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Let [Formula: see text] be a Morita ring which is an Artin algebra and has zero bimodule homomorphisms. Assume that [Formula: see text] and [Formula: see text] are projective modules. For any positive integer [Formula: see text], it is proved that [Formula: see text] is [Formula: see text]-Igusa–Todorov (respectively, [Formula: see text]-syzygy-finite) if and only if [Formula: see text] and [Formula: see text] are [Formula: see text]-Igusa–Todorov (respectively, [Formula: see text]-syzygy-finite).

In this article, we prove that a precovering class F is closed under direct limits provided that F is closed under direct products and pure submodules. As a corollary, we get that a ring R ...

Let 𝒲,𝒴,𝒳 be three classes of left R-modules. In this paper, we introduce and study (𝒲,𝒴,𝒳)-Gorenstein complexes as a common generalization of completely 𝒲-resolved complexes [26], Gorenstein projective (resp., injective) complexes [8], Ding projective (resp., injective) complexes [32] and Gorenstein AC-projective (resp., AC-injective) complexes [4]. It is shown that under certain hypotheses, a complex C is (𝒲,𝒴,𝒳)-Gorenstein if and only if each Cn is a (𝒲,𝒴,𝒳)-Gorenstein module and ℋomR(Y,C),ℋomR(C,X) are exact for any Y∈𝒴,X∈𝒳. This result unifies the corresponding results of the aforementioned complexes. As applications, the stability of (𝒲,𝒴,𝒳)-Gorenstein complexes and modules are explored.

Let $A$, $B$ be two rings and $T=\left(\begin{smallmatrix} A & M 0 & B \end{smallmatrix}\right)$ with $M$ an $A$-$B$-bimodule. Given two complete hereditary cotorsion pairs $(\mathcal{A}_{A},\mathcal{B}_{A})$ and $(\mathcal{C}_{B},\mathcal{D}_{B})$ in $A$-Mod and $B$-Mod respectively. We define two cotorsion pairs $(��(\mathcal{A}_{A},\mathcal{C}_{B}), \mathrm{Rep}(\mathcal{B}_{A},\mathcal{D}_{B}))$ and $(\mathrm{Rep}(\mathcal{A}_{A},\mathcal{C}_{B}), ��(\mathcal{B}_{A},\mathcal{D}_{B}))$ in $T$-Mod and show that both of these cotorsion pairs are complete and hereditary. Given two cofibrantly generated model structures $\mathcal{M}_{A}$ and $\mathcal{M}_{B}$ on $A$-Mod and $B$-Mod respectively. Using the result above, we investigate when there exist a cofibrantly generated model structure $\mathcal{M}_{T}$ on $T$-Mod and a recollement of $\mathrm{Ho}(\mathcal{M}_{T})$ relative to $\mathrm{Ho}(\mathcal{M}_{A})$ and $\mathrm{Ho}(\mathcal{M}_{B})$. Finally, some applications are given in Gorenstein homological algebra.

We study Tate–Vogel cohomology of complexes by applying the model structure induced by a complete hereditary cotorsion pair ( A , B ) of modules. Vanishing of Tate–Vogel cohomology characterizes the finiteness of A dimension and B dimension of complexes defined by Yang and Ding [57] . Applications go in three directions. The first is to characterize when a left and right Noetherian ring is Gorenstein. The second is to obtain some criteria for the validity of the Finitistic Dimension Conjecture. The third is to investigate the relationships between flat dimension and Gorenstein flat dimension for complexes.

The classification of affine prime regular Hopf algebras of GK-dimension one is completed. As consequences, 1) we give a negative answer to an open question posed by Brown-Zhang and 2) we show that there do exist prime regular Hopf algebras of GK-dimension one which are not pointed., 39pages

Let R be a commutative ring. We investigate the relationship between (pre)covers ((pre)envelopes) of an R-module and the counterparts of the corresponding homomorphism module or tensor product module. Some applications are also given.

Let C be a semidualizing module over a commutative Noetherian ring R. We investigate duality pairs induced by C-Gorenstein projective modules. It is proven that R is Artinian if and only if \((\mathcal {GPC},\mathcal {GIC})\) is a duality pair if and only if \((\mathcal {GIC},\mathcal {GPC})\) is a duality pair and \(M^{+}\in {\mathcal {GIC}}\) whenever \(M\in {\mathcal {GPC}}\), where \(\mathcal {GPC}\) (\(\mathcal {GIC}\)) is the class of \(\mathcal {GC}\)-Gorenstein projective (\(\mathcal {GC}\)-Gorenstein injective) R-modules. In particular, we give a necessary and sufficient condition for a commutative Artinian ring to be virtually Gorenstein. Moreover, we get that R is Artinian if and only if the class \(\mathcal {GP}\) of Gorentein projective R-modules is preenveloping. As applications, some new criteria for a semidualizing module to be dualizing are given provided that R is a commutative Artinian ring.

In this paper complexes with N-nilpotent differentials are considered. We describe the loop functor Ω and the suspension functor Σ in the category C N ( A ) of N-complexes on an abelian category A to provide an effective construction of left and right triangles in the homotopy category K N ( A ) of N-complexes, and prove that K N ( A ) together with the functors Ω, Σ and the classes of left and right triangles mentioned above is a pretriangulated category. We also investigate different homologies and get another example of pretriangulated categories, i.e., the derived category D N ( A ) of N-complexes. Some basic properties are given.

Let R be a commutative ring and U an R-module. The aimof this paper is to study the duality between U-reflexive (pre)envelopesand U-reflexive (pre)covers of R-modules. 1. Introduction and preliminariesLet Rbe a commutative ring and Uan R-module. For R-modules MandN, Hom(M,N) means Hom R (M,N), and Ext n (M,N) means Ext nR (M,N) foran integer n≥ 1. For an R-module M, Hom(M,U) is called the dual moduleof Mwith respect to Uand denoted by M ∗ . For a homomorphism fbetweenR-modules, we put f ∗ = Hom(f,U). Let δ M : M→ M ∗∗ via δ M (x)(f) = f(x)for any x∈ Mand f∈ M ∗ be the canonical evaluation homomorphism. If δ M is an isomorphism, then Mis called a U-reflexive module. We denote by R U the class of U-reflexive modules.Let Rbe a Noetherian ring. A finitely generated R-module Mis said to haveGorensteindimension (abbr. G-dimension) zero[2]if M∼=Hom(Hom(M,R),R)(i.e., M is R-reflexive); and Ext i (M,R) = 0 = Ext i (Hom(M,R),R) for alli≥ 1. We denote by G(R) the class of R-modules having G-dimension zero.Let F be a class of R-modules and M an R-module. A homomorphismφ : M → F with F ∈ F is called an F-preenvelope of M [10] if for anyhomomorphism f: M→ F

We study Tate cohomology of modules over a commutative Noetherian ring with respect to semidualizing modules. First, we show that the class of modules admitting a Tate $\mathcal{F}_C $-resolution is exactly the class of modules in $\mathcal{B}_{C} $ with finite $\mathcal{GF}_{C} $-projective dimension. Then, the interaction between the corresponding relative and Tate cohomologies of modules is given. Finally, we give some new characterizations of modules with finite $\mathcal{F}_C $-projective dimension.

We study Gorenstein right derived functors of − ⊗ −with respect to semidualizing modules. As applications, some new criteria for a semidualizing module to be dualizing are given provided that R is a ring with a dualizing complex.

A question of Gillespie concerning the completeness of the induced cotorsion pairs is settled in the general setting of any bicomplete abelian category. That is, given a complete hereditary cotorsion pair ( 𝒜 , ℬ ) ${(\mathcal {A},\mathcal {B})}$ in a bicomplete abelian category, we show that the cotorsion pairs of chain complexes ( 𝒜 ˜ , dg ℬ ˜ ) ${(\mathcal {\protect \widetilde{A}},\operatorname{dg}\mathcal {\protect \widetilde{B}})}$ and ( dg 𝒜 ˜ , ℬ ˜ ) ${(\operatorname{dg}\mathcal {\protect \widetilde{A}},\mathcal {\protect \widetilde{B}})}$ are complete, compatible and hereditary. This immediately puts a model structure on the chain complex category. As an application of this result, we establish homological dimensions of unbounded complexes in a Grothendieck category by means of dg 𝒜 ${\operatorname{dg}\mathcal {A}}$ complexes and dg ℬ ${\operatorname{dg}\mathcal {B}}$ complexes respectively. Some examples of complete hereditary cotorsion pairs in Grothendieck categories are given.

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.

Let R be any ring. A right R-module M is called n-copure projective if Ext1(M, N) = 0 for any right R-module N with fd(N) ≤ n, and M is said to be strongly copure projective if Ext i (M, F) = 0 for all flat right R-modules F and all i ≥ 1. In this article, firstly, we present some general properties of n-copure projective modules and strongly copure projective modules. Then we define and investigate copure projective dimensions of modules and rings. Finally, more properties and applications of n-copure projective modules, strongly copure projective modules and copure projective dimensions are given over coherent rings with finite self-FP-injective dimension.

It is well known that tilting modules are of particular importance in the general theory of modules. One of the roles of a tilting module R ω is that it provides a connection between the categories of modules over R and S = End( R ω). This connection is described in the Tilting Theorem. In this article, we get a generalized version of the Tilting Theorem for a Wakamatsu tilting module R ω.

Let R be a left coherent ring. We first prove that a right R-module M is strongly copure flat if and only if Ext i (M, C) = 0 for all flat cotorsion right R-modules C and i ≥ 1. Then we define and investigate copure flat dimensions of left coherent rings. Finally, we give some new characterizations of n-FC rings.

Gorenstein injective modules and dimensions have been studied extensively by many authors. In this paper, we investigate Gorenstein injective modules and dimensions relative to a Wakamatsu tilting module.

Let R be a ring and U a left R-module with S=End(RU). The aim of this paper is to characterize when S is coherent. We first show that a left R-module F is TU-flat if and only if HomR(U,F) is a flat left S-module. This removes the unnecessary hypothesis that U is Σ-quasiprojective from Proposition 2.7 of Gomez Pardo and Hernandez [‘Coherence of endomorphism rings’, Arch. Math. (Basel)48(1) (1987), 40–52]. Then it is shown that S is a right coherent ring if and only if all direct products of TU-flat left R-modules are TU-flat if and only if all direct products of copies of RU are TU-flat. Finally, we prove that every left R-module is TU-flat if and only if S is right coherent with wD(S)≤2 and US is FP-injective.

In this article, the notion of pure hereditary rings is introduced as a nontrivial generalization of the notion of both pure semisimple rings and of hereditary rings. Some properties and examples of pure hereditary rings are given. For a positive integer n, let 𝒫 n be the class of all left R-modules with projective dimension ≤ n. It is shown that over a left pure hereditary ring is a perfect and hereditary cotorsion pair, which gives a partial positive answer to a question of Gobel and Trlifaj. The modules in the class are also studied. Special attention is paid to the case n = 1.

Let R be a ring. Recall that a left R-module M is coherent if every finitely generated submodule of M is finitely presented. R is a left coherent ring if the left R-module RR is coherent. In this paper, we say that R is left J-coherent if its Jacobson radical J(R) is a coherent left R-module. J-injective and J-flat modules are introduced to investigate J-coherent rings. Necessary and sufficient conditions for R to be left J-coherent are given. It is shown that there are many similarities between coherent and J-coherent rings. J-injective and J-flat dimensions are also studied.

In this article, we first study the existence of envelopes and covers by modules of finite divisible and torsionfree dimensions. Then we investigate divisible and torsionfree dimensions as well as localizations of divisible and torsionfree modules over commutative rings. Finally, Gorenstein divisible and torsionfree modules are introduced and studied.

A ring R is called left P-coherent in case each principal left ideal of R is finitely presented. A left R-module M (resp. right R-module N) is called D-injective (resp. D-flat) if Ext1(G, M) = 0 (resp. Tor1(N, G) = 0) for every divisible left R-module G. It is shown that every left R-module over a left P-coherent ring R has a divisible cover; a left R-module M is D-injective if and only if M is the kernel of a divisible precover A → B with A injective; a finitely presented right R-module L over a left P-coherent ring R is D-flat if and only if L is the cokernel of a torsionfree preenvelope K → F with F flat. We also study the divisible and torsionfree dimensions of modules and rings. As applications, some new characterizations of von Neumann regular rings and PP rings are given.

In this article, we study the weak global dimension of coherent rings in terms of the left FP-injective resolutions of modules. Let R be a left coherent ring and ℱ ℐ the class of all FP-injective left R-modules. It is shown that wD(R) ≤ n (n ≥ 1) if and only if every nth ℱ ℐ-syzygy of a left R-module is FP-injective; and wD(R) ≤ n (n ≥ 2) if and only if every (n − 2)th ℱ ℐ-syzygy in a minimal ℱ ℐ-resolution of a left R-module has an FP-injective cover with the unique mapping property. Some results for the weak global dimension of commutative coherent rings are also given.

A ring R is called “left generalized morphic” if for every element a in R, there exists b ∈ R such that l(a)≅ R/Rb, where l(a) denotes the left annihilator of a in R. The aim of this article is to investigate these rings. Several examples are given. They include left morphic rings and left p.p. rings. As applications, some homological dimensions over these rings are defined and studied.

Let H and K be normal subgroups of a finite group G and let K ≤ H . If A is a subgroup of G such that A H = A K or A ∩ H = A ∩ K , we say that A covers or avoids H / K respectively. The purpose of this paper is to investigate factor groups of a finite group G using this concept. We get some characterizations of a finite group being solvable or supersolvable and generalize some known results.

It is well known that the right global dimension of a ring R is usually computed by the right derived functors of Hom and the left projective resolutions of right R-modules. In this paper, for a left coherent and right perfect ring R, we characterize the right global dimension of R, from another point of view, using the left derived functors of Hom and the right projective resolutions of right R-modules. It is shown that rD(R) ⩽ n (n ⩾ 2) if and only if the gl right Proj-dim M R ⩽ n−2 if and only if Ext n−1(N,M) = 0 for all right R-modules N and M if and only if every (n−2)th Projcosyzygy of a right R-module has a projective envelope with the unique mapping property. It is also proved that rD(R) ⩽ n (n ⩾ 1) if and only if every (n−1)th Proj-cosyzygy of a right R-module has an epic projective envelope if and only if every nth Proj-cosyzygy of a right R-module is projective. As corollaries, the right hereditary rings and the rings R with rD(R) ⩽ 2 are characterized.

Let R be a ring. A left R-module M (respectively right R-module N) is called FI-injective (respectively FI-flat) if Ext 1 ( G , M ) = 0 (respectively Tor 1 ( N , G ) = 0 ) for any FP-injective left R-module G. Suppose R is a left coherent ring. It is shown that a left R-module M is FI-injective if and only if M is a direct sum of an injective left R-module and a reduced FI-injective left R-module; a finitely presented right R-module M is FI-flat if and only if M is a cokernel of a flat preenvelope of a right R-module. These modules together with the left derived functors of Hom are used to study the FP-injective dimensions of modules and rings.

Let R be a ring, n a fixed non-negative integer and ℱ ℐ n (ℱ n ) the class of all right (left) R-modules of FP-injective (flat) dimension at most n. We prove that ( is a perfect cotorsion theory if R is a right coherent ring with FP-id(R R ) ≤ n. This result was proven by Aldrich, Enochs, Jenda, and Oyonarte in Noetherian case. The modules in are also studied. Some applications are given.

Let R be a ring, n a fixed nonnegative integer and I n ( F n ) the class of all left (right) R -modules of injective (flat) dimension at most n . A left R -module M (resp., right R -module F ) is called n -copure injective (resp., n -copure flat) if Ext 1 ( N , M ) = 0 (resp., Tor 1 ( F , N ) = 0 ) for any N ∈ I n . It is shown that a left R -module M over any ring R is n -copure injective if and only if M is a kernel of an I n -precover f : A → B of a left R -module B with A injective. For a left coherent ring R , it is proven that every right R -module has an F n -preenvelope, and a finitely presented right R -module M is n -copure flat if and only if M is a cokernel of an F n -preenvelope K → F of a right R -module K with F flat. These classes of modules are also used to construct cotorsion theories and to characterize the global dimension of a ring under suitable conditions.

Let M R be a right R-module over a ring R with S = End(M R ). We study the coherence of the left S-module S M relative to a hereditary torsion theory for the category of right R-modules. Various results are developed, many extending known results.

Let R be a ring, and n and d fixed non-negative integers. An R-module M is called (n, d)-injective if Ext d+1 R (P, M) = 0 for any n-presented R-module P. M is said to be (n, d)-projective if Ext1 R (M, N) = 0 for any (n, d)-injective R-module N. We use these concepts to characterize n-coherent rings and (n, d)-rings. Some known results are extended.

Let R be a ring and ℒ a class of R-modules. An R-module N is called ℒ-injective if for all L ∈ ℒ. An ℒ-injective hull of an R-module M is defined to be a homomorphism φ: M → F with F ℒ-injective such that for any monomorphism f: M → F′ with F′ ℒ-injective, there is a monomorphism g: F → F′ satisfying gφ = f. The aim of this paper is to study ℒ-injective hulls and their relations with ℒ-injective envelopes in Enochs' sense.

Let m and n be fixed positive integers and M a right R-module. Recall that M is said to be (m,n)-injective if for any (m,n)-presented right R-module P; M is called (m,n)-flat if for any (m,n)-presented left R-module Q. In this article, M is defined to be (m,n)-projective (resp. (m,n)-cotorsion) if (resp. ) for any (m,n)-injective (resp. (m,n)-flat) right R-module N. These concepts are used to characterize von Neumann regular rings and (m,n)-coherent rings. Some known results are extended.

Let R be a ring, M a right R-module, and n a fixed non-negative integer. M is called n-cotorsion if for any flat right R-module N. M is said to be n-flat if for any n-cotorsion right R-module N. We prove that (ℱ n ,  n ) is a complete hereditary cotorsion theory, where ℱ n (resp.  n ) denotes the class of all n-flat (resp. n-cotorsion) right R-modules. Several applications are given.

Let C be a class of R -modules closed under isomorphisms and finite direct sums. It is first shown that the finite direct sum of almost C -precovers is an almost C -precover and the direct sum of an almost C -cover and a weak C -cover is a weak C -cover. Then the notion of almost C -preenvelopes is introduced and studied.

We introduce the concepts of lifting modules and (quasi-)discrete modules relative to a given left module. We also introduce the notion of SSRS-modules. It is shown that (1) ifMis an amply supplemented module and0→N′→N→N″→0an exact sequence, thenMisN-lifting if and only if it isN′-lifting andN″-lifting; (2) ifMis a Noetherian module, thenMis lifting if and only ifMisR-lifting if and only ifMis an amply supplemented SSRS-module; and (3) letMbe an amply supplemented SSRS-module such thatRad(M)is finitely generated, thenM=K⊕K′, whereKis a radical module andK′is a lifting module.

Let R be a ring and M a right R -module. M is called n - FP -projective if Ext 1 ( M , N ) = 0 for any right R -module N of FP -injective dimension ≤ n , where n is a nonnegative integer or n = ∞. ν R ( M ) is defined as sup{ n : M is n - FP -projective} and ν R ( M ) = − 1 if Ext 1 ( M , N ) ≠ 0 for some FP -injective right R -module N . The right ν-dimension r .ν-dim( R ) of R is defined to be the least nonnegative integer n such that ν R ( M ) ≥ n implies ν R ( M ) = ∞ for any right R -module M . If no such n exists, set r .ν-dim( R ) = ∞. The aim of this paper is to investigate n - FP -projective modules and the ν-dimension of rings. #Communicated by A. Facchini.

We define a dimension, called an FP-projective dimension, for modules and rings. It measures how far away a finitely generated module is from being finitely presented, and how far away a ring is from being Noetherian. This dimension has nice properties when the ring in question is coherent. The relations between the FP-projective dimension and other homological dimensions are discussed.