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.

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.

Weakly-injective modules over hereditary noetherian prime rings

1995 ◽  
Vol 58 (3) ◽  
pp. 287-297 ◽  
Author(s):  
S. K. Jain ◽  
S. R. López-Permouth
Keyword(s):  
Simple Module ◽  
Injective Module ◽  
Injective Hull ◽  
Prime Rings ◽  
Direct Sum ◽  
Injective Modules ◽  
Hereditary Noetherian Prime Ring ◽  
Free Modules ◽  
Torsion Modules ◽  
Torsion Free

AbstractA module M is said to be wealdy-injective if and only if for every finitely generated submodule N of the injective hull E(M) of M there exists a submodule X of E(M), isomorphic to M such that N ⊂ X. In this paper we investigate weakly-injective modules over bounded hereditary noetherian prime rings. In particular we show that torsion-free modules over bounded hnp rings are always wealdy-injective, while torsion modules with finite Goldie dimension are weakly-injective only if they are injective.As an application, we show that weakly-injective modules over bounded Dedekind prime rings have a decomposition as a direct sum of an injective module B, and a module C satisfying that if a simple module S is embeddable in C then the (external) direct sum of all proper submodules of the injective hull of S is also embeddable in C. Indeed, we show that over a bounded hereditary noetherian prime ring every uniform module has periodicity one if and only if every weakly-injective module has such a decomposition.

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.

Injective modules over Mori domains

2003 ◽  
Vol 40 (1-2) ◽  
pp. 33-40
Author(s):  
L. Fuchs
Keyword(s):  
Injective Module ◽  
The Other ◽  
Injective Hull ◽  
Maximum Condition ◽  
Direct Sum ◽  
Injective Modules ◽  
Special Case ◽  
False Claim

Injective modules are considered over commutative domains. It is shown that every injective module admits a decomposition into two summands, where one of the summands contains an essential submodule whose elements have divisorial annihilator ideals, while the other summand contains no element with divisorial annihilator. In the special case of Mori domains (i.e., the divisorial ideals satisfy the maximum condition), the first summand is a direct sum of a S-injective module and a module that has no such summand. The former is a direct sum of indecomposable injectives, while the latter is the injective hull of such a direct sum. Those Mori domains R are characterized for which the injective hull of Q/R is S-injective (Q denotes the field of quotients of R) as strong Mori domains, correcting a false claim in the literature.

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.

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.

scholarly journals Relative injectivity and CS-modules

1994 ◽  
Vol 17 (4) ◽  
pp. 661-666
Author(s):  
Mahmoud Ahmed Kamal
Keyword(s):  
Direct Decomposition ◽  
Injective Hull ◽  
Extra Condition ◽  
Direct Sums ◽  
Direct Sum ◽  
Injective Modules ◽  
Semisimple Module ◽  
Continuous Module

In this paper we show that a direct decomposition of modulesM⊕N, withNhomologically independent to the injective hull ofM, is a CS-module if and only ifNis injective relative toMand both ofMandNare CS-modules. As an application, we prove that a direct sum of a non-singular semisimple module and a quasi-continuous module with zero socle is quasi-continuous. This result is known for quasi-injective modules. But when we confine ourselves to CS-modules we need no conditions on their socles. Then we investigate direct sums of CS-modules which are pairwise relatively inective. We show that every finite direct sum of such modules is a CS-module. This result is known for quasi-continuous modules. For the case of infinite direct sums, one has to add an extra condition. Finally, we briefly discuss modules in which every two direct summands are relatively inective.

scholarly journals Injective Modules Over Non-Artinian Serial Rings

1988 ◽  
Vol 44 (2) ◽  
pp. 242-251 ◽  
Author(s):  
David A. Hill
Keyword(s):  
Affirmative Answer ◽  
Primitive Idempotent ◽  
Injective Hull ◽  
Finitely Generated ◽  
Serial Ring ◽  
Direct Sum ◽  
Injective Modules

AbstractA module is uniserial if its lattice of submodules is linearly ordered, and a ring R is left serial if R is a direct sum of uniserial left ideals. The following problem is considered. Suppose the injective hull of each simple left R-module is uniserial. When does this imply that the indecomposable injective left R-modules are uniserial? An affirmative answer is known when R is commutative and when R is Artinian. The following result is proved.Let R be a left serial ring and suppose that for each primitive idempotent e, eRe has indecomposable injective left modules uniserial. The following conditions are equivalent. (a) The injective hull of each simple left R-module is uniserial. (b) Every indecomposable injective left R-module is univerial. (c) Every finitely generated left R-module is serial.The rest of the paper is devoted to a study of some non-Artinian serial rings which serve to illustrate this theorem.

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.

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