Infinitely generated projective modules over pullbacks of rings

We use pullbacks of rings to realize the submonoids $M$ of $(\N_0\cup{\infty})^k$ which are the set of solutions of a finite system of linear diophantine inequalities as the monoid of isomorphism classes of countably generated projective right $R$-modules over a suitable semilocal ring. For these rings, the behavior of countably generated projective left $R$-modules is determined by the monoid $D(M)$ defined by reversing the inequalities determining the monoid $M$. These two monoids are not isomorphic in general. As a consequence of our results we show that there are semilocal rings such that all its projective right modules are free but this fails for projective left modules. This answers in the negative a question posed by Fuller and Shutters \cite{FS}. We also provide a rich variety of examples of semilocal rings having non finitely generated projective modules that are finitely generated modulo the Jacobson radical.

💡 Research Summary

The paper investigates the structure of infinitely generated projective modules over rings obtained as pull‑backs, focusing on the interplay between right and left projective modules in the non‑noetherian semilocal setting. The authors start from the observation that for any semilocal ring R the Jacobson radical J(R) determines projective modules up to isomorphism: two projective modules are isomorphic precisely when their quotients modulo J(R) are isomorphic. Consequently, the monoid V*(R) of isomorphism classes of countably generated projective right R‑modules (and analogously V*(Rᵒ) for left modules) can be embedded into (ℕ₀∪{∞})ᵏ, where k is the number of simple components of the semisimple artinian factor R/J(R).

The central aim is to realize, for a given integer k, any submonoid M⊆(ℕ∗₀)ᵏ that is defined by a finite system of linear Diophantine inequalities as the dimension monoid dim_φ(V*(R)) of a suitable semilocal ring R. A monoid defined by inequalities means that there exist matrices D∈M_{n×k}(ℕ₀), E₁,E₂∈M_{ℓ×k}(ℕ₀) and integers m₁,…,m_n≥2 such that   D·t∈(m₁ℕ∗₀,…,m_nℕ∗₀) and E₁·t≥E₂·t for a vector t=(t₁,…,t_k)∈(ℕ∗₀)ᵏ. The “dual” monoid D(M) is obtained by reversing the inequality sign. Theorem 1.6 asserts that if M∩D(M) contains at least one ordinary k‑tuple of natural numbers, then there exists a field F, a suitable extension field E, and a semilocal F‑algebra R together with a surjective homomorphism φ:R→S (S=∏{i=1}^k M{n_i}(E), Ker φ=J(R)) such that

 dim_φ(V*(R)) = M and dim_φ(V*(Rᵒ)) = D(M).

Moreover, the submonoids of modules that are finitely generated modulo the Jacobson radical satisfy  dim_φ(W(R)) = M∩ℕᵏ, dim_φ(W(Rᵒ)) = D(M)∩ℕᵏ, and the intersection V(R) = V*(R)∩V*(Rᵒ) corresponds to M∩D(M)∩ℕᵏ.

The construction of R relies on pull‑backs of two semilocal rings A and B over a common quotient C, a technique pioneered by Milnor. By carefully choosing A, B, and the maps to C so that the dimension data of projective modules over A and B encode the given inequalities, the pull‑back R inherits precisely the desired monoid of projective modules. The authors also prove auxiliary lifting results (Theorem 2.2) showing that isomorphisms modulo an ideal contained in J(R) lift to genuine isomorphisms, which is crucial for controlling the structure of projective modules in the pull‑back.

Several concrete examples illustrate the theory. In Example 3.6 the authors construct a semilocal ring R with R/J(R) ≅ D₁×D₂ such that every countably generated projective right R‑module is free, while there exists a non‑free, infinitely generated projective left R‑module. This provides a negative answer to a question of Fuller and Shutters concerning whether the property “all right projectives are free” forces the same property on the left. Other examples exhibit rings where non‑finitely generated projective modules become finitely generated after factoring out J(R), extending earlier isolated constructions by Gerasimov‑Sakhaev and Sakhaev.

Section 2 develops the relationship between projective modules and the Jacobson radical, establishing that for any ideal I⊆J(R) a homomorphism between projective modules is a pure monomorphism (or an isomorphism) precisely when its reduction modulo I has the same property. Corollary 2.3 shows that any endomorphism of a countably generated projective module that is the identity modulo J(R) can be adjusted on a finite set to become the identity on the whole module, a technical tool used later in the pull‑back construction.

Section 4 studies the algebraic properties of monoids defined by inequalities, proving that they are always finitely generated and that the “full affine” condition (closure under subtraction of elements that stay in the monoid) characterizes those monoids that arise from solutions of homogeneous linear equations. This analysis guarantees that the monoids M and D(M) appearing in Theorem 1.6 are well‑behaved and suitable for realization as dimension monoids.

Finally, Section 5 contains the proof of the main realization theorem. The authors first translate the inequality system into a presentation of a commutative monoid, then build two auxiliary semilocal algebras whose projective module monoids correspond to the two sides of the inequality. Using Milnor’s pull‑back theorem, they glue these algebras along a common semisimple quotient, obtaining the desired ring R. The Jacobson radical of R is precisely the kernel of the natural map to the semisimple factor, and the dimension maps dim_φ recover the original inequality solutions.

In summary, the paper provides a comprehensive framework for encoding finite systems of linear Diophantine inequalities into the structure of countably generated projective modules over semilocal pull‑back rings. It demonstrates that right and left projective module monoids can differ dramatically, supplies explicit counterexamples to longstanding conjectures, and expands the catalogue of rings possessing infinitely generated projective modules that become finitely generated modulo the Jacobson radical. The work bridges module theory, ring pull‑backs, and monoid combinatorics, opening new avenues for constructing and classifying projective modules in non‑noetherian contexts.