Search

AbstractWe provide necessary and/or sufficient conditions for the stable Hom functors to be finitely presented. When the covariant Hom functor modulo projectives is finitely presented, its defect is isomorphic to the Bass torsion of the fixed argument. When the contravariant Hom functor modulo injectives is finitely presented, its defect is isomorphic to the cotorsion of the fixed argument. We also give a sufficient condition for the sub-stabilization of the tensor product to be finitely presented. A finite presentation of the tensor product leads to an unexpected application to injective modules. [ABSTRACT FROM AUTHOR]

Assume that B is a finite dimensional algebra, and A = B [ P 0 ] is the one-point extension algebra of B using a finitely generated projective B-module P0. The categories of B-modules and A-modules are connected via two adjoint functors known as the restriction and extension functors, denoted by R and E , respectively. These functors have nice homological properties and have been studied in the category mod- A of finitely presented modules that we extend to the category Mod- A of all A-modules. Our main focus is to investigate the behavior of important subcategories (tilting and τ-tilting subcategories) and objects (finendo quasi-tilting modules, silting modules, and cosilting modules) under these functors. [ABSTRACT FROM AUTHOR]

We extend a result due to Kawai on block varieties for blocks with abelian defect groups to blocks with arbitrary defect groups. Kawai's result is a tool to calculate the cohomology variety of a module in a block B of a finite group algebra kG restricted to subgroups of a defect group P, provided that P is abelian. Kawai's result coincides with a Theorem of Avrunin and Scott specialized to modules in the principal block and their restrictions to p-subgroups. J. Rickard raised the question whether Kawai's result can be extended to modules in blocks with arbitrary defect groups. We show that this is indeed the case for modules whose corresponding module over some almost source algebra is fusion stable. We show that this fusion stability hypothesis is automatically satisfied for principal blocks and blocks with abelian defect groups. [ABSTRACT FROM AUTHOR]

Let R ⋉ M be a trivial extension of a ring R by an R-R-bimodule M. We first give some homological formulas over R ⋉ M . Then we investigate how to construct (hereditary, complete) cotorsion pairs and (hereditary, cofibrantly generated) Hovey triples over R ⋉ M . Finally, some applications are given in some Morita context rings. [ABSTRACT FROM AUTHOR]

We give a sufficient condition that a certain homological epimorphism between two finite dimensional algebras induces a triangle equivalence between their Gorenstein defect categories as well as between their stable categories of Gorenstein projective modules. [ABSTRACT FROM AUTHOR]

Let R be a ring and X a complex of R-modules. We give some characterizations of the dimension of X related to special Gorenstein projective precovers. Let R be a ring such that (G P , G P ⊥) forms a cotorsion pair cogenerated by a set, where G P denotes the category consisting of all Gorenstein projective R-modules. Then the dimension related to special Gorenstein projective precovers Gppd (X) is equal to the infimum of the set { n ∈ Z | there exists a complete Gorenstein projective resolution of X T → σ G → X such that σi is bijective for each i ≥ n }. Forthmore, if Gppd (X) is finite and n an integer then Gppd (X) ≤ n if and only if Ext G P i (X , Q) = 0 for any i > n and any projective R-module Q, where Ext G P i (− , −) is the relative cohomology functor. [ABSTRACT FROM AUTHOR]

Let R be a ring and n be a non-negative integer. In this paper, we introduce and study the notions of Gorenstein n-weak injective and Gorenstein n-weak flat modules by using the notion of special super finitely presented modules. On an arbitrary ring, we investigate the relationships between Gorenstein n-weak injective and Gorenstein n-weak flat modules. Among other results, we prove that any R-module admits a Gorenstein n-weak injective (resp. Gorenstein n-weak flat) cover and pre-envelope. Then, we deduce that the class of Gorenstein n-weak injective (resp. Gorenstein n-weak flat) R-modules coincides with that of two-degree Gorenstein n-weak injective (resp. two-degree Gorenstein n-weak flat) R-modules. [ABSTRACT FROM AUTHOR]

Let S and R be rings, S C R be a semidualizing bimodule and n ≥ 0 be an integer or n = ∞. We introduce and study C-FPn-injective and C-FPn-flat modules as a common generalization of some known modules such as C-injective (resp. C-FP-injective, C-weak injective) and C-flat (resp. C-projective, C-weak flat) modules. Suppose that S C R is a faithfully semidualizing bimodule. We give some equivalent characterizations of left n-coherent rings in terms of C-FPn-flat left S-modules and C-FPn-injective left R-modules. Then we show that the pairs (F F C n (S) , F I C n (S o p)) and (F I C n (R) , F F C n (R o p)) are coproduct-closed and product-closed duality pairs and both F F C n (S) and F I C n (R) are covering and preenveloping, where F F C n (S) and F I C n (R) denote the classes of C-FPn-flat left S-modules and C-FPn-injective left R-modules respectively. Finally, we investigate Foxby equivalence relative to C-FPn-injective and C-FPn-flat modules. [ABSTRACT FROM AUTHOR]

Let R be a ring and n, k two non-negative integers. In this paper, we introduce the concepts of n-weak injective and n-weak flat modules and via the notion of special super finitely presented modules, we obtain some characterizations of these modules. We also investigate two classes of modules with richer contents, namely W I k n (R) and W F k n (R o p) which are larger than that of modules with weak injective and weak flat dimensions at most k. Then over any arbitrary ring, we study the existence of W I k n (R) and W F k n (R o p) covers and pre-envelopes. [ABSTRACT FROM AUTHOR]

Let R be a Gorenstein ring. We first study Gorenstein orthogonal classes and Gorenstein cotorsion pairs of R-modules. Then, we introduce the concept of Gorenstein weak tilting R-modules, which is a "Gorenstein analogue" of weak tilting modules. It is proven that a left R-module W is Gorenstein weak n-tilting if and only if its character module W+ is Gorenstein n-cotilting if and only if its Gorenstein flat dimension is at most n and its left Gorenstein Tor-orthogonal class consists precisely of all right R-modules with a G-exact right Prod (W +) -resolution of finite length. Finally, we explore the connections between Gorenstein weak tilting modules and Gorenstein tilting modules. [ABSTRACT FROM AUTHOR]

We give a series of infinite dimensional noncommutative and noncocommutative pointed Hopf algebras, which are Artin-Schelter Gorenstein Hopf algebras with injective dimensions 3. Radford's Hopf algebra and Gelaki's Hopf algebra are homomorphic images of these Hopf algebras. We determine the irreducible representations of them. We describe the Grothendieck rings of them. We obtain that two non-isomorphic Hopf algebras could have isomorphic Grothendieck rings. [ABSTRACT FROM AUTHOR]

Let B be the one-point extension algebra of A by an A-module X. We proved that every support τ-tilting A-module can be extended to be a support τ-tilting B-module by two different ways. As a consequence, it is shown that there is an inequality | s τ ‐ tilt B | ⩾ 2 | s τ ‐ tilt A |. [ABSTRACT FROM AUTHOR]

An R-module M is called strongly FP-injective if Ext R i (P , M) = 0 for any finitely presented R-module P and any i > 0. Denoted by S F I the class of all strongly FP-injective R-modules and by ⊥ S F I the left orthogonal class with respect to S F I. Comparing with some classical results of Noetherian rings, we show that a ring is coherent if and only if S F I is closed under pure submodules; if and only if any finitely generated module in ⊥ S F I is finitely presented; if and only if R is ℵ 0 -coherent and S F I is closed under direct sums; if and only if Tor n R ( Hom T (M , E) , X) ≅ Hom T ( Ext R n (X , M) , E) for any nonnegative integer n, any finitely presented left R-module X, any R-T-bimodule M and any injective right T-module E. In addition, we show that a module is injective if and only if it is a pure I -periodic module, where I denotes the class of all injective modules. [ABSTRACT FROM AUTHOR]

In this paper, (higher) Gorenstein flat phantom morphisms over rings will be introduced and studied. To study their relationship, a characterization of Gorenstein flat objects in morphism category is given. Communicated by Alberto Facchini [ABSTRACT FROM AUTHOR]

In this article, we investigate the Gorenstein global dimension with respect to the recollements of abelian categories. With the invariants spli and silp of the categories, we give some upper bounds of Gorenstein global dimensions of the categories involved in a recollement of abelian categories. We apply our results to some rings and artin algebras, especially to the triangular matrix artin algebras. [ABSTRACT FROM AUTHOR]

Hartmann et al. defined the concept of cellularly stratified algebras that combine the features of both cellular algebras and stratified algebras. Many important diagram algebras in mathematics and physics, such as some Brauer, partition and BMW algebras, are cellularly stratified algebras, and each of these forms a tower of algebras. This article gives the concept of towers of cellularly stratified algebras in an axiomatic manner, and studies it in terms of induction and restriction functors. In particular, for certain towers of cellularly stratified algebras, we provide a criterion for semi-simplicity by using the cohomology groups of cell modules. [ABSTRACT FROM AUTHOR]

Let X be a class of left R-modules, Y be a class of right R-modules. In this article, we introduce and study Gorenstein (X , Y) -flat modules as a common generalization of some known modules such as Gorenstein flat modules, Gorenstein n-flat modules, Gorenstein B -flat modules, Gorenstein AC-flat modules, Ω-Gorenstein flat modules and so on. We show that the class of all Gorenstein (X , Y) -flat modules have a strong stability. In particular, when (X , Y) is a perfect (symmetric) duality pair, we construct a hereditary abelian model structure on R-Mod whose cofibrant objects are exactly the Gorenstein (X , Y) -flat modules. Finally, we investigate when the class of Gorenstein (X , Y) -flat modules is closed under extensions. These results unify the corresponding results of the aforementioned modules. [ABSTRACT FROM AUTHOR]

Assume that A is a connected cochain DG algebra such that the trivial DG algebra is a Calabi-Yau DG algebra. We show that A is also a Calabi-Yau DG algebra. As applications, we study the Calabi-Yau property of a connected cochain DG algebra A when H(A) belongs to certain types of 3-Calabi-Yau graded algebras. [ABSTRACT FROM AUTHOR]

In this article, we study a natural generalization of finitistic n-self-cotilting modules (and hence also of costar-modules) by introducing the notion of ∞-costar modules over any ring R. The most important results about finitistic n-self-cotilting modules and costar modules are generalized. Also, we introduce a subclass of ∞-costar modules, which is a natural generalization of cotilting modules of finite injective dimensions to infinite injective dimensions. When R is a finite dimensional K-algebra, a finitely generated ∞-costar module is an ∞-cotilting module. [ABSTRACT FROM AUTHOR]

For a linear category over a field K, the morphism space of any two objects admits a bimodule structure over the endomorphism algebras of the objects, so it induces a Morita context between those two algebras. In this article, we use Morita context functors to study cellular categories and give relationships between the cell modules which belong to different endomorphism algebras in a cellular category. [ABSTRACT FROM AUTHOR]

LetRbe a ring. Then a leftR-moduleNis pure-injective if and only ifHomR(M,N) is a pure-injective leftS-module for any ringSand any (R,S)-bimodule. IfRis a commutative ring andM,NareR-modules withNpure-injective, thenis a pure-injectiveR-module for anyn≥0. LetRandSbe rings and letbe an (S,R)-bimodule andMa finitely presented leftR-module. IfNis pure-injective as a leftS-module, then the leftS-moduleN⊗RMis pure-injective; and ifRis left coherent, then the leftS-moduleis pure-injective for anyn≥1. [ABSTRACT FROM PUBLISHER]

A rightR-moduleMis called glat if any homomorphism from any finitely presented rightR-module toMfactors through a finitely presented Gorenstein projective rightR-module. The concept of glat modules may be viewed as another Gorenstein analogue of flat modules. We first prove that the class of glat rightR-modules is closed under direct sums, direct limits, pure quotients and pure submodules for arbitrary ringR. Then we obtain that a rightR-moduleMis glat if and only ifMis a direct limit of finitely presented Gorenstein projective rightR-modules. In addition, we explore the relationships between glat modules and Gorenstein flat (Gorenstein projective) modules. Finally we investigate the existence of preenvelopes and precovers by glat and finitely presented Gorenstein projective modules. [ABSTRACT FROM AUTHOR]

Let𝒲,𝒴,𝒳be three classes of leftR-modules. In this paper, we introduce and study (𝒲,𝒴,𝒳)-Gorenstein complexes as a common generalization of completely𝒲-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 complexCis (𝒲,𝒴,𝒳)-Gorenstein if and only if eachCnis a (𝒲,𝒴,𝒳)-Gorenstein module andare exact for any. This result unifies the corresponding results of the aforementioned complexes. As applications, the stability of (𝒲,𝒴,𝒳)-Gorenstein complexes and modules are explored. [ABSTRACT FROM PUBLISHER]

In the article “When does the Associated graded Lie algebra of an Arrangement Group Decompose?” by Stefan Papadima and Alexander Suciu [7], it is proved that the holonomy Lie algebra of an arrangement of hyperplanes through origin decomposes as a direct product of Lie algebras in degree at least two if and only if a certain (computable) condition is fulfilled. We prove similar results for a class of Lie algebras which is a generalization of the holonomy Lie algebras. The proof methods are the same as in the article cited above. [ABSTRACT FROM PUBLISHER]

A morphismf:M → Nof leftR-modules is called a phantom morphism if the induced morphismfor every (finitely presented) rightR-moduleA. Similarly, a morphismg:M → Nof leftR-modules is said to be an Ext-phantom morphism if the induced morphismfor every finitely presented leftR-moduleB. We prove that every leftR-module has a phantom preenvelope ifRis a right coherent ring and every leftR-module has an Ext-phantom cover ifRis a left coherent ring. In addition, we investigate the properties of precovers and preenvelopes by phantom and Ext-phantom morphisms under change of rings. [ABSTRACT FROM PUBLISHER]

In this article, we give a new characterization of Gorenstein projective modules. As applications of our result, we prove that a strongly Gorenstein projective module of countable type is Gorenstein flat, and each leftR-module has a special Gorenstein projective precover whenever all projective leftR-modules have finite injective dimension. [ABSTRACT FROM PUBLISHER]

In this article, we define and study a notion of strongly Gorenstein flat dimensions for complexes of left modules over associative rings. In particular, we consider the class of homologically bounded below complexes of leftR-modules, and show that strongly Gorenstein flat dimension has a nice functorial description. In addition, we will investigate the strongly Gorenstein flat properties of complexes under change of rings. As an application, we study the Tate cohomology theories with respect to ℱ-complete resolutions, where ℱ is the class of all flat modules. [ABSTRACT FROM PUBLISHER]

LetRbe a ring. A rightR-moduleMis said to be neat-flat if the kernel of any epimorphismY → Mis neat inY, i.e., the induced map Hom(S,Y) → Hom(S,M) is surjective for any simple rightR-moduleS. Neat-flat rightR-modules are projective if and only ifRis a right-CSring. Every cyclic neat-flat rightR-module is projective if and only ifRis rightCSand rightC-ring. It is shown that, over a commutative Noetherian ringR, (1) every neat-flat module is flat if and only if every absolutely coneat module is injective if and only ifR ≅ A × B, whereinAis aQF-ring andBis hereditary, and (2) every neat-flat module is absolutely coneat if and only if every absolutely coneat module is neat-flat if and only ifR ≅ A × B, whereinAis aQF-ring andBis Artinian withJ2(B) = 0. [ABSTRACT FROM AUTHOR]

In this article, ive study the characterizations of Gorenstein injective left S-moJutes and finitely generated Gorenstein projective left R-modutes when there is a dualizing S-R-himodiile associated with a right noetherian ring R und a left noetherian ring S. [ABSTRACT FROM AUTHOR]

LetRbe a ring. A leftR-moduleM(resp., rightR-moduleN) is called weak injective (resp., weak flat) if(resp.,) for every super finitely presented leftR-moduleF. By replacing finitely presented modules by super finitely presented modules, we may generalize many results of a homological nature from coherent rings to arbitrary rings. Some examples are given to show that weak injective (resp., weak flat) modules need not be FP-injective (resp., not flat) in general. In addition, we introduce and study the super finitely presented dimension (denote byl.sp.gldim(R)) ofRthat are defined in terms of only super finitely presented leftR-modules. Some known results are extended. [ABSTRACT FROM AUTHOR]

LetRbe a ring with an identity element. We prove thatRis right Kasch if and only if injective hull of every simple rightR-modules is neat-flat if and only if every absolutely pure rightR-module is neat-flat. A commutative ringRis hereditary and noetherian if and only if every absolutelys-pureR-module is injective andRis nonsingular. If every simple rightR-module is finitely presented, then (1)RRis absolutelys-pure if and only ifRis right Kasch and (2)Ris a right-CSring if and only if every pure injective neat-flat rightR-module is projective if and only if every absolutelys-pure leftR-module is injective andRis right perfect. We also study enveloping and covering properties of absolutelys-pure and neat-flat modules. The rings over which every simple module has an injective cover are characterized. [ABSTRACT FROM AUTHOR]

LetAbe a (not necessarily commutative) monoid object in an abelian symmetric monoidal category (C, ⊗,1) satisfying certain conditions. In this paper, we continue our study of the localizationMSof anyA-moduleMwith respect to a subsetS ⊆ HomA−Bimod(A,A) that is closed under composition. In particular, we prove the following theorem: ifPis anA-bimodule such thatPis symmetric as a bimodule over the centerZ(A) ofA, we have isomorphismsHH*(A,P)S ≅ HH*(A,PS) ≅ HH*(AS,PS) of Hochschild homology groups. [ABSTRACT FROM AUTHOR]

Vatne [13] and Green and Marcos [9] have independently studied the Koszul-like homological properties of graded algebras that have defining relations in degree 2 and exactly one other degree. We contrast these two approaches, answer two questions posed by Green and Marcos, and find conditions that imply the corresponding Yoneda algebras are generated in the lowest possible degrees. [ABSTRACT FROM PUBLISHER]

LetAbe a noetherian complete basic semiperfect algebra over an algebraically closed field, andC = A°be its dual coalgebra. IfAis Artin–Schelter regular, then the local cohomology ofAis isomorphic to a shift of twisted bimodule1Cσ*with σ a coalgebra automorphism. This yields that the balanced dualinzing complex ofAis a shift of the twisted bimoduleσ*A1. If σ is an inner automorphism, thenAis Calabi–Yau. An appendix is included to prove a duality theorem of the bounded derived category of quasi-finite comodules over an artinian coalgebra. [ABSTRACT FROM AUTHOR]

We extend the cotorsion dimension ofR-modules to unboundedR-complexes by applying the flat model structure onCh(R) proposed by J. Gillespie. This is not natural because there has been no sufficiently general result available for the existence of proper “cotorsion” resolutions of unbounded complexes, for which one would be able to define the derived functors. The global cotorsion dimension of ring is discussed in our present framework, and the relations between it and other dimensions are investigated as well. Some rings are characterized and some known results are extended. [ABSTRACT FROM PUBLISHER]

The results of [7] and [2] gave a recursive construction for all quasi-hereditary and standardly stratified algebras starting with local algebras and suitable bimodules. Using the notion of stratifying pairs of subcategories, introduced in [3], we generalize these earlier results to construct recursively all CPS-stratified algebras. [ABSTRACT FROM AUTHOR]

We study the existence of covers and envelopes by some special functors on the category of finitely presented modules. As an application, we characterize some important rings using these functors. We also investigate homological properties of some functors on the stable module category. The relationship between phantoms and Ext-phantoms is obtained. It is shown that every leftR-moduleMhas an Ext-phantom preenvelopef:M → Nwith coker(f) pure-projective. Finally, we prove that, as a torsionfree class of (mod-R, Ab), (mod-R, Ab) is generated by theFP-injective objects. [ABSTRACT FROM AUTHOR]

In this article, the concept of Gorenstein FP-injective modules and some related known results are generalized to Gorenstein FP-injective complexes. Moreover, some new characterizations of Gorenstein flat complexes are given. It is also proved that every complex has a Gorenstein flat preenvelope over coherent rings with finite self-FP-injective dimension. [ABSTRACT FROM AUTHOR]

LetRbe a ring and 𝒲 a self-orthogonal class of leftR-modules which is closed under finite direct sums and direct summands. A complexCof leftR-modules is called a 𝒲-complexif it is exact with each cycleZn(C) ∈ 𝒲. The class of such complexes is denoted by 𝒞𝒲. A complexCis calledcompletely𝒲-resolvedif there exists an exact sequence of complexesD·= … → D−1 → D0 → D1 → … with each termDiin 𝒞𝒲such thatC = ker(D0 → D1) andD·is both Hom(𝒞𝒲, −) and Hom(−, 𝒞𝒲) exact. In this article, we show thatC= … → C−1 → C0 → C1 → … is a completely 𝒲-resolved complex if and only ifCnis a completely 𝒲-resolved module for alln∈ ℤ. Some known results are obtained as corollaries. [ABSTRACT FROM AUTHOR]

LetRbe a ring. A leftR-moduleMis called strongly max-flat iffor every simple rightR-moduleSand everyi ≥ 1. In this article, we show that the class of strongly max-flatR-modules and its Ext-orthogonal class form a perfect and hereditary cotorsion pair. This result has some applications to the characterizations of rightSF, semisimple, left perfect, von Neumann regular and quasi-Frobenius rings. [ABSTRACT FROM AUTHOR]

Let R be a ring. A left R-module M is called GI-injective if for any Gorenstein injective left R-module N. It is shown that a left R-module M over any ring R is GI-injective if and only if M is a kernel of a Gorenstein injective precover f: A → B of a left R-module B with A injective. Suppose R is an n-Gorenstein ring, we prove that a left R-module M is GI-injective if and only if M is a direct sum of an injective left R-module and a reduced GI-injective left R-module. Then we investigate GI-injective dimensions of modules and rings. As applications, some new characterizations of the weak (Gorenstein) global dimension of coherent rings are given. [ABSTRACT FROM PUBLISHER]

Let R be a ring, n a fixed non-negative integer and ℱℐ n (ℱ n ) the class of all left (right) R-modules of FP-injective (flat) dimension at most n. A left R-module M (resp., right R-module N) is called n-FI-injective (resp., n-FI-flat) if (resp., ) for any F ∈ ℱℐ n . It is proved that a left R-module M is n-FI-injective if and only if M is a kernel of an ℱℐ n -precover f: A → B of a left R-module B with A injective. For a left coherent ring R, it is shown that a finitely presented right R-module M is n-FI-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. Some known results are extended. Finally, we investigate n-FI-injective and n-FI-flat modules over left coherent rings with FP-id( R R) ≤ n. [ABSTRACT FROM AUTHOR]

Let A be a connected-graded algebra with trivial module 𝕜, and let B be a graded Ore extension of A. We relate the structure of the Yoneda algebra E(A): = Ext A (𝕜, 𝕜) to E(B). Cassidy and Shelton have shown that when A satisfies their 𝒦2 property, B will also be 𝒦2. We prove the converse of this result. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM PUBLISHER]

We present a method for constructing the minimal injective resolution of a simple comodule of a path coalgebra of quivers with relations. Dual to the Calabi–Yau condition of algebras, we introduce the concept of a Calabi–Yau coalgebra, and then describe the Calabi–Yau coalgebras of low global dimensions. [ABSTRACT FROM PUBLISHER]

Let R be a right perfect ring, and let (ℱ, 𝒞) be a cotorsion theory in the category of right R-modules ℳ R . In this article, it is shown that every right R-module has a superfluous ℱ-cover if and only if there exists a torsion theory (𝒜, ℬ) such that (ℱ, 𝒞) is cogenerated by ℬ. It is also proved that if (𝒜, ℬ) is a cosplitting torsion theory, then (⊥ℬ, (⊥ℬ)⊥) is a hereditary and complete cotorsion theory, and if (𝒜, ℬ) is a centrally splitting torsion theory, then (⊥ℬ, (⊥ℬ)⊥) is a hereditary and perfect cotorsion theory. [ABSTRACT FROM AUTHOR]

In this article, we study the relation between m-strongly Gorenstein projective (resp., injective) modules and n-strongly Gorenstein projective (resp., injective) modules whenever m ≠ n, and the homological behavior of n-strongly Gorenstein projective (resp., injective) modules. We introduce the notion of n-strongly Gorenstein flat modules. Then we study the homological behavior of n-strongly Gorenstein flat modules, and the relation between these modules and n-strongly Gorenstein projective (resp., injective) modules. [ABSTRACT FROM AUTHOR]

A ring is of finite type if it has only finitely many maximal right ideals, all two-sided. In this article, we give a complete set of invariants for finite direct sums of cyclically presented modules over a ring R of finite type. More generally, our results apply to finite direct sums of direct summands of cyclically presented right R-modules (DCP modules). Using a duality, we obtain as an application a similar set of invariants for kernels of morphisms between finite direct sums of pair-wise non-isomorphic indecomposable injective modules over an arbitrary ring. This application motivates the study of DCP modules. [ABSTRACT FROM AUTHOR]

Let R be a left coherent ring. We first prove that a right R-module M is strongly copure flat if and only if Exti(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. [ABSTRACT FROM AUTHOR]

In this article, we generalize the characterization of Gorenstein flat complexes over Gorenstein rings to coherent rings with finite self-FP-injective dimension. [ABSTRACT FROM AUTHOR]