Copure direct injective modules

2021 ◽  
pp. 2250187
Author(s):  
Sanjeev Kumar Maurya ◽  
Sultan Eylem Toksoy
Keyword(s):  
Direct Summand ◽  
Projective Module ◽  
Injective Modules

In this paper, we have introduced copure-direct-injective modules. A right [Formula: see text]-module [Formula: see text] is said to be copure-direct-injective if every copure submodule of [Formula: see text] isomorphic to a direct summand of [Formula: see text] is itself a direct summand. We have studied properties of copure-direct-injective modules. We characterized rings over which every (cofinitely generated, free, projective) module is copure-direct-injective. We have examined for which rings or under what conditions copure-direct-injective modules are direct-injective, quasi-injective, copure-injective, injective. Also we have compared copure-direct-injective modules with pure-direct-injective modules.

Semiregular Modules and Rings

1976 ◽  
Vol 28 (5) ◽  
pp. 1105-1120 ◽  
Author(s):  
W. K. Nicholson
Keyword(s):  
Direct Summand ◽  
Homomorphic Image ◽  
Projective Module ◽  
Structure Theorem ◽  
Projective Cover ◽  
Finitely Generated

Mares [9] has called a projective module semiperfect if every homomorphic image has a projective cover and has shown that many of the properties of semiperfect rings can be extended to these modules. More recently Zelmanowitz [16] has called a module regular if every finitely generated submodule is a projective direct summand. In the present paper a class of semiregular modules is introduced which contains all regular and all semiperfect modules. Several characterizations of these modules are given and a structure theorem is proved. In addition several theorems about regular and semiperfect modules are extended.

scholarly journals Rings characterised by semiprimitive modules

1995 ◽  
Vol 52 (1) ◽  
pp. 107-116
Author(s):  
Yasuyuki Hirano ◽  
Dinh Van Huynh ◽  
Jae Keol Park
Keyword(s):  
Direct Summand ◽  
Projective Module ◽  
Injective Module ◽  
Direct Sum

A module M is called a CS-module if every submodule of M is essential in a direct summand of M. It is shown that a ring R is semilocal if and only if every semiprimitive right R-module is CS. Furthermore, it is also shown that the following statements are equivalent for a ring R: (i) R is semiprimary and every right (or left) R-module is injective; (ii) every countably generated semiprimitive right R-module is a direct sum of a projective module and an injective module.

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 totally projective QTAG-modules characterized by its submodules

2018 ◽  
Vol 36 (4) ◽  
pp. 77-86
Author(s):  
Ayazul Hasan ◽  
Mohd Rafiquddin
Keyword(s):  
Direct Summand ◽  
Projective Module ◽  
Special Cases ◽  
Nice System

A $QTAG$-module $M$ is called almost totally projective if it has a weak nice system. Here we show that the isotype submodules of a totally projective module which are almost totally projective are precisely those that are separable. From this characterization it follows that every balanced submodule of a totally projective module is almost totally projective. Finally, in some special cases we settle the question of whether a direct summand of an almost totally projective module is again almost totally projective.

ON MODULES WHICH ARE SUBISOMORPHIC TO THEIR PURE-INJECTIVE ENVELOPES

2002 ◽  
Vol 01 (03) ◽  
pp. 289-294
Author(s):  
MAHER ZAYED ◽  
AHMED A. ABDEL-AZIZ
Keyword(s):  
Direct Summand ◽  
Global Dimension ◽  
The Other ◽  
Direct Products ◽  
Direct Sums ◽  
Elementary Equivalence ◽  
Injective Modules ◽  
Reduced Products ◽  
Coherent Ring ◽  
Pure Injective Module

In the present paper, modules which are subisomorphic (in the sense of Goldie) to their pure-injective envelopes are studied. These modules will be called almost pure-injective modules. It is shown that every module is isomorphic to a direct summand of an almost pure-injective module. We prove that these modules are ker-injective (in the sense of Birkenmeier) over pure-embeddings. For a coherent ring R, the class of almost pure-injective modules coincides with the class of ker-injective modules if and only if R is regular. Generally, the class of almost pure-injective modules is neither closed under direct sums nor under elementary equivalence. On the other hand, it is closed under direct products and if the ring has pure global dimension less than or equal to one, it is closed under reduced products. Finally, pure-semisimple rings are characterized, in terms of almost pure-injective modules.

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 Direct sums of indecomposable injective modules

2000 ◽  
Vol 62 (1) ◽  
pp. 57-66
Author(s):  
Sang Cheol Lee ◽  
Dong Soo Lee
Keyword(s):  
Direct Summand ◽  
Direct Sums ◽  
Direct Sum ◽  
Injective Modules

This paper proves that every direct summand N of a direct sum of indecomposable injective submodules of a module is the sum of a direct sum of indecomposable injective submodules and a sum of indecomposable injective submodules of Z2(N).

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.

FPn-Injective, FPn-Flat Covers and Preenvelopes, and Gorenstein AC-Flat Covers

2018 ◽  
Vol 25 (02) ◽  
pp. 319-334
Author(s):  
Daniel Bravo ◽  
Sergio Estrada ◽  
Alina Iacob
Keyword(s):  
Direct Summand ◽  
Flat Module ◽  
Injective Modules ◽  
Flat Modules ◽  
Gorenstein Flat ◽  
Strongly Gorenstein Flat Modules

We prove that, for any n ≥ 2, the classes of FPn-injective modules and of FPn-flat modules are both covering and preenveloping over any ring R. This includes the case of FP∞-injective and FP∞-flat modules (i.e., absolutely clean and, respectively, level modules). Then we consider a generalization of the class of (strongly) Gorenstein flat modules, i.e., the (strongly) Gorenstein AC-flat modules (cycles of exact complexes of flat modules that remain exact when tensored with any absolutely clean module). We prove that some of the properties of Gorenstein flat modules extend to the class of Gorenstein AC-flat modules; for example, we show that this class is precovering over any ring R. We also show that (as in the case of Gorenstein flat modules) every Gorenstein AC-flat module is a direct summand of a strongly Gorenstein AC-flat module. When R is such that the class of Gorenstein AC-flat modules is closed under extensions, the converse is also true. Moreover, we prove that if the class of Gorenstein AC-flat modules is closed under extensions, then it is covering.

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