr o of. Let N be a subgr oup of the gr oup GL ( n, R ), inv ariant with re- sp ect to E ( n, R ) and not co ntaining non- ident ity transvections. If Λ J ( R ) N contains non-identit y tra nsvection Λ J ( R ) t ij ( r ), r / ∈ J ( R ), then t ij ( r ) ∈ hC J ( R ) where h ∈ N . Thus, h ∈ t ij ( r ) C J ( R ) and at least one element of the ma- trix h is con tained in J ( R ). According to Theorem 2, h ∈ ξ GL ( n, R ) and r ∈ J ( R ) , contradicting the assumption. Therefore, the gr oup Λ J ( R ) N do es not contain non-identit y tra ns vections. Since R / J ( R ) is a partially norma l ring , then Λ J ( R ) N ∈ ξ GL  n, R / J ( R )  . This means tha t N ⊆ C ( n, J ( R )). By Theorem 1, N ⊆ ξ GL ( n, R ) . Thus, it is pr ov ed that R is a partia lly normal ring. Lemma 10 L et R b e an asso ciative ring with identity, J ( R ) - ra dic al, R / J ( R ) - normal ring. Then al l t he quotient rings of the ring R ar e p artial ly normal rings. Pr o of. L e t R b e s o me quotien t ring of the ring R . Since under the epimor- phism of rings, preimag es of maximal one-sided idea ls are maximal o ne-sided ideals, then J ( R ) ⊆ J  R  . Hence, the e pimorhism of rings R → R → R / J  R  induces the epimorhism R / J ( R ) → R / J  R  and, as a consequence, R  J  R  is a quotient ring of the normal ring R / J ( R ) . Since all quotients of normal ring are partially norma l rings, then R  J  R  is a par tia lly normal ring. In vie w of Lemma 9, R is a partially nor mal ring. F ro m Theorem 1 o ne has Corollary 5 [22] L et R b e an asso ciative ring with ide ntity, J ( R ) - r adic al, R / J ( R ) - a stable ring. Then R is a st able ring. Pr o of. Since R / J ( R ) is a sta ble ring, then R / J ( R ) is commutator and normal ring. F ro m commutatorness of the ring R / J ( R ) it follows that for arbitrar y ideals I and J o f the r ing R one has the inclusions [ C ( n, I ) , E ( n, J )] ⊆  E ( n, I ) \ E ( n, J )  C I T J ( R ) . According to Theo rem 2  C I T J ( R ) , E ( n, J )  ⊆ E ( n, I ) \ E ( n, J ) . Therefore, [ C ( n, I ) , E ( n, J ) , E ( n, J )] ⊆ h E ( n, I ) \ E ( n, J )  C I T J ( R ) , E ( n, J ) i ⊆ ⊆ E ( n, I ) \ E ( n, J ) . 20 In particula r, if J = R , then [ C ( n, I ) , E ( n, R ) , E ( n, R )] ⊆ E ( n, R ) . This means tha t R is a w eakly-co mm utator ring. Since R / J ( R ) is a normal r ing , then, in view of Lemma 9, all the quotient rings of the ring R are pa rtially normal. According to Lemma 2, w eakly-co mmu tator r ings a ll q uotients of which are partially normal, a re stable. Definition 10 V e ct or ( r 1 , . . . , r n ) is c al le d unimo dular in R n , if ther e ar e ele- ments t 1 , . . . , t n in the ring R such that t 1 r 1 + · · · + t n r n = 1 . Obviously , in the c a se n = 1 unimodula r vectors ar e exa ctly the left in vertible elements of the ring R . F ro m the inclusio n 1 + J ( R ) ⊆ R ∗ one has tha t vectors, unimo dular modulo the radica l J ( R ), a re in fact unimo dula r . Definition 11 L et n ≥ 2 . Th e asso ciative ring R is said to satisfy the c ondition of s tability of r ank n − 1 , if for an arbitr ary unimo dular ve ctor ( r 1 , . . . , r n ) ther e ar e elements s 2 , . . . , s n in R such that the ve ctor ( r 2 + s 2 r 1 , . . . , r n + s n r 1 ) is unimo dular in R n − 1 . It is no t hard to s e e that the direct sum of rings, which sa tisfy the condition of stability o f rank n − 1, satisfies the condition of stability of rank n − 1 a s well. Analogously , rings which mo dulo the radical satisfy the condition of s tability of rank n − 1 , satisfy the co ndition of stability of rank n − 1. It is known [2 7], that if R satisfies the condition of s tability of rank m − 1 , then R satisfie s the condition of stabilit y of r ank n − 1 fo r all n ≥ m . Let us consider a sso ciative rings R with identit y such that for an arbitra ry element e ∈ R ther e exist elements r 1 , r 2 ∈ R ∗ such that r 1 er 2 is an idemp otent of the ring R . In particula r, matrix r ings over skew fields satisfy the a forementioned con- dition. Indeed, by means of elementary transforma tions an arbitr ary matrix in the ring o f matrices ov er skew field ca n b e br ought to the idempo tent diag (1 , . . . , 1 , 0 , . . . , 0 ). Let us now prov e that the co nsidered ring s satis fy the condition of stability of ra nk 1. Let ( r, e ) b e a unimo dular element. Find an element s ∈ R such that e + sr ∈ R ∗ . At first we co nsider the c a se when e is an idempo tent of R . Let 1 = αr + β e , wher e α and β a r e so me elemen ts of the ring R , e 2 = e is an idempo ten t of the r ing R . Since ( 1 − e ) β e is a nilp otent element then by putting s = (1 − e ) α we obtain e + sr = e + (1 − e ) αr = e + (1 − e ) (1 − β e ) = 1 − (1 − e ) β e ∈ R ∗ . If e is an ar bitrary element of the r ing R such that ( r , e ) is unimodula r and ele- men t r 1 er 2 is an idempo tent of the ring R , wher e r 1 , r 2 ∈ R ∗ , then ( r 1 rr 2 , r 1 er 2 ) is unimo dular and, ac c ording to the a rgument ab ove, there exists an element 21 s 1 ∈ R suc h that r 1 er 2 + s 1 r 1 rr 2 ∈ R ∗ . By taking s = r − 1 1 s 1 r 1 we obtain e + sr ∈ R ∗ . In the end, it is pr ov e n that the as so ciative rings with identit y which mo dulo the radical are the dir ect sum o f r ings with the prop er t y that their elements, m ultiplied b y in vertible elements of the ring, can b e turned into idemp otents, satisfy the co ndition of stability of rank 1. Definition 12 A sso ciative ring R with identity is c al le d semilo c al if R / J ( R ) is a dir e ct sum of ful l matrix rings over skew fields. Semilo cal ring is a classic ex a mple of the r ing which s a tisfies the condition of stability of rank 1. Asso ciative rings with identity which satisfy the c ondition of stability of r ank n − 1 > 1 ar e st able [5]. Indeed, let g ∈ GL ( n, R ). V ector ( g 1 n , g 2 n , . . . , g nn ) is obviously unimo du- lar. Ther efore, there a re elemen ts k 2 , . . . , k n such that ( g 2 n + k 2 g 1 n , . . . , g nn + k n g 1 n ) is also a unimo dular vector. Hence, there are elements s 2 , . . . , s n in g 1 n R s uch that g 1 n + s 2 ( g 2 n + k 2 g 1 n ) + · · · + s n ( g nn + k n g 1 n ) = 0 . Let e 1 = t 21 ( k 2 ) · · · t n 1 ( k n ), e 2 = t 12 ( s 2 ) · · · t 1 n ( s n ), g 1 = e 2 g e 1 , g 2 = [ g e 1 , t n 2 (1)] e 2 = t n 2 (1) g 1 t n 2 ( − 1) e 2 . Then ( g 1 ) 1 n = ( g 2 ) 1 n = 0 . By Theorem 2, elemen ts g 1 and g 2 are stable. Obviously , e 2 ∈ E I if g ∈ C I and g 2 ∈ N if g ∈ N . Since elemen ts g 1 and g 2 , up to trans vections, approximate the matrix g , then, according to Lemma 8, g is a stable element. Thus, it is prov ed that R is a stable ring . W e shall need the useful Lemma 11 L et R b e an asso ciative ring with identity, I, J - ide als of the ring R, N - sub gr oup of GL ( n, R ) inva riant wi th r esp e ct to E ( n , R ) and not c on- taining non-identity tr ansve ctions, g ∈ GL ( n, R ) and ther e exists an element e ∈ R s uch that e 2 − e ∈ J ( R ) , g j k ∈ eR , e ∈ g j k R for some 1 ≤ k , j ≤ n. Then  g − 1 , t ij ( J )  ⊆ E ( n, I ) T E ( n, J ) if g ∈ C I , 1 ≤ i 6 = k, j ≤ n and g ∈ ξ GL ( n, R ) if g ∈ N and j = k . Pr o of. Let e = g j k r , g e = t ki ( rg j i ) g − 1 , where r ∈ R , 1 ≤ i 6 = k , j ≤ n . It is not har d to see that g − 1 e = g t ki ( − r g j i ) and  g − 1 e  j k = g j k ∈ e R,  g − 1 e  j i = g j i − g j k rg j i = (1 − g j k r ) g j i ∈ (1 − e ) R . Since (1 − e )  g − 1 e  j k ( g e ) ki ∈ J ( R ) and e  g − 1 e  j i ( g e ) ii ∈ J ( R ) , then R (1 − e ) ⊆ R ( g e ) and Re ⊆ R ( g e ) . Therefore, R ⊆ Re + R (1 − e ) ⊆ R ( g e ) ⊆ R . Hence, R ( g e ) = R a nd g e is an ( R, i, j )-stable element for all 1 ≤ i 6 = k , j ≤ n . If g ∈ C I , then g j i ∈ I , g e ∈ C I , [ g e , t ij ( J )] ⊆ E ( n, I ) T E ( n, J ) and, as a consequence, 22  g − 1 , t ij ( J )  ⊆ E ( n, I ) T E ( n, J ) . If g ∈ N and j = k , then as was shown in Lemma 9, Λ J ( R ) N does not contain non-identit y transvections. In accordance with Lemma 3, diagona l elements of matrices Λ J ( R ) N ar e zero divisors free. Hence, 1 − e ∈ J ( R ) , e ∈ R ∗ and g j j ∈ R ∗ . According to Corollar y 4, g ∈ ξ GL ( n, R ) . Let’s note that if Lemma 11 holds for a ll 1 ≤ i 6 = k , j ≤ n , then  g − 1 , E J  ⊆ E ( n, I ) T E ( n, J ) and, as a consequence, when J = R  g − 1 , E ( n, R )  ⊆ E ( n, I ) \ E ( n, J ) . Thu s,  g − 1 , E ( n, J )  ⊆ E ( n, I ) \ E ( n, J ) , if g ∈ C I and g ∈ ξ GL ( n, R ) , if g ∈ N . This means tha t g is a sta ble element. Definition 13 A sso ciative ring R with identity is c al le d von Neumann r e gu lar if for an arbitr ary element a ∈ R ther e exists an element a ′ ∈ R such that aa ′ a = a . It turns out [19, 30] that von Neumann r e gular rings ar e stable . Indeed, let R b e a von Neumann regular ring and g ∈ GL ( n, R ) . Let a = g j k for ar bitrary 1 ≤ k , j ≤ n . Then ther e exists an elemen t a ′ ∈ R suc h that aa ′ a = a . Let e = aa ′ . Obviously , ea = a, e 2 = eaa ′ = e, g j k = a ∈ eR and e ∈ g j k R . According to Lemma 11,  g − 1 , t ij ( J )  ⊆ E ( n, I ) T E ( n, J ) if g ∈ C I and g ∈ ξ GL ( n, R ) if g ∈ N , where I , J - idea ls of the ring R and N is a s ubgroup of GL ( n, R ) inv ariant with resp ect to E ( n, R ) a nd do es not contain non-identit y trans vections. Thu s, g is a stable element and, as a consequence, R is a stable ring . Let us present an author ’s Definition 14 A sso ciative ring R with identity is c al le d ne arly lo c al if for an arbitr ary a ∈ R ther e ex ists an element a ′ ∈ R su ch that (1 + a ′ a ) (1 − a ′ + aa ′ ) = 0 . Obviously , lo cal rings with ident ity and their direct and Cartesian pro ducts are nearly lo ca l rings. Theorem 3 Ne arly lo c al rings ar e stable. Pr o of. L e t g ∈ GL ( n, R ), r ∈ R , a =  g − 1  ii g ii where 1 ≤ i ≤ n. Define g 0 = t 1 i  g 1 i r  g − 1  ii  · · · t ni  g ni r  g − 1  ii  , g 1 = g − 1 g − 1 0 , g 2 =  g − 1  g 0 . It is no t hard to see that ( g 1 ) ii = (1 − r + ar )  g − 1  ii = ( g 2 ) ii . 23 Let a ′ be an element of R, for whic h (1 + a ′ a ) (1 − a ′ + aa ′ ) = 0 . Define e = (1 − a ′ + aa ′ ) a . Then 1 − e = (1 − a ) (1 + a ′ a ). O bviously , (1 + a ′ a ) e = 0 , (1 − e ) (1 − a ′ + aa ′ ) = 0 and (1 − e ) e = 0 . This means tha t e 2 = e a nd (1 − e ) 2 = 1 − e . Therefore, eR = { t ∈ R | (1 − e ) t = 0 } and (1 − e ) R = { t ∈ R | et = 0 } . Hence, 1 − a ′ + aa ′ ∈ eR . Suppose that in the definition of elements g 0 , g 1 and g 2 element r = a ′ . Then ( g 1 ) ii = ( g 2 ) ii = (1 − a ′ + aa ′ )  g − 1  ii ∈ eR . If a =  g − 1  ii g ii , then e = (1 − a ′ + aa ′ ) a = (1 − a ′ + aa ′ )  g − 1  ii g ii = ( g 1 ) ii g ii = ( g 2 ) ii g ii . Thu s, Lemma 11 can be applied to ele ment s g 1 and g 2 . In particula r, if g ∈ C I , then g 0 ∈ C I , g 1 ∈ C I ,  g − 1 1 , E ( n, J )  ⊆ E ( n, I ) T E ( n, J ) and [ g , E ( n, J )] ⊆ E ( n, I ) T E ( n, J ) , where I, J – ideals of R . If g ∈ N , then g 2 ∈ N and g 2 ∈ ξ GL ( n, R ), g ∈ ξ GL ( n, R ), where N is a subgroup of GL ( n, R ) in v aria nt with r esp ect to E ( n, R ) and do es not contain non-identit y tra nsvections. Thu s, it is proved that a ll elements of the gr oup GL ( n, R ) are stable. Ac- cording to Theo rem 1, R is a stable ring . Similarly , one can pr ove that the asso ciative rings with identity, which ar e algebr aic over the field (even A rtinian subrings) of own c enters [1 9], ar e stable . As in this case for an arbitrar y element a ∈ R a nd an Ar tinian subring K ⊆ ξ R the chain of ideals ( a ) ⊇  a 2  ⊇  a 3  ⊇ · · · of an Ar tinian commutativ e ring K [ a ] is stabilized as well. Hence, ther e exists a po sitive in teger m a nd an element a ′ ∈ R , which commutes with a , such that a m = a m +1 a ′ . Then e = a m ( a ′ ) m = a m +1 ( a ′ ) m +1 = · · · = a 2 m ( a ′ ) 2 m = e 2 and a m = a m +1 a ′ = · · · = a 2 m ( a ′ ) m = ea m . In this ca se 1 − r + ar = a m . Supp ose r = 1 + a + · · · + a m − 1 . If g 1 = g − 1 g − 1 0 , g 2 = ( g − 1 ) g 0 , where g 0 = Q l t li  g li r  g − 1  ii  , then ( g 1 ) ii = ( g 2 ) ii = (1 − r + ar )  g − 1  ii = a m  g − 1  ii ⊆ eR . If a =  g − 1  ii g ii , then e = a m ( a ′ ) m = a m +1 ( a ′ ) m +1 = a m a ( a ′ ) m +1 = a m  g − 1  ii g ii ( a ′ ) m +1 ⊆ ⊆ ( g 1 ) ii R = ( g 2 ) ii R. 24 Thu s, Lemma 11 can be a pplied to the elements g 1 and g 2 . Just as in Theo rem 3 we obta in g – stable element. Let S be a m ultiplicatively clos ed subset, with identit y , of the cent er ξ R of the ring R which do es not co nt ain 0, R S – the classical ring of fra ctions of the ring R by S . The natura l homomorphism Λ : R → R S , defined by the rule Λ : r → r 1 induces a ho momorphism Λ : GL ( n, R ) → GL ( n, R S ). Lemma 12 L et R b e an asso ciative ring with identity, N – a sub gr oup, invariant under E ( n, R ) and not c ontaining non-identity tr ansve ctions. Then Λ N do es n ot c ontain non-identity tr ansve ctions. Pr o of. If Λ N contains non-identit y tr ansvection τ , then there exists a transvection t ∈ E S such that for some r ∈ R, rS 6 = 0 the following inclus io n holds Λ t ij ( r ) = [ τ , Λ t ] ∈ Λ N . This means that t ij ( r ) h ∈ N for some h ∈ ker Λ. In such a case there exists s ∈ S , s uch that ( h − 1) s = 0 and s annihilates some non-dia gonal element of the matrix t ij ( r ) h . Acco rding to Lemma 3 , element s annihilates all non- diagonal ele ments of the ma trix t ij ( r ) h . Th us, t ij ( r ) s = t ij ( r ) hs is a diagona l matrix and rs = 0. The contradiction thus obtained shows that Λ N do es not contain non- identit y transvections. Lemma 13 L et R b e an asso ciative ring with identity, I – ide al of R, N - a su b gr oup, invaria nt with r esp e ct to E ( n, R ) and not c ontaining non-identity tr ansve ctions. If Λ ( g ) is a stable element of the gr oup GL ( n, R S ) , then ther e exists an element s ∈ S such that for an arbitr ary e ∈ E ( n, R ) [ g , e, E sR ] ⊆ E ( n, I ) if g ∈ C I and g ∈ C ( n, Anns ) if g ∈ N . Pr o of. L e t g ∈ C I . Then Λ ( g ) ⊆ C I S , [Λ ( g ) , E ( n, R S )] ⊆ E ( n, I S ) and there exists a n element s 0 ∈ S suc h that [ g , e, t ij ( cs 0 r ) ] ∈ E ( n , c I ) ker Λ for arbitra ry e ∈ E ( n, R ), r ∈ R , c ∈ ξ R , 1 ≤ i 6 = j ≤ n . The inc lus ion thus obtained ho lds if the r ing R s hould b e int erchanged with the ring R [ x, y ], in which the v a riables x and y commute, x commutes with the elements of R , a nd y - with the elements of ξ R . Hence, it ca n b e viewed as a p oly no mial in ter ms of v a riable x with the co efficients from the ring R n [ y ], which are annihilated b y s o me element s 1 ∈ S . Let s ij = s o s 1 . Then [ g , e, t ij ( s ij y )] ∈ E ( n, I [ y ]). Th us, [ g , e, t ij ( s ij R )] ⊆ E ( n, I ) and [ g , e , E sR ] ⊆ E ( n, I ) , 25 where s = T s ij for all pair s 1 ≤ i 6 = j ≤ n . Let g ∈ N . In view of Lemma 12, the group Λ N doe s not contain no n- ident ity transvections. Since Λ ( g ) is a stable element, then Λ ( g ) ∈ ξ GL ( n, R S ). Therefore, there exists s ∈ S that a nnihilates non- dia gonal element o f matr ix g . In acco rdance w ith Lemma 3, g ∈ C ( n, Anns ) . Corollary 6 L et R b e an asso ciative ring with identity, R S – stable rings for al l maximal ide als J 0 of the subring K ⊂ ξ R , 1 ∈ K , S = K \ J 0 . Then R is a stable ring. Pr o of. Let e 0 , e b e arbitrar y elemen ts of the group E ( n, R ) , G – subgroup of the group GL ( n, R ) inv ariant with res p e c t to E ( n, R ) a nd I 0 – largest ideal of the ring R such that E ( n, I 0 ) ⊂ G . Let J ( I ) = { s ∈ K | [ g , e 0 , e, E sR ] ⊆ E ( n, I ) if g ∈ C ( n, I ) } and J ( G ) = { s ∈ K | Λ 0 ( g ) ∈ C ( n, Ann Λ I 0 ( s )) if g ∈ G } . It is understa nda ble that J ( I ) and J ( G ) – ideals of the ring K . If J ( I ) 6 = K , then ther e exis ts a maxima l ideal J 0 ( I ) of the ring K such that J ( I ) ⊆ J 0 ( I ) , S = K \ J 0 ( I ) . Similarly one ca n define S = K \ J 0 ( G ), if J ( G ) 6 = K , where J ( G ) ⊆ J 0 ( G ). Let g ∈ C ( n, I ). Since R S is a commutator ring, then Λ I 0 [ g , e 0 ] ⊆ E ( n, I S ). According to Lemma 1 3, there exists an elemen t in S , which is contained in J ( I ). The co ntradiction thus o btained shows that J ( I ) = K , 1 ∈ J ( I ), R – weakly-comm utator ring. Let g ∈ G , R = R / I 0 . As in Lemma 1 2 we pr ov e that Λ I 0 ( G ) do es not c ontain non-identit y transvections. Since I 0 T K ⊆ J ( G ), then S do es not contain ze ro element. As a quotient of the nor mal ring R S , the r ing R S is par tially normal. If Λ : R → R S , then the group ΛΛ I 0 ( G ), according to Lemma 12, do es not contain non- identit y tr ansvections. In view of the par tial normality of the ring R S and Lemma 1 3 there exis ts an ele ment in S , which is contained in J ( G ). Therefore, J ( G ) = K , 1 ∈ J ( G ), Λ I 0 ( g ) ∈ ξ GL ( n, R ), g ∈ C ( n, I 0 ). This means that R is a norma l a nd, as a consequence, stable r ing. In the par ticular case, w he n R S - r ings which satisfy the c ondition of stability of r ank n − 1 > 1 for al l maximal ide als J of t he subring K ⊂ ξ R under t he c ondition 1 ∈ K , S = K \ J the stability of t he ring R is pr ove d in [29]. Corollary 7 [22] L et R b e an asso ciative ring with identity, which is inte ger- algebr aic over the subring K ⊂ ξ R , 1 ∈ K . Then R is a st able ring. Pr o of. Let I 0 be a maximal ideal of the ring K , S = K \ I 0 , r ∈ R. Then K S ( r ) is a finitely-gene r ated mo dule ov er K S . Accor ding to Nak ay ama’s Lemma J ( K S ) ⊆ J ( K S ( r ) ) and, as a consequence, J ( K S ) ⊆ J ( R S ). Therefor e, R S / J ( R S ) - is a ring, algebr aic over the field K S / J ( K S ) . In such a case, as 26 was mentioned a b ove, the r ing R S / J ( R S ) is stable. In view of Coro lla ry 5, R S is a stable ring and, accor ding to the coro llary 6, R is a stable ring . As is known [8] not every asso cia tive r ing with identit y is stable. F or in- stance, algebr a over field with 2 n 2 gener ating elements x ij , y ij , 1 ≤ i, j ≤ n and the defining r elations ( x ij ) ( y ij ) = ( y ij ) ( x ij ) = 1 is not a s t able ring . How ever, the cla ss of stable r ings is quite wide. The mo st vividly it was demonstrated in the work [13]. Definition 15 L et R b e an asso ciative ring with identity. Ide al F of the ring R is c al le d we akly No etherian (r esp e ctively inte ger-algebr aic) if for arbitr ary el- ements y , z ∈ F , m ≥ 1 left and right mo dules P m Rz y m and P m y m z R (re sp e ctively P m ξ R z y m and P m ξ R y m z ) ar e finitely gener ate d as mo dules over R (over ξ R r esp e ctively). Definition 16 A sso ciative ring R with identity is c al le d we akly No etherian (re - sp e ctively int e ger-algebr aic) if ther e ex ists a chain of ide als 0 = I 0 ⊆ I 1 ⊆ . . . ⊆ I q +1 = R such that t he ide als I i +1 / I i in the rings R / I i ar e we akly No etherian (inte ger- algebr aic r esp e ctively) for al l 1 ≤ i ≤ q . Obviously , the blo ck integer-algebra ic rings a r e w eakly No ether ia n. It is known [3, 2, 10] that PI – ring s that are blo ck integer-algebra ic. O b- viously , rings that are algebr aic ov er subrings of own c e nt ers ar e blo ck int eger- algebraic . Let g ∈ GL ( n, R ) and l - maximal in teger such that I l g 1 n = 0. If l < q + 1 , then w e c ho ose g 1 ∈ [ g , t n 1 ( I l +1 )], y = ( g 1 ) 11 , z = ( g 1 ) 1 n . Then y − 1 and z are contained in I l +1 T g 1 n R a nd I l ( y − 1) = 0 . Therefore, there exists a p ositive integer m suc h that z ( y − 1) m − m − 1 P p =1 s p z ( y − 1) p ∈ I l , z y m +1 = m P p =0 r p z y p , r p , s p ∈ R . Let N b e a s ubg roup of GL ( n, R ), inv a riant with re s pe ct to E ( n, R ) and not containing non-ide ntit y transvections. If g ∈ N , then g 1 ∈ N a nd, according to Lemma 3, y is not a divisor of zero . According to Lemma 4, from the equality r y + r 0 z = 0, where r - some element of the ring R , it follows that r 0 z = r = 0 . Similarly one can prov e 0 = r 0 z = · · · = r m z = z . Hence, [ g , t n 1 ( I l +1 )] ⊆ ξ GL ( n, R ) . In view of Lemma 3, I l +1 g 1 n = 0. The contradiction th us obtained shows that l = q + 1 , g 1 n = 0, R - partially nor mal ring. Therefore, we akly No etherian rings ar e p artial ly normal . Since the pr o p- erty of b eing weakly No e therian is pres erved under factoriza tion, then al l the quotients of we akly No et herian rings ar e p artial ly normal as wel l . 27 Let I be a n ideal of R , which is cont ained in some weakly No e ther ian idea l of the r ing R , y - arbitr ary element of I . Then z y ∈ I a nd there exis ts a p ositive int eger m s uch that z y m +1 = P p r p z y p , where z , r p ∈ R , 1 ≤ p ≤ m. Let λ ∈ ξ R. Multiply the equality a b ove by λ m +1 . Since λy = λy − 1 + 1 , then there exists a p olynomial ψ ( λ ) such that the following e q uality holds ψ ( λ ) z + a (1 − λy ) = 0 , where ψ (0) = 1 . Let g ∈ C ( n, I ), g λ = t pq ( λr ) g , c λ = [ t pq ( − λr ) , g ], y = 1 − ( g 1 ) ii , z =  g − 1 1  j k ( g 1 ) ki , where i, j, k - pair wise distinct num ber s, r ∈ R . Then g λ = λg 1 − λ + 1 , c λ ∈ C I , g λ = t pq ( λr ) c λ . Since ( g λ ) ii = λ ( g 1 ) ii − λ + 1 = 1 − λy , ( g λ ) ki = λ ( g 1 ) ki , ( g − 1 λ ) j k = ( g − λ ) j k = − λ ( g 1 ) j k = λ ( g − 1 1 ) j k and t pq ( − λr ) j k t pq ( λr ) ki = 0 , then ψ ( λ )  g − 1 λ  j k ( g λ ) ki + λ 2 a ( g λ ) ii = 0 , a ∈ I . In acco rdance w ith Lemma 7, [ c λ , t ij ( J ψ ij ( λ ))] ⊂ E ( n, I ) T E ( n, J ) , where J is a n arbitrar y deal of R. F ollowing similar “right-sided” a r guments and taking into a c c ount the ma- trix commutator for mulas for each pair there exists a p olynomia l such that [ c λ , t ij ( J ψ ij ( λ ) J )] ⊂ E ( n, I ) T E ( n, J ), where ψ ij (0) = 1 . Let f ( λ ) = Q i,j ψ ij ( λ ) for a ll pairs 1 ≤ i 6 = j ≤ n. T hen  c λ , E J f ( λ ) J  ⊆ E ( n, I ) \ E ( n, J ) . Define I 1 = R f ( λ ) R , I 2 = I 2 1 f (1 − λ ) I 2 1 . Obviously I 1 , I 2 are ideals of the ring R and [ c λ , E I 1 ] ⊆ E ( n, I ) , [ c 1 − λ , E I 2 ] ⊆ E ( n, I ) . Since E I 2 ⊆ E ( n, I 2 ) ⊆ E  n, I 2 1  ⊆ E I 1 , then for an arbitrar y elemen t e ∈ E ( n, R ) the following inclusions hold [ c e λ , E I 1 ] ⊆ [ c λ , E ( n, I 2 )] e ⊆ E ( n, I ) and [ c λ , E I 1 ] e ⊆ E  n, I 2 I  ⊆ E I 1 . T ak ing in to acc ount that c 1 = c t pq ( λ − 1) r λ c 1 − λ we obtain [ c 1 , E I 2 ] ⊆ E ( n, I ). Let us put I 0 = P λ I 2 for all λ ∈ ξ R . If I 0 6 = R then, due to the eq ua lity f (0) = 1, the image of the poly nomial 28 f ( λ ) 2 f (1 − λ ) f ( λ ) 2 is non-zero over the ring R / I 0 , and imag e s of the elements of ξ R a re its ro o ts. If ξ R contains a n infinite field, then I 0 = R , [ c 1 , E ( n, R )] ⊆ E ( n, I ), R - weakly-comm utator ring. In a c cordance with L e mma 2, R is a stable r ing. Thus, the we akly No et herian rings, which c ontain infinite fields in own c ent ers, ar e stable [13]. In the particula r cas e, the blo ck inte ger-algebr aic rings ar e stable without the demand of existence of infinite fields in own ce nt ers. Indeed, if elemen ts of the idea l I of the ring R are integer-algebr aic over the subring K ⊆ ξ R , 1 ∈ K , r ∈ I , I 0 is a maximal ide a l of K , S = K \ I 0 , then K S ( r ) is a finitely generated mo dule over K S . Due to Nak ay ama’s Lemma, J ( K S ) ⊆ J ( K S ( r ) ) and J ( K S ) ⊆ J ( K S ( I )). Since the ring K S ( I ) / J ( K S ( I )) is algebra ic ov er the field K S / J ( K S ) ∼ = K / I 0 , the rings K S ( I ) and, res pe c tively , K ( I ) are stable. In such a ca se the groups [ C I , E K ] and [ C I , E ( n, R )] are contained in E ( n, I ). Hence, if R is a blo ck integer-algebr aic ring then, accor ding to the fact pr oved ab ov e ,  C I i +1 , E ( n, R )  ⊂ E ( n, I i +1 ) C I i for all 0 ≤ i ≤ q . Therefore, R is a weakly-commutator ring with partially normal quotients. 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