Flat R-module Research Articles

Ding Projective and Ding Injective Modules

An R-module M is called Ding projective if there exists an exact sequence ⋯ → P1→ P0→ P0→ P1→ ⋯ of projective R-modules with M= Ker (P0→ P1) such that Hom (-,F) leaves the sequence exact whenever F is a flat R-module. In this paper, we develop some basic properties of such modules. Also, properties of Ding injective modules are discussed.

On Stability of Gorenstein Categories

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.

N-almost prime submodules

Let R be a commutative ring with identity. A proper submodule N of an R-module M will be called prime [resp. n-almost prime], if for r ∈ R and a ∈ M with ra ∈ N [resp. ra ∈ N \ (N: M) n−1 N], either a ∈ N or r ∈ (N: M). In this note we will study the relations between prime, primary and n-almost prime submodules. Among other results it is proved that: (1) If N is an n-almost prime submodule of an R-module M, then N is prime or N = (N: M)N, in case M is finitely generated semisimple, or M is torsion-free with dim R = 1. (2) Every n-almost prime submodule of a torsion-free Noetherian module is primary. (3) Every n-almost prime submodule of a finitely generated torsion-free module over a Dedekind domain is prime. (4) There exists a finitely generated faithful R-module M such that every proper submodule of M is n-almost prime, if and only if R is Von Neumann regular or R is a local ring with the maximal ideal m such that m 2 = 0. (5) If I is an n-almost prime ideal of R and F is a flat R-module with IF ≠ F, then IF is an n-almost prime submodule of F.

LOCAL HOMOLOGY AND GORENSTEIN FLAT MODULES

Let R be a commutative Noetherian ring, 𝔞 be an ideal of R and [Formula: see text] denote the derived category of R-modules. We investigate the theory of local homology in conjunction with Gorenstein flat modules. Let X be a homologically bounded to the right complex and Q be a bounded to the right complex of Gorenstein flat R-modules such that Q and X are isomorphic in [Formula: see text]. We establish a natural isomorphism LΛ𝔞(X) ≃ Λ𝔞(Q) in [Formula: see text] which immediately asserts that sup LΛ𝔞(X) ≤ Gfd RX. This isomorphism yields several conseQuences. For instance, in the case R possesses a dualizing complex, we show that Gfd RLΛ𝔞(X) ≤ Gfd RX. Also, we establish a criterion for regularity of Gorenstein local rings.

On Copure Projective Modules and Copure Projective Dimensions

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.

STABILITY OF GORENSTEIN FLAT MODULES

AbstractSather-Wagstaff et al. proved in [8] (S. Sather-Wagsta, T. Sharif and D. White, Stability of Gorenstein categories, J. Lond. Math. Soc.(2), 77(2) (2008), 481–502) that iterating the process used to define Gorenstein projective modules exactly leads to the Gorenstein projective modules. Also, they established in [9] (S. Sather-Wagsta, T. Sharif and D. White, AB-contexts and stability for Goren-stein at modules with respect to semi-dualizing modules, Algebra Represent. Theory14(3) (2011), 403–428) a stability of the subcategory of Gorenstein flat modules under a procedure to build R-modules from complete resolutions. In this paper we are concerned with another kind of stability of the class of Gorenstein flat modules via-à-vis the very Gorenstein process used to define Gorenstein flat modules. We settle in affirmative the following natural question in the setting of a left GF-closed ring R: Given an exact sequence of Gorenstein flat R-modules G = ⋅⋅⋅ G2G1G0G−1G−2 ⋅⋅⋅ such that the complex H ⊗RG is exact for each Gorenstein injective right R-module H, is the module M:= Im(G0 → G−1) a Gorenstein flat module?

On Strongly Copure Flat Modules and Copure Flat Dimensions

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.

When Are Covers Relative to Two Classes Isomorphic

Let ℒ1 ⊇ ℒ2 be classes of right R-modules, we introduce ℒ1-covering modules and projective modules relative to ℒ2 to characterize the relations between the existences of ℒ1-covers, ℒ2-covers and the classes ℒ1, ℒ2. As corollaries, every module in ℱ n is an ℱ-covering module if and only if every flat cover is an ℱ n -cover for each right module if and only if R is a von Neumann regular ring whenever wD(R) < ∞; every flat right R-module is projective relative to 𝒫 if and only if every flat cover is a projective cover for each right module if and only if R is right perfect.

AB-Contexts and Stability for Gorenstein Flat Modules with Respect to Semidualizing Modules

We investigate the properties of categories of GC-flat R-modules where C is a semidualizing module over a commutative noetherian ring R. We prove that the category of all GC-flat R-modules is part of a weak AB- context, in the terminology of Hashimoto. In particular, this allows us to deduce the existence of certain Auslander-Buchweitz approximations for R- modules of finite GC-flat dimension. We also prove that two procedures for building R-modules from complete resolutions by certain subcategories of GC- flat R-modules yield only the modules in the original subcategories.

Almost-Perfect Rings and Modules

Following [1], a ring R is called right almost-perfect if every flat right R-module is projective relative to R. In this article, we continue the study of these rings and will find some new characterizations of them in terms of decompositions of flat modules. Also we show that a ring R is right almost-perfect if and only if every right ideal of R is a cotorsion module. Furthermore, we prove that over a right almost-perfect ring, every flat module with superfluous radical is projective. Moreover, we define almost-perfect modules and investigate some properties of them.

STRONGLY GORENSTEIN FLAT MODULES

AbstractIn this paper, strongly Gorenstein flat modules are introduced and investigated. An R-module M is called strongly Gorenstein flat if there is an exact sequence ⋯→P1→P0→P0→P1→⋯ of projective R-modules with M=ker (P0→P1) such that Hom(−,F) leaves the sequence exact whenever F is a flat R-module. Several well-known classes of rings are characterized in terms of strongly Gorenstein flat modules. Some examples are given to show that strongly Gorenstein flat modules over coherent rings lie strictly between projective modules and Gorenstein flat modules. The strongly Gorenstein flat dimension and the existence of strongly Gorenstein flat precovers and pre-envelopes are also studied.

Rings Over Which Flat Covers of Finitely Generated Modules are Projective

In Bican et al. (2001), it is proved that all modules over an arbitrary ring have flat covers. In this article, we shall study rings over which flat covers of finitely generated modules are projective. We call a ring R right almost-perfect if every flat right R-module is projective relative to R. It turns out that a ring is right almost-perfect if and only if flat covers of finitely generated modules are projective. We shall show that the class of almost-perfect rings is properly between the class of perfect and semiperfect rings. We also outline some new characterizations of perfect rings. For example, we show that a ring R is right perfect if every finitely cogenerated right R-module has a projective cover.

Acyclicity of Complexes of Flat Modules

AbstractLet R be a noetherian commutative ring, anda complex of flat R-modules. We prove that if is acyclic for every ρ ϵ Spec R, then is acyclic, and H0() is R-flat. It follows that if is a (possibly unbounded) complex of flat R-modules and is exact for every ρ ϵ Spec R, then is exact for every R-complex . If, moreover, is a complex of projective R-modules, then it is null-homotopic (follows from Neeman’s theorem).

On Generalized Perfect Rings

We call a ring R “right generalized perfect” if every right R-module is a superfluous epimorphic image of a flat right R-module, that is, every right R-module has a flat cover. We shall study such rings and find some classes of modules which have flat covers.

Relative Cotorsion Modules and Relative Flat Modules

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 , 𝒞 n ) is a complete hereditary cotorsion theory, where ℱ n (resp. 𝒞 n ) denotes the class of all n-flat (resp. n-cotorsion) right R-modules. Several applications are given.

LEFT COTORSION RINGS

It is proved that if R is an associative ring that is cotorsion as a left module over itself, and J is the Jacobson radical of R, then the quotient ring R/J is a left self-injective von Neumann regular ring and idempotents lift modulo J. In particular, if R is indecomposable, then it is a local ring. Let R be an associative ring with identity. A left R-module RM is called cotorsion if Ext 1(F, M) = 0 for every flat left R-module RF . Cotorsion modules were introduced by Harrison [4], and are a generalization of pure-injective modules. Their interest derives from the recent result of Bican, El Bachir and Enochs (see [1] and [8, Theorem 3.4.6]), that every module admits a cotorsion envelope. The ring R is called left cotorsion if it is cotorsion when considered as a left R-module over itself. Left cotorsion rings are closely related to pure-injectivity from another point of view. The category R -Flat of flat left R-modules is a locally finitely presented additive category, and therefore admits a theory of purity, as propounded in [2]. By [5, Lemma 3], R is pure-injective as an object of R -Flat if and only if it is left cotorsion. The point of this paper is to extend to these kinds of rings, results of Zimmermann and Zimmermann-Huisgen [10] about rings that are pure-injective as left modules over themselves. It is proved (see Theorem 6) that if R is a left cotorsion ring and J is the Jacobson radical of R, then the quotient ring R/J is a von Neumann regular ring. Furthermore, we show (see Corollary 9) that R/J is injective as a left module over itself, and that idempotents lift modulo J (see Corollary 4). Every left pure-injective ring R may be realized as the endomorphism ring of an injective object of some Grothendieck category, so these results about left cotorsion rings may be seen as generalizations of classical results concerning the endomorphism ring of an injective module. However, let us remark that our results are a proper extension, since there exist many left cotorsion rings that are not left pure-injective. Take, for example, any left perfect ring that is not left pureinjective (see [9]). We should also point out that, as the endomorphism ring of any injective object of a Grothendieck category is always left pure-injective, standard functor category techniques cannot be used in the study of left cotorsion rings. We develop in this paper new techniques based on the behaviour of direct limits, whose ¯ ¯

On (ℵ, U)-Coherence of Modules and Rings

Let U be a flat right R-module and ℵ an infinite cardinal number. A left R-module M is said to be (ℵ, U)-coherent if every finitely generated submodule of every finitely generated M-projective module in σ[M] is (ℵ, U)-finitely presented in σ[M]. It is proved under some additional conditions that a left R-module M is (ℵ, U)-coherent if and only if $$ {\prod\nolimits_{i \in 1}^{\aleph } U } $$ is M-flat as a right R-module if and only if the (ℵ, U)-coherent dimension of M is equal to zero. We also give some characterizations of left (ℵ, U)-coherent dimension of rings and show that the left ℵ-coherent dimension of a ring R is the supremum of (ℵ, U)-coherent dimensions of R for all flat right R-modules U.

On representable linearly compact modules

For a flat R-module F, we prove that Hom R (F, -) is a functor from the category of linearly compact R-modules to itself and is exact. Moreover, Hom R (F, M) is representable when M is linearly compact and representable. This gives an affirmative answer to a question of L. Melkersson (1995) for linearly compact modules without the condition of finite Goldie dimension. The set of attached prime ideals of the co-localization Hom R (R S , M) of a linearly compact representable R-module M with respect to a multiplicative set S in R is described.

Flat and Stably Flat Modules

We study Kropholler's generalisation of Lazard's criterion for a module to be flat. The context is complete cohomology and modules of type (FP)∞. Let G be a group in the class h1F of groups which act on a finite dimensional contractible cell complex with finite stabilisers and let R be a strongly G-graded algebra. We provide a characterisation of the stably flat R-modules under the assumption that R1 is coherent of finite global dimension.

Complete flat modules

Let R be a commutative and noetherian ring. It is known tht if R is local with maximal ideal M and F is a flat R-module, then the Hausdorff completion F of F with the M-adic topology is flat. We show that if we assume that the Krull dimension of R is finite, then for any ideal I C R, the Hausdorff completion F* of a flat module F with the I-adic topology is flat. Furthermore, for a flat module F over such R, there is a largest ideal I such that F is Hausdorff and complete with the I-adic topology. For this I, the flat R/I-module F/IF will not be Hausdorff and complete with respect to the topology defined by any non-zero ideal of R/I. As a tool in proving the above, we will show that when R has finite Krull dimension, the I-adic Hausdorff completion of a minimal pure injective resolution of a flat module F is a minimal pure injective resolution of its completion F*. Then it will be shown that flat modules behave like finitely generated modules in the sense that on F* the I-adic and the completion topologies coincide, so F* is I-adically complete.