Class Of Left R-modules Research Articles

Unifying some classical results on Artinian rings and modules

In this note, we introduce a very crude but natural notion of measure on the class of left R-modules over a ring R. We use this notion to give short proofs of some classical theorems on (left) Artinian rings and modules, due to Akizuki, Anderson, Hopkins, and Levitzki, as well as of some new results.

Superstability, noetherian rings and pure-semisimple rings

We uncover a connection between the model-theoretic notion of superstability and that of noetherian rings and pure-semisimple rings.We characterize noetherian rings via superstability of the class of left modules with embeddings. Theorem 0.1For a ring R the following are equivalent.(1)R is left noetherian.(2)The class of left R-modules with embeddings is superstable.(3)For everyλ≥|R|+ℵ0, there isχ≥λsuch that the class of left R-modules with embeddings has uniqueness of limit models of cardinality χ.(4)Every limit model in the class of left R-modules with embeddings is Σ-injective.We characterize left pure-semisimple rings via superstability of the class of left modules with pure embeddings. Theorem 0.2For a ring R the following are equivalent.(1)R is left pure-semisimple.(2)The class of left R-modules with pure embeddings is superstable.(3)There existsλ≥(|R|+ℵ0)+such that the class of left R-modules with pure embeddings has uniqueness of limit models of cardinality λ.(4)Every limit model in the class of left R-modules with pure embeddings is Σ-pure-injective.Both equivalences provide evidence that the notion of superstability could shed light in the understanding of algebraic concepts.As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both.

Gorenstein flat modules with respect to duality pairs

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

Resolutions and Stability of Gorenstein Classes of Modules

Let $$\mathcal {X}$$, $$\mathcal {U}$$ and $$\mathcal {Z}$$ be classes of left R-modules, $$\mathcal {W}$$ be a class of right R-modules and $$\mathcal {X}\subseteq \mathcal {U}$$. In this paper, we investigate the relationship between $$\mathcal {Z}$$-proper (resp. $$\mathcal {Z}$$-coproper, $$\mathcal {W}$$-pure) $$\mathcal {U}$$-(co)resolutions and $$\mathcal {Z}$$-proper (resp. $$\mathcal {Z}$$-coproper, $$\mathcal {W}$$-pure) $$\mathcal {X}$$-(co)resolutions. As applications, we give an affirmative answer to the stability question on Gorenstein classes of modules, which unify the corresponding results possessed by some known Gorenstein classes of modules.

On Right Orthogonal Classes and Cohomology Over Ding–Chen Rings

In this paper, we investigate the properties of right orthogonal modules of \({{\mathscr {C}}}\), where \({{\mathscr {C}}}\) is a class of left R-modules. As an application, we investigate the properties of right orthogonal modules of Ding injective left R-modules, and present various characterizations of semisimple and von Neumann regular rings and so on. Moreover, we also consider another cohomology, strong Tate cohomology, which connects the usual cohomology with the Ding cohomology.

(𝒲,𝒴,𝒳)-Gorenstein complexes

ABSTRACTLet 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 are 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.

Completely 𝒲-Resolved Complexes

Let R be a ring and 𝒲 a self-orthogonal class of left R-modules which is closed under finite direct sums and direct summands. A complex C of left R-modules is called a 𝒲-complex if it is exact with each cycle Z n (C) ∈ 𝒲. The class of such complexes is denoted by 𝒞𝒲. A complex C is called completely 𝒲-resolved if there exists an exact sequence of complexes D · = … → D −1 → D 0 → D 1 → … with each term D i in 𝒞𝒲 such that C = ker(D 0 → D 1) and D · is both Hom(𝒞𝒲, −) and Hom(−, 𝒞𝒲) exact. In this article, we show that C = … → C −1 → C 0 → C 1 → … is a completely 𝒲-resolved complex if and only if C n is a completely 𝒲-resolved module for all n ∈ ℤ. Some known results are obtained as corollaries.

RESOLUTIONS AND DIMENSIONS OF RELATIVE INJECTIVE MODULES AND RELATIVE FLAT MODULES

Abstract. 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 Ext 1 (P,M) = 0for any (m,n)-presented right R-module P; M is said to be (m,n)-flatif Tor 1 (N,P) = 0 for any (m,n)-presented left R-module P. In termsof some derived functors, relative injective or relative flat resolutions anddimensions are investigated. As applications, some new characterizationsof von Neumann regular rings and p.p. rings are given. 1. IntroductionLet C be a class of left R-modules and M a left R-module. Following ([7]),we say that a homomorphism ϕ : M → C is a C-preenvelope of M if C ∈ C andthe abelian group homomorphism Hom(ϕ,C ′ ) : Hom (C,C ′ ) → Hom (M,C ′ ) issurjective for each C ′ ∈ C. A C-preenvelope ϕ : M → C is called a C-envelopeif every endomorphism f : C → C such that fϕ = ϕ is an isomorphism. AC-envelope ϕ : M → C is said to have the unique mapping property (see[6]) if for any homomorphism f : M → C ′ with C ′ ∈ C, there is a uniquehomomorphism g : C → C

NOTES ON STABILITY OF GORENSTEIN CATEGORIES

Let [Formula: see text] be a class of left R-modules that contains all projective left R-modules. We study a kind of stability of [Formula: see text]-Gorenstein projective modules and settle in affirmative the following question: given an exact sequence of [Formula: see text]-Gorenstein projective left R-modules G = ⋯ → G1 → G0 → G0 → G1 → ⋯ such that the complex Hom R(G, F) is exact for each left R-module [Formula: see text], is the module [Formula: see text]-Gorenstein projective? Some applications are also given.

Baer cotorsion Pairs

LetR be a unital associative ring and $$\mathfrak{V},\mathfrak{W}$$ two classes of leftR-modules. In [St3] the notion of a ( $$\mathfrak{V},\mathfrak{W}$$ ) pair was introduced. In analogy to classical cotorsion pairs, a pair (V,W) of subclasses $$\mathcal{V} \subseteq \mathfrak{V} and \mathcal{W} \subseteq \mathfrak{W}$$ is called a ( $$\mathfrak{V},\mathfrak{W}$$ ) pair if it is maximal with respect to the classes $$\mathfrak{V},\mathfrak{W}$$ and the condition Ext R 1 (V, W)=0 for all $$V \in \mathcal{V} and W \in \mathcal{W}$$ . In this paper we study $$\mathfrak{T}\mathfrak{f},\mathfrak{T}$$ pairs whereR = ℤ and $$\mathfrak{T}\mathfrak{f}$$ is the class of all torsion-free abelian groups andT is the class of all torsion abelian groups. A complete characterization is obtained assumingV=L. For example, it is shown that every $$\mathfrak{T}\mathfrak{f},\mathfrak{T}$$ pair is singly cognerated underV=L.

$${\left( {\mathfrak{V},\mathfrak{W}} \right)}{\text{-cotorsion pairs}}$$

Let R be a unital associative ring and \(\mathfrak{V},\mathfrak{W}\) two classes of left R-modules. In this paper we introduce the notion of a \({\left( {\mathfrak{V},\mathfrak{W}} \right)}{\text{-cotorsion pair}}{\text{.}}\) In analogy to classical cotorsion pairs as defined by Salce [10], a pair \({\left( {\mathcal{V},\mathcal{W}} \right)}\) of subclasses \(\mathcal{V} \subseteqq \mathfrak{V}\) and \(\mathcal{W} \subseteqq \mathfrak{W}\) is called a \({\left( {\mathfrak{V},\mathfrak{W}} \right)}{\text{-cotorsion pair}}\) if it is maximal with respect to the classes \({\mathfrak{V},\mathfrak{W}}\) and the condition \({\text{Ext}}^1_R (V, W) = 0\) for all \(V \in \mathcal{V}\) and \(W \in \mathcal{W}.\) Basic properties of \({\left( {\mathfrak{V},\mathfrak{W}} \right)}{\text{-cotorsion pairs}}\) are stated and several examples in the category of abelian groups are studied.

Balance in Generalized Tate Cohomology #

ABSTRACT We consider two preenveloping classes of left R-modules ℐ, ℰ such that Inj ⊂ ℐ ⊂ ℰ, and a left R-module N. For any left R-module M and n ≥ 1 we define the relative extension modules (M, N) and prove the existence of an exact sequence connecting these modules and the modules (M, N) and (M, N). We show that there is a long exact sequence of (M, −) associated with a Hom(−, ℰ) exact sequence 0 → N′ → N → N′′ → 0 and a long exact sequence of (−, N) associated with a Hom(−, ℰ) exact sequence 0 → M′ → M → M′′ → 0. Using these properties we prove that for two complete hereditary cotorsion theories (𝒞, ℒ), (ℒ, ℰ) we have (M, N) for any left R modules M, N and for any n ≥ 1, where (M, N) are the generalized Tate cohomology modules (see Section 1 for the definition). So in this case we have an occurrence of balance, i.e. the generalized Tate cohomology can be computed either using a left 𝒞-resolution and a projective resolution of M or using a right ℰ-resolution and an injective resolution of N.

INJECTIVE PRECOVERS AND MODULES OF GENERALIZED INVERSE POLYNOMIALS

This paper is motivated by S. Park [10] in which the injective cover of left R[x]-module M[x-1] of inverse polynomials over a left R-module M was discussed. The author considers the Ω-covers of modules and shows that if η: P → M is an Ω-cover of M, then [ηS,≤] : [PS,≤] → [MS,≤] is an [ΩS,≤]-cover of left [[RS,≤]]-module [MS,≤], where Ω is a class of left R-modules and [MS,≤] is the left [[RS,≤]]-module of generalized inverse polynomials over a left R-module M. Also some properties of the injective cover of left [[RS,≤]]-module [MS,≤] are discussed.

THE LARGEST NON-TRIVIAL TORSION CLASS OF MODULES

ABSTRACT In this paper we investigate the following two classes of left R-modules: N(P) ={A|A has no non-zero direct summand P e P} and H(p) = {A} if B ⋚ A with B e N(P), then B = 0}, where P is a class of projective R-modules. We demonstrate that N(p) is, in general, not a torsion class but that H(P) is always a torsionfree class. We also investigate those classes P and rings R for which N(P) is the largest non-trivial torsion class of R-modules.

FILTERED-INJECTIVE AND COFLAT MODULES

Abstract For an arbitrary left R-module M, we denote by F(M) the class of left R-modules F such that for any exact sequence 0 → A α→ B of left R-modules and any R-homomorphism β: A → M factoring through F, there exists an R- homomorphism γ: B → M such that β = γα. For any given class R of left R-modules, we denote ∩EϵR F(M) by F(R) or simply by 9 if the context is clear. The class of short exact sequences E of left R-modules relative to which each ME'JR has the injective property, is denoted by E(R) or just &. Relative properties of RR, F and E are investigated for a given class R. The special case where JR is the class of all pure-injective left R-modules is explored. In this way the class F of coflat left R-modules is introduced and it is pointed out that a module is coflat if and only if it is absolutely pure.

Injective Hulls of Torsion Free Modules

In § 1, we begin with a basic theorem which describes a convenient embedding of a nonsingular left R-module into a complete direct product of copies of the left injective hull of R (Theorem 2). Several applications follow immediately. Notably, the injective hull of a finitely generated nonsingular left R-module is isomorphic to a direct sum of injective hulls of closed left ideals of R (Corollary 4). In particular, when R is left self-injective, every finitely generated nonsingular left R-module is isomorphic to a finite direct sum of injective left ideals (Corollary 6).In § 2, where it is assumed for the first time that rings have identity elements, we investigate more generally the class of left R-modules which are embeddable in direct products of copies of the left injective hull Q of R. Such modules are called torsion free, and can also be characterized by the property that no nonzero element is annihilated by a dense left ideal of R (Proposition 12).