scholarly journals The New Results in Injective Modules

2021 ◽  
pp. 145-159
Author(s):  
Samira Hashemi ◽  
Feysal Hassani ◽  
Rasul Rasuli
Keyword(s):  
Injective Module ◽  
Injective Modules ◽  
Divisible Module

In this paper, we introduce and clarify a new presentation between the divisible module and the injective module. Also, we obtain some new results about them.

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.

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].

Quasi-pseudo Principally Injective Modules

2009 ◽  
Vol 16 (03) ◽  
pp. 397-402 ◽  
Author(s):  
Avanish Kumar Chaturvedi ◽  
B. M. Pandeya ◽  
A. J. Gupta
Keyword(s):  
Injective Module ◽  
Injective Modules

In this paper, the concept of quasi-pseudo principally injective modules is introduced and a characterization of commutative semi-simple rings is given in terms of quasi-pseudo principally injective modules. An example of pseudo M-p-injective module which is not M-pseudo injective is given.

scholarly journals On (co)pure Baer injective modules

2021 ◽  
Vol 31 (2) ◽  
pp. 219-226
Author(s):  
M. F. Hamid ◽  
Keyword(s):  
Left Ideal ◽  
Injective Module ◽  
Injective Modules

For a given class of R-modules Q, a module M is called Q-copure Baer injective if any map from a Q-copure left ideal of R into M can be extended to a map from R into M. Depending on the class Q, this concept is both a dualization and a generalization of pure Baer injectivity. We show that every module can be embedded as Q-copure submodule of a Q-copure Baer injective module. Certain types of rings are characterized using properties of Q-copure Baer injective modules. For example a ring R is Q-coregular if and only if every Q-copure Baer injective R-module is injective.

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.

Some remarks on Nonnil-coherent rings and ϕ-IF rings

2021 ◽  
pp. 2250211
Author(s):  
Wei Qi ◽  
Xiaolei Zhang
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Injective Module ◽  
Nilpotent Radical ◽  
Injective Modules ◽  
Flat Modules ◽  
Coherent Rings

Let [Formula: see text] be a commutative ring. If the nilpotent radical [Formula: see text] of [Formula: see text] is a divided prime ideal, then [Formula: see text] is called a [Formula: see text]-ring. In this paper, we first distinguish the classes of nonnil-coherent rings and [Formula: see text]-coherent rings introduced by Bacem and Ali [Nonnil-coherent rings, Beitr. Algebra Geom. 57(2) (2016) 297–305], and then characterize nonnil-coherent rings in terms of [Formula: see text]-flat modules, nonnil-injective modules and nonnil-FP-injective modules. A [Formula: see text]-ring [Formula: see text] is called a [Formula: see text]-IF ring if any nonnil-injective module is [Formula: see text]-flat. We obtain some module-theoretic characterizations of [Formula: see text]-IF rings. Two examples are given to distinguish [Formula: see text]-IF rings and IF [Formula: see text]-rings.

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.

SP-INJECTIVITY OF MODULES AND RINGS

2012 ◽  
Vol 05 (04) ◽  
pp. 1250053
Author(s):  
A. J. Gupta ◽  
B. M. Pandeya ◽  
A. K. Chaturvedi
Keyword(s):  
Injective Module ◽  
Endomorphism Rings ◽  
Injective Modules

Let >M and N be two R-modules. NR is called singular M-p-injective if for every singular M-cyclic submodule X of MR, every homomorphism from X to N can be extended to a homomorphism from M to N. MR is quasi-singular prinicipally injective if M is a singular M-p-injective module. It is shown that a ring R is right non-singular if and only if every right R-module is singular R-p-injective if and only if factors of singular R-p-injective modules are singular R-p-injective. A singular R-module M is injective if and only if M is N-sp-injective for every R-module N. Finally, we characterize quasi-sp-injective modules in terms of their endomorphism rings.

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.

On Divisibility and Injectivity

1966 ◽  
Vol 18 ◽  
pp. 901-919 ◽  
Author(s):  
Garry Helzer
Keyword(s):  
Integral Domain ◽  
Abelian Groups ◽  
Dedekind Domain ◽  
Divisible Group ◽  
Injective Module ◽  
Commutative Case ◽  
Zero Divisors ◽  
Divisible Module

In the category of abelian groups the concepts of divisible group and injective group coincide. In (4) this is generalized to modules over an integral domain and it is proved for a (commutative) integral domain that the concepts of divisible module and injective module coincide if and only if the ring is hereditary if and only if the ring is a Dedekind domain.In (8) the assumption of commutativity is dropped and the ring is assumed to have a one-sided field of quotients. The result obtained is essentially the same as in the commutative case; see the theorem following 6.2. In (13) the requirement that the ring have no zero-divisors is also dropped and the ring is assumed to possess what we have called an Ore ring (see the definition following 6.2). The result obtained is the equivalent of parts (a) and (b) of our 6.13.

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