scholarly journals The New Results in n-injective Modules and n-projective Modules

2021 ◽  
pp. 271-285
Author(s):  
Samira Hashemi ◽  
Feysal Hassani ◽  
Rasul Rasuli
Keyword(s):  
Exact Sequence ◽  
Projective Module ◽  
Injective Module ◽  
Projective Modules ◽  
Injective Modules

In this paper, we introduce and clarify a new presentation between the n-exact sequence and the n-injective module and n-projective module. Also, we obtain some new results about them.

scholarly journals On injectiveL-modules

2005 ◽  
Vol 2005 (5) ◽  
pp. 747-754 ◽  
Author(s):  
Paul Isaac
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Injective Module ◽  
Projective Modules ◽  
Module Theory ◽  
Direct Sum ◽  
Injective Modules ◽  
Free Modules

The concepts of free modules, projective modules, injective modules, and the like form an important area in module theory. The notion of free fuzzy modules was introduced by Muganda as an extension of free modules in the fuzzy context. Zahedi and Ameri introduced the concept of projective and injectiveL-modules. In this paper, we give an alternate definition for injectiveL-modules and prove that a direct sum ofL-modules is injective if and only if eachL-module in the sum is injective. Also we prove that ifJis an injective module andμis an injectiveL-submodule ofJ, and if0→μ→fv→gη→0is a short exact sequence ofL-modules, thenν≃μ⊕η.

Strict Mittag–Leffler modules over Gorenstein injective modules

2019 ◽  
Vol 19 (03) ◽  
pp. 2050050 ◽  
Author(s):  
Yanjiong Yang ◽  
Xiaoguang Yan
Keyword(s):  
Noetherian Ring ◽  
Direct Limit ◽  
Injective Module ◽  
Noetherian Rings ◽  
Projective Modules ◽  
Injective Modules ◽  
Pure Submodules ◽  
Covering Class ◽  
Gorenstein Injective ◽  
Finitely Presented

In this paper, we study the conditions under which a module is a strict Mittag–Leffler module over the class [Formula: see text] of Gorenstein injective modules. To this aim, we introduce the notion of [Formula: see text]-projective modules and prove that over noetherian rings, if a module can be expressed as the direct limit of finitely presented [Formula: see text]-projective modules, then it is a strict Mittag–Leffler module over [Formula: see text]. As applications, we prove that if [Formula: see text] is a two-sided noetherian ring, then [Formula: see text] is a covering class closed under pure submodules if and only if every injective module is strict Mittag–Leffler over [Formula: see text].

Stability of Gorenstein Classes of Modules

2013 ◽  
Vol 20 (04) ◽  
pp. 623-636 ◽  
Author(s):  
Samir Bouchiba
Keyword(s):  
Exact Sequence ◽  
Projective Modules ◽  
Natural Question ◽  
Alternate Proof ◽  
Similar Treatment ◽  
Arbitrary Ring ◽  
Injective Modules ◽  
Gorenstein Projective Modules ◽  
Category Of Modules ◽  
Gorenstein Injective

The purpose of this paper is to give, via totally different techniques, an alternate proof to the main theorem of [18] in the category of modules over an arbitrary ring R. In effect, we prove that this theorem follows from establishing a sequence of equalities between specific classes of R-modules. Actually, we tackle the following natural question: What notion emerges when iterating the very process applied to build the Gorenstein projective and Gorenstein injective modules from complete resolutions? In other words, given an exact sequence of Gorenstein injective R-modules G= ⋯ → G1→ G0→ G-1→ ⋯ such that the complex Hom R(H,G) is exact for each Gorenstein injective R-module H, is the module Im (G0→ G-1) Gorenstein injective? We settle such a question in the affirmative and the dual result for the Gorenstein projective modules follows easily via a similar treatment to that used in this paper. As an application, we provide the Gorenstein versions of the change of rings theorems for injective modules over an arbitrary ring.

scholarly journals Faithfully Exact Functors and Their Applications to Projective Modules and Injective Modules

1964 ◽  
Vol 24 ◽  
pp. 29-42 ◽  
Author(s):  
Takeshi Ishikawa
Keyword(s):  
General Theory ◽  
Noetherian Ring ◽  
Projective Module ◽  
Special Kind ◽  
Projective Modules ◽  
Flat Module ◽  
Commutative Case ◽  
Injective Modules ◽  
Equivalent Conditions

The aim of this paper is to study a property of a special kind of exact functors and give some applications to projective modules and injective modules.In section 1 we introduce the notion of faithfully exact functors [Definition 1] as a generalization of the functor T(X) = X⊗M, where M is a faithfully flat module, and give a property of this class of functors [Theorem 1.1]. Next, applying this general theory to functors ⊗ and Horn, we define the notion of faithfully projective modules [Definition 2] and faithfully injective modules [Definition 3]. In the commutative case “faithfully projective” means, however, simply “projective and faithfully flat” [Proposition 2.3]. In section 2, equivalent conditions for a projective module P to be faithfully projective are given [Theorem 2.2, Proposition 2. 3 and 2.4]. And a simpler proof is given to Y. Hinohara’s result [6] asserting that projective modules over an indecomposable weakly noetherian ring are faithfully flat [Proposition 2.5]. In section 3, we consider faithfully injective modules.

scholarly journals Decompositions of modules into projective modules and CS-modules

2000 ◽  
Vol 62 (1) ◽  
pp. 159-164
Author(s):  
Somyot Plubtieng
Keyword(s):  
Projective Module ◽  
Injective Module ◽  
Projective Modules ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Noetherian Module ◽  
Direct Sum ◽  
Continuous Module

Let M be a right R-module. It is shown that M is a locally Noetherian module if every finitely generated module in σ[M] is a direct sum of a projective module and a CS-module. Moreover, if every module in σ[M] is a direct sum of a projective module and a CS-module, then every module in σ[M] is a direct sum of modules which are either indecomposable projective or uniform Σ-quasi-injective. In particular, if every module in σ[M] is a direct sum of a projective module and a quasi-continuous module, then every module in σ[M] is a direct sum of a projective module and a quasi-injective module.

PERIODIC SHORT EXACT SEQUENCES AND PERIODIC PURE-EXACT SEQUENCES

2010 ◽  
Vol 09 (06) ◽  
pp. 859-870 ◽  
Author(s):  
SAMIR BOUCHIBA ◽  
MOSTAFA KHALOUI
Keyword(s):  
Exact Sequence ◽  
Finite Groups ◽  
Projective Module ◽  
Homological Algebra ◽  
Flat Module ◽  
Injective Modules ◽  
Flat Modules ◽  
Gorenstein Homological Algebra ◽  
Exact Sequences

Benson and Goodearl [Periodic flat modules, and flat modules for finite groups, Pacific J. Math.196(1) (2000) 45–67] proved that if M is a flat module over a ring R such that there exists an exact sequence of R-modules 0 → M → P → M → 0 with P a projective module, then M is projective. The main purpose of this paper is to generalize this theorem to any exact sequence of the form 0 → M → G → M → 0, where G is an arbitrary module over R. Moreover, we seek counterpart entities in the Gorenstein homological algebra of pure projective and pure injective modules.

ON M-C-INJECTIVE AND SELF-C-INJECTIVE MODULES

2010 ◽  
Vol 03 (03) ◽  
pp. 387-393 ◽  
Author(s):  
A. K. Chaturvedi ◽  
B. M. Pandeya ◽  
A. M. Tripathi ◽  
O. P. Mishra
Keyword(s):  
Endomorphism Ring ◽  
Projective Module ◽  
Injective Module ◽  
Injective Modules ◽  
Factor Module ◽  
Closed Submodule

Let M1 and M2 be two R-modules. Then M2 is called M1-c-injective if every homomorphism α from K to M2, where K is a closed submodule of M1, can be extended to a homomorphism β from M1 to M2. An R-module M is called self-c-injective if M is M-c-injective. For a projective module M, it has been proved that the factor module of an M -c-injective module is M -c-injective if and only if every closed submodule of M is projective. A characterization of self-c-injective modules in terms endomorphism ring of an R-module satisfying the CM-property is given.

Psq-injective modules and rings

2021 ◽  
pp. 2250101
Author(s):  
Avanish Kumar Chaturvedi ◽  
Sandeep Kumar
Keyword(s):  
Injective Module ◽  
Injective Modules

For any two right [Formula: see text]-modules [Formula: see text] and [Formula: see text], [Formula: see text] is said to be a ps-[Formula: see text]-injective module if, any monomorphism [Formula: see text] can be extended to [Formula: see text]. Also, [Formula: see text] is called psq-injective if [Formula: see text] is a ps-[Formula: see text]-injective module. We discuss some properties and characterizations in terms of psq-injective modules.

Model structures, recollements and duality pairs

2021 ◽  
Author(s):  
Wenjing Chen ◽  
Zhongkui Liu
Keyword(s):  
Projective Modules ◽  
Cotorsion Pair ◽  
Gorenstein Rings ◽  
Injective Modules ◽  
Flat Modules ◽  
Duality Pairs ◽  
Cotorsion Pairs ◽  
Gorenstein Flat ◽  
Model Structures

In this paper, we construct some model structures corresponding Gorenstein [Formula: see text]-modules and relative Gorenstein flat modules associated to duality pairs, Frobenius pairs and cotorsion pairs. By investigating homological properties of Gorenstein [Formula: see text]-modules and some known complete hereditary cotorsion pairs, we describe several types of complexes and obtain some characterizations of Iwanaga–Gorenstein rings. Based on some facts given in this paper, we find new duality pairs and show that [Formula: see text] is covering as well as enveloping and [Formula: see text] is preenveloping under certain conditions, where [Formula: see text] denotes the class of Gorenstein [Formula: see text]-injective modules and [Formula: see text] denotes the class of Gorenstein [Formula: see text]-flat modules. We give some recollements via projective cotorsion pair [Formula: see text] cogenerated by a set, where [Formula: see text] denotes the class of Gorenstein [Formula: see text]-projective modules. Also, many recollements are immediately displayed through setting specific complete duality pairs.

Modules close to the automorphism-invariant and coinvariant

2019 ◽  
Vol 18 (12) ◽  
pp. 1950235 ◽  
Author(s):  
Truong Cong Quynh ◽  
Adel Nailevich Abyzov ◽  
Nguyen Thi Thu Ha ◽  
Tülay Yildirim
Keyword(s):  
General Theory ◽  
General Setting ◽  
Projective Modules ◽  
Injective Modules

The aim of this paper is to introduce a general setting where some well-known results on essentially injective modules, automorphism-(co)invariant modules and small projective modules can be obtained by developing a general theory of modules which are (co)invariant under automorphisms of their covers and envelopes.

Sign in / Sign up

Export Citation Format

Share Document