Search

Let D be a domain with its quotient field K. Let K [ θ ] = K [ x ] / (x n) where x is an indeterminate and θ = x + (x n) . Denote by R n = D + θ K [ θ ] , which is a subring of K [ θ ] . This paper studies several properties of Rn: coherence, weakly factorial rings, strong Mori rings, rings with Noetherian spectrum, weak normality, and t-closedness. [ABSTRACT FROM AUTHOR]

We show that an N-complex of modules C is FP-injective if and only if C is N-exact and Z n i (C) is an FP-injective module for each i = 1 , 2 , ⋯ , N and each n ∈ Z by virtue of Gaussian binomial coefficients. Applications of this result go in three directions. Firstly, over a coherent ring, we prove that a bounded above N-complex C is FP-injective if and only if C is N-exact and C n is an FP-injective module for each n ∈ Z . Secondly, we obtain some examples of FP-injective N-complexes for some fixed integer N. Finally, we give a characterization of coherent rings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generalization of the well-known notion of weak coherent rings, which we call weak P -coherent rings. We investigate the stability of this property under direct products and homomorphic image, and its transfer to various contexts of constructions such as trivial ring extensions and amalgamated algebras along an ideal. We conclude with various examples of weak P -coherent rings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the FI-projective module and the FI-projective dimensions of modules and rings. An R-module M is called FI-projective if Ext1/R(M; F) = 0 for any FP-projective module F. We show that a module is FI-projective if and only if it is a cokernel of an FP-projective preenvelope f: A → B with B is projective. Also we see that rings over which all modules are FI-projective are exactly the semisimple rings. Furthermore, we study the FI-projective dimension of modules over short exact sequences and we characterize the hereditary ring using the FI-projective dimension of rings. Finally, we see the relation between these dimensions and other homological dimensions. [ABSTRACT FROM AUTHOR]

This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented modules over a right coherent ring is the completion of the category of perfect complexes. The result extends to non-affine noetherian schemes and gives rise to a direct construction of the singularity category. The parallel theory of completion for abelian categories is compatible with the completion of derived categories. There are three appendices. The first one by Tobias Barthel discusses the completion of perfect complexes for ring spectra. The second one by Tobias Barthel and Henning Krause refines for a separated noetherian scheme the description of the bounded derived category of coherent sheaves as a completion. The final appendix by Bernhard Keller introduces the concept of a morphic enhancement for triangulated categories and provides a foundation for completing a triangulated category. [ABSTRACT FROM AUTHOR]

We give, in a more general case than the Noetherian case, an answer to the question posed by Shaul on the flatness of the completed tensor product. [ABSTRACT FROM AUTHOR]

We investigate preenvelopes and precovers in the category R E of short exact sequences of left R-modules. Let C be a class of left R-modules. We prove that: (1) A 1 → δ 1 B 1 is a C -preenvelope in the category R-Mod of left R-modules if and only if any exact sequence 0 → A 1 → A 2 → A 3 → 0 has a C L -preenvelope (0 → A 1 → A 2 → A 3 → 0) → (δ 1 , δ 2 , δ 3) (0 → B 1 → B 2 → B 3 → 0) in R E , where C L = { 0 → X → Y → Z → 0 ∈ R E : X ∈ C } ; (2) If C is closed under pure submodules and direct products, then C LM = { 0 → X → Y → Z → 0 ∈ R E : X ∈ C , Y ∈ C } is a preenveloping class in R E ; (3) If C is a coresolving class and an enveloping class in R-Mod, then C LM is an enveloping class in R E ; (4) If C is closed under finite direct sums, then C is a preenveloping (resp. enveloping) class if and only if every short exact sequence in R E satisfying the functor Hom (- , C) leaves it exact has an C SE -preenvelope (resp. C SE -envelope), where C SE = { split exact sequences 0 → X → Y → Z → 0 ∈ R E : X ∈ C , Y ∈ C , Z ∈ C } . Similarly, we obtain the connections between precovers in R-Mod and precovers in R E . As applications, we characterize many rings such as perfect rings, coherent rings and Noetherian rings in terms of preenvelopes and precovers by some special short exact sequences. [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, we investigate the transfer of Matlis' semi-regularity and semi-coherence in trivial ring extensions issued from rings (with zero-divisors). We use the obtained results to enrich the literature with new examples of semi-regular or semi-coherent rings issued as trivial extensions and validate some questions left open in the literature. [ABSTRACT FROM AUTHOR]

In this article, we prove that a precovering class is closed under direct limits provided that is closed under direct products and pure submodules. As a corollary, we get that a ring R is left coherent if and only if the class of absolutely pure left R-modules is a precovering class. [ABSTRACT FROM AUTHOR]

The category of morphisms of R-modules is an expansion of the category of R-modules. Using some special morphisms of modules such as phantom, -phantom, flat and absolutely pure morphisms, we give some new characterizations of left coherent rings, von Neumann regular rings, left semihereditary rings, left Noetherian rings, and right perfect rings. [ABSTRACT FROM AUTHOR]

In this paper, a new homological dimension of modules, co-presented dimension, is defined. We study some basic properties of this homological dimension. Some ring extensions are considered, too. For instance, we prove that if S ≥ R is a finite normalizing extension and SR is a projective module, then for each right S-module MS, the cop-resented dimension of MS does not exceed the copresented dimension of HomR(S,M). [ABSTRACT FROM AUTHOR]

Let f: A → B be a ring homomorphism and let J be an ideal of B. In this paper, we inverstigate the weakly coherent property that the amalgamation A ...f J might inherit from the ring A for some classes of ideals J and homomorphisms f. Our results generates original examples which enrich the current literature with new families of examples of non-coherent weakly coherent rings. [ABSTRACT FROM AUTHOR]

In this paper, we prove that a ring R is left coherent if and only if the class 풜bs of absolutely pure left R-modules is a covering class. [ABSTRACT FROM AUTHOR]

In this article, we extend the notion of FP-injective modules to that of Cartan–Eilenberg complexes. We show that a complex $C$ is Cartan–Eilenberg FP-injective if and only if $C$ and $\text{Z}(C)$ are complexes consisting of FP-injective modules over right coherent rings. As an application, coherent rings are characterized in various ways, using Cartan–Eilenberg FP-injective and Cartan–Eilenberg flat complexes. [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]

A moduleMis called stronglyFP-injective if Exti(P,M) = 0 for any finitely presented modulePand alli≥1. (Pre)envelopes and (pre)covers by stronglyFP-injective modules are studied. We also use these modules to characterize coherent rings. An example is given to show that (strongly)FP-injective (pre)covers may fail to be exist in general. We also give an example of a module that isFP-injective but not stronglyFP-injective. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring. If P is a maximal ideal of R with a finitely generated power then we prove that P is finitely generated if R is either locally coherent or arithmetical or a polynomial ring over a ring of global dimension ≤ 2. And, if P is a prime ideal of R with a finitely generated power then we show that P is finitely generated if R is either a reduced coherent ring or a polynomial ring over a reduced arithmetical ring. These results extend a theorem of Roitman, published in 2001, on prime ideals of coherent integral domains. [ABSTRACT FROM AUTHOR]

In this paper, we prove an extension of Zaks' conjecture on integral domains with semi-regular proper homomorphic images (with respect to finitely generated ideals) to arbitrary rings (i.e., possibly with zero-divisors). The main result extends and recovers Levy's related result on Noetherian rings [23, Theorem] and Matlis' related result on Prüfer domains [26, Theorem] . It also globalizes Couchot's related result on chained rings [10, Theorem 11] . New examples of rings with semi-regular proper homomorphic images stem from the main result via trivial ring extensions. [ABSTRACT FROM AUTHOR]

It is shown that a ring is left semihereditary if and only each homomorphic image of its injective hull as left module is FP-injective. It is also proven that a commutative ring R is reduced and arithmetical if and only if E / U if FP-injective for any FP-injective R -module E and for any submodule U of finite Goldie dimension. A characterization of commutative rings for which each module of finite Goldie dimension is of injective dimension at most one is given. Let R be a chain ring and Z its subset of zerodivisors. It is proven that E / U is FP-injective for each FP-injective R -module E and each pure polyserial submodule U of E if R / I is complete in its f.c. topology for each ideal I whose the top prime ideal is Z . The converse holds if each indecomposable injective module whose the bottom prime ideal is Z contains a pure uniserial submodule. For some chain ring R we show that E / U is FP-injective for any FP-injective module E and any its submodule U of finite Goldie dimension, even if R is not coherent. It follows that any Archimedean chain ring is either coherent or maximal if and only if each factor of any injective module of finite Goldie dimension modulo a pure submodule is injective. [ABSTRACT FROM AUTHOR]

Let f : A → B be a ring homomorphism and let J be an ideal of B. In this paper, we investigate the transfer of the property of coherence to the amalgamation A ▷◁f J. We provide necessary and sufficient conditions for A ▷◁f J to be a coherent ring. [ABSTRACT FROM AUTHOR]

Let $${\mathcal{P}}$$ be an ideal of closed quotients of a completely regular frame L and $${\mathcal{R}_{\mathcal{P}}(L)}$$ the collection of all functions in the ring $${\mathcal{R}(L)}$$ whose support belong to $${\mathcal{P}}$$ . We show that $${\mathcal{R}(L)}$$ is a Noetherian ring if and only if $${\mathcal{R}(L)}$$ is an Artinian ring if and only if L is a finite frame. Using this result, we next show that if $${\mathcal{P}}$$ is the ideal of all compact closed quotients of L and L is $${\mathcal{P}}$$ -continuous, then $${\mathcal{R}_{\mathcal{P}}(L)}$$ is a Noetherian ring if and only if L is finite. Moreover, we show that L is a P-frame if and only if each ideal of $${\mathcal{R}(L)}$$ is of the form $${\mathcal{R}_{\mathcal{P}}(L)}$$ for some choice of $${\mathcal{P}}$$ . We furnish equivalent conditions for $${\mathcal{R}_{\mathcal{P}}(L)}$$ to be a prime ideal, a free ideal, and an essential ideal of $${\mathcal{R}(L)}$$ separately in terms of the cozero elements of L. Finally, we show that L is basically disconnected if and only if $${\mathcal{R}(L)}$$ is a coherent ring. [ABSTRACT FROM AUTHOR]

LetRbe a ring.Ris left coherent if each of its finitely generated left ideals is finitely presented.Ris called left generalized morphic if for every elementainR,l(a) = Rbfor someb ∈ R, wherel(a) denotes the left annihilator ofainR. The main aim of this article is to investigate the coherence and the generalized morphic property of the upper triangular matrix ringTn(R) (n ≥ 1). It is shown thatRis left coherent if and only ifTn(R) is left coherent for eachn ≥ 1 if and only ifTn(R) is left coherent for somen ≥ 1. And an equivalent condition is obtained forTn(R) to be left generalized morphic. Moreover, it is proved thatRis left coherent and left Bézout if and only ifTn(R) is left generalized morphic for eachn ≥ 1. [ABSTRACT FROM AUTHOR]

A ring R is called Gorenstein hereditary (G-hereditary) if every submodule of a projective module is Gorenstein projective (i.e. Ggldim(R) < 1). In this paper, we settle a question raised by Mahdou and Tamekkante in [On (strongly) Gorenstein (semi)hereditary rings, Arab. J. Sei. Eng. 36 (2011) 436] about the coherence of G-hereditary rings. It is shown that a ring R is Gorenstein semihereditary if and only if every finitely generated sub-module of a projective module is Gorenstein projective. As a consequence of this result, we have that every G-hereditary ring R is coherent. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and let M be an R -module which is generated by μ elements but not fewer. We denote by SL n (R) the group of the n × n matrices over R with determinant 1. We denote by E n (R) the subgroup of SL n (R) generated by the matrices which differ from the identity by a single off-diagonal coefficient. Given n ≥ μ and G ∈ { SL n (R) , E n (R) } , we study the action of G by matrix right-multiplication on V n (M) , the set of elements of M n whose components generate M. Assuming that M is finitely presented and that R is an elementary divisor ring or an almost local-global coherent Prüfer ring, we obtain a description of V n (M) / G which extends the author's earlier result on finitely generated modules over quasi-Euclidean rings [22]. [ABSTRACT FROM AUTHOR]

Let H be a finite dimensional Hopf algebra and A be an algebra over a fixed field k. Firstly, it is proved that the left FP-projective dimension is invariant under cleft extensions when H is semisimple and A is left coherent. Secondly, using (co)induction functors, we study the relations between FP-projective dimensions in A # H-Mod and the counterparts in AH-Mod. Finally, we characterize the FP-projective preenvelopes (resp., precovers) under H*-extensions and cleft extensions, respectively. [ABSTRACT FROM AUTHOR]

Using the facts that the disk algebra and the Wiener algebra are not coherent, we prove that the polydisc algebra, the ball algebra, and the Wiener algebra of the polydisc are not coherent. [ABSTRACT FROM AUTHOR]

This article continues to investigate a particular case of Gorenstein FP-injective modules, called strongly Gorenstein FP-injective modules. Some examples are given to show that strongly Gorenstein FP-injective modules lie strictly between FP-injective modules and Gorenstein FP-injective modules. Various results are developed, many extending known results in [1]. We also characterize FC rings in terms of strongly Gorenstein FP-injective, projective, and flat modules. [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]