English

A left R-module M is called almost F-injective, if every R-homomorphism from a finitely presented left ideal to M extends to a homomorphism of R to M. A right R-module V is said to be almost flat, if for every finitely presented left ideal I, the canonical map V ⊗ I → V ⊗R is monic. A ring R is called left almost semihereditary, if every finitely presented left ideal of R is projective. A ring R is said to be left almost regular, if every finitely presented left ideal of R is a direct summand of RR. We observe some characterizations and properties of almost F-injective modules and almost flat modules. Using the concepts of almost F-injectivity and almost flatness of modules, we present some characterizations of left coherent rings, left almost semihereditary rings, and left almost regular rings.

Let R be a ring. A subclass T of left R-modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let T be a weak torsion class of left R-modules and n a positive integer. Then a left R-module M is called T-finitely generated if there exists a finitely generated submodule N such that M/N ∈ T; a left R-module A is called (T,n)-presented if there exists an exact sequence of left R-modules $$0 \to {K_{n - 1}} \to {F_{n - 1}} \to \cdots \to {F_1} \to {F_0} \to M \to 0$$ such that F0,..., Fn−1 are finitely generated free and Kn−1 is T-finitely generated; a left R-module M is called (T,n)-injective, if Ext n R (A,M) = 0 for each (T, n+1)-presented left R-module A; a right R-module M is called (T,n)-flat, if Tor R n (M,A) = 0 for each (T, n+1)-presented left R-module A. A ring R is called (T,n)-coherent, if every (T, n+1)-presented module is (n + 1)-presented. Some characterizations and properties of these modules and rings are given.

Let R be a left coherent ring, S a right coherent ring and RU a generalized tilting module, with S=End(RU) satisfying the condition that each finitely presented left R-module X with ExtRi(X,U)=0 for any i≥1 is U-torsionless. If M is a finitely presented left R-module such that ExtRi(M,U)=0 for any i≥0 with $i \neq n$ (where n is a nonnegative integer), then $\mathrm {Ext}_S^n( \mathrm {Ext}_R^n(M, U), U)\cong M$ and ExtSi(ExtRn(M,U),U)=0 for any i≥0 with $i\neq n$. A duality is thus induced between the category of finitely presented holonomic left R-modules and the category of finitely presented holonomic right S-modules.

FI-injective and FI-flat modules

Let R be a ring and n be a positive integer. R is called left n-coherent, if every n-presented left R-module is $$(n+1)$$ -presented. A left R-module M is called (n, 0)-injective if Ext $$^1_R(V, M)=0$$ for every n-presented module V, a right R-module F is called (n, 0)-flat if Tor $$^R_1(F, V)=0$$ for every n-presented module V, a left R-module M is called (n, 0)-projective if $$\mathrm{Ext}^1_R(M, N)=0$$ for any (n, 0)-injective module N, and a right R-module M is called (n, 0)-cotorsion if $$\mathrm{Ext}^1_R(F, M)=0$$ for any (n, 0)-flat module F. We give some characterizations and properties of (n, 0)-projective modules and (n, 0)-cotorsion modules. n-coherent rings, n-hereditary rings and n-regular rings are characterized by (n, 0)-projective modules, (n, 0)-cotorsion modules, (n, 0)-injective modules and (n, 0)-flat modules.

We show that an iteration of the procedure used to define the Gorenstein projective modules over a ring R yields exactly the Gorenstein projective modules. Specifically, given an exact sequence of Gorenstein projective left R-modules G = … → G 1 → G 0 → G 0 → G 1 → … such that the complex Hom R (G, H) is exact for each projective left R-module H, the module Im(G 0 → G 0) is Gorenstein projective. We also get similar results for Gorenstein flat left R-modules when R is a right coherent ring. As applications, we obtain the corresponding results for Gorenstein complexes.

<abstract><p>Let $ N $ be a left $ R $-module with the endomorphism ring $ S = \text{End}(_{R}N) $. Given two cardinal numbers $ \alpha $ and $ \beta $ and a matrix $ A\in S^{\beta\times\alpha} $, $ N $ is called flat relative to $ A $ in case, for each $ x\in l_{N^{(\beta)}}(A) = \{u\in N^{(\beta)} \mid uA = 0\} $, there are a positive integer $ k $, $ y\in N^{k} $ and a $ k\times \beta $ row-finite matrix $ C $ over $ S $ such that $ CA = 0 $ and $ x = yC $. It is shown that $ N_{S} $ is flat relative to a matrix $ A $ if and only if $ l_{N^{(\beta)}}(A) $ is generated by $ N $. $ S $ is called left coherent relative to $ A $ if Ker$ (_{S}S^{(\beta)}\to _{S}S^{(\beta)}A) $ is finitely generated. It is shown that $ S $ is left coherent relative to $ A $ if and only if Hom$ _{R}(N, l_{N^{n}}(A)) $ is a finitely generated left $ S $-module if and only if $ l_{N^{n}}(A) $ has an add$ (N) $-precover (add$ (N) $ denotes the category of all direct summands of finite direct sums of copies of $ _{R}N $). Regarding applications, new necessary and sufficient conditions for epic (monic, having the unique mapping property) add$ (N) $-precovers of $ l_{N^{(\beta)}}(A) $ are investigated. Also, some new characterizations of left $ n $-semihereditary rings and von Neumann regular rings are given.</p></abstract>

A ring R is called a left (right) SF-ring if all simple left (right) R-modules are flat. It is known that von Neumann regular rings are left and right SF-rings. In this article, we study the regularity of left SF-rings and we prove the following: 1) if R is a left SF-ring whose all complement left (right) ideals are W-ideals, then R is strongly regular; 2) if R is a left SF-ring whose all maximal essential right ideals are GW-ideals, then R is regular.

Throughout this paper it is assumed that rings are associative, have the identity element, and all modules are left unital. R will denote a ring with identity, R-Mod the category of left R-modules, and for each left R-module M, E(M) (resp. J(M)) will represent the injective hull (resp. Jacobson radical) of M. Also, for a module M, A ⊆' M will mean that A is an essential submodule of M, and Z(M) denotes the singular submodule of M. M is called singular if Z(M) = M, and it is called non-singular in case Z(M) = 0. For fundamental definitions and results related to torsion theories, we refer to [12] and [14]. In this paper we shall deal mainly with Goldie torsion theory. Recall that a pair (G, F) of classes of left R-modules is known as Goldie torsion theory if G is the smallest torsion class containing all modules B/A, where A ⊆' B, and the torsion free class F is precisely the class of non-singular modules.

We prove that each almost local-global semihereditary ring R has the stacked bases property and is almost Bézout. More precisely, if M is a finitely presented module, its torsion part tM is a direct sum of cyclic modules where the family of annihilators is an ascending chain of invertible ideals. These ideals are invariants of M. Moreover, M/tM is a projective module which is isomorphic to a direct sum of finitely generated ideals. These ideals allow us to define a finitely generated ideal whose isomorphism class is an invariant of M. The idempotents and the positive integers defined by the rank of M/tM are invariants of M too. It follows that each semihereditary ring of Krull-dimension one or of finite character, in particular each hereditary ring, has the stacked base property. These results were already proved for Prüfer domains by Brewer, Katz, Klinger, Levy, and Ullery. It is also shown that every semihereditary Bézout ring of countable character is an elementary divisor ring.

The flat stable module category of a coherent ring

Suppose that [Formula: see text] and [Formula: see text] are elements of a, not necessarily unital, ring [Formula: see text] such that [Formula: see text] is regular. We prove that if [Formula: see text], then [Formula: see text] implies [Formula: see text] if and only if [Formula: see text]. This salvages Theorem 17 in [R. B. Bapat, S. K. Jain, K. M. P. Karantha and M. D. Raj, Outer inverses: Characterization and applications, Linear Algebra Appl. 528 (2017) 171–184] and extends Theorem 1 in [S. K. Jain and K. M. Prasad, Right-left symmetry of [Formula: see text] in regular rings, J. Pure Appl. Algebra 133 (1998) 141–142] to non-unital rings. For any positive integer [Formula: see text], if [Formula: see text] for [Formula: see text] and [Formula: see text] is regular, then we prove that [Formula: see text] implies [Formula: see text]. For any positive integer [Formula: see text], if [Formula: see text] for [Formula: see text] are such that [Formula: see text] and [Formula: see text], then we prove that the regularity of any [Formula: see text] elements of the set [Formula: see text] implies the regularity of the remaining element. Let [Formula: see text] and [Formula: see text] be regular elements of a ring [Formula: see text]. We prove that [Formula: see text] if and only if [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the set of all inner inverses of the regular element [Formula: see text] in [Formula: see text]. This extends Theorem 2.2 in [T. K. Lee, Unit-regularity of elements in rings, J. Algebra Appl. 22 (2023) 2350135] where the result is proved assuming [Formula: see text] to be semiprime. We also give some applications of Jain–Prasad symmetry result Theorem 1 in [S. K. Jain and K. M. Prasad, Right-left symmetry of [Formula: see text] in regular rings, J. Pure Appl. Algebra 133 (1998) 141–142] by giving new characterizations of unit-regular and special clean elements. This leads to quick proofs of some known results.

Let $n$ be a fixed positive integer. A ring $R$ is called left $n$-$\Pi$-coherent if every $n$-generated torsionless left $R$-module is finitely presented, some characterizations and applications of $n$-$\Pi$-coherent rings are obtained.

A full subcategory of modules over a commutative ring R is wide if it is abelian and closed under extensions. Hovey (2001) gave a classification of the wide subcategories of finitely presented modules over regular coherent rings in terms of certain specialization closed subsets of Spec(R). We use this classification theorem to study K-theory and Krull–Schmidt type decompositions for wide subcategories. It is shown that the K-group, in the sense of Grothendieck, of a wide subcategory 𝒲 of finitely presented modules over a regular coherent ring is isomorphic to that of the thick subcategory of perfect complexes whose homology groups belong to 𝒲. We also show that the wide subcategories of finitely generated modules over a Noetherian regular ring can be decomposed uniquely into indecomposable ones. This result is then applied to obtain a decomposition for the K-groups of wide subcategories.

Let M M be a right R R -module with S = End ( M R ) S={\rm{End}}\left({M}_{R}) . Given two cardinal numbers α \alpha and β \beta and a row-finite matrix A ∈ RFM β × α ( S ) A\in {{\rm{RFM}}}_{\beta \times \alpha }\left(S) , S M {}_{S}M is called injective relative to A A if every left S S -homomorphism from S ( β ) A {S}^{\left(\beta )}A to M M extends to one from S ( α ) {S}^{\left(\alpha )} to M M . It is shown that S M {}_{S}M is injective relative to A A if and only if the right R R -module M β ∕ A M α {M}^{\beta }/A{M}^{\alpha } is cogenerated by M M . S S is called left coherent relative to A ∈ S β × α A\in {S}^{\beta \times \alpha } if Ker ( S ( β ) S → S ( β ) S A ) \left({}_{S}S^{\left(\beta )}\to {}_{S}S^{\left(\beta )}A) is finitely generated. It is shown that S S is left coherent relative to A A if and only if M n ∕ A M α {M}^{n}/A{M}^{\alpha } has an add ( M ) {\rm{add}}\left(M) -preenvelope. As applications, we obtain the necessary and sufficient conditions under which M n ∕ A M α {M}^{n}/A{M}^{\alpha } has an add ( M ) {\rm{add}}\left(M) -preenvelope, which is monic (resp., epic, having the unique mapping property). New characterizations of left n n -semihereditary rings and von Neumann regular rings are given.