scholarly journals Strongly K-nonsingular Modules

2019 ◽  
Vol 32 (1) ◽  
pp. 173
Author(s):  
Tha'ar Younis Ghawi
Keyword(s):  
Direct Summand ◽  
Direct Sums ◽  
Nonzero Intersection ◽  
Small Submodule

       A submodule N of a module M  is said to be s-essential if it has nonzero intersection with any nonzero small submodule in M. In this article, we introduce and study a class of modules in which all its nonzero endomorphisms have non-s-essential kernels, named, strongly -nonsigular. We investigate some properties of strongly -nonsigular modules. Direct summand, direct sums and some connections of such modules are discussed.        

Direct Sums of Hollow-lifting Modules

2012 ◽  
Vol 19 (01) ◽  
pp. 87-97
Author(s):  
Yongduo Wang ◽  
Dejun Wu
Keyword(s):  
Direct Summand ◽  
Direct Sums ◽  
Direct Sum ◽  
Small Submodule

A module M over a ring R is called hollow-lifting if every submodule N of M with M/N hollow contains a direct summand K of M such that N/K is a small submodule of M/K. It is known that a (finite) direct sum of hollow-lifting modules need not be hollow-lifting. In this paper, we show that a quotient of a hollow-lifting module is not hollow-lifting in general, and discuss when a direct sum of hollow-lifting modules is hollow-lifting. We also study relative properties of hollow-lifting modules and investigate direct sums of X-hollow-lifting modules.

scholarly journals Semi-T-Hollow Modules and Semi-T-Lifting Modules

2021 ◽  
pp. 2357-2361
Author(s):  
Alaa A. Elewi
Keyword(s):  
Direct Summand ◽  
Associative Ring ◽  
Left Module ◽  
Small Submodule

Let be an associative ring with identity and let be a unitary left -module. Let  be a non-zero submodule of .We say that  is a semi- - hollow module if for every submodule  of  such that  is a semi- - small submodule ( ). In addition, we say that  is a semi- - lifting module if for every submodule  of , there exists a direct summand  of  and  such that   The main purpose of this work was to develop the properties of these classes of module.  

Strongly Gorenstein categories

2021 ◽  
pp. 2250213
Author(s):  
Wan Wu ◽  
Zenghui Gao
Keyword(s):  
Direct Summand ◽  
Full Subcategory ◽  
Sufficient Conditions ◽  
Abelian Category ◽  
Direct Products ◽  
Direct Sums ◽  
Gorenstein Categories

We introduce and study strongly Gorenstein subcategory [Formula: see text], relative to an additive full subcategory [Formula: see text] of an abelian category [Formula: see text]. When [Formula: see text] is self-orthogonal, we give some sufficient conditions under which the property of an object in [Formula: see text] can be inherited by its subobjects and quotient objects. Then, we introduce the notions of one-sided (strongly) Gorenstein subcategories. Under the assumption that [Formula: see text] is closed under countable direct sums (respectively, direct products), we prove that an object is in right (respectively, left) Gorenstein category [Formula: see text] (respectively, [Formula: see text]) if and only if it is a direct summand of an object in right (respectively, left) strongly Gorenstein subcategory [Formula: see text] (respectively, [Formula: see text]). As applications, some known results are obtained as corollaries.

On coseparable and 𝔪-coseparable modules

2020 ◽  
pp. 2150031
Author(s):  
Rachid Ech-chaouy ◽  
Abdelouahab Idelhadj ◽  
Rachid Tribak
Keyword(s):  
Direct Summand ◽  
Noetherian Rings ◽  
Direct Products ◽  
Finitely Generated ◽  
Direct Sums ◽  
Direct Sum ◽  
Free Modules

A module [Formula: see text] is called coseparable ([Formula: see text]-coseparable) if for every submodule [Formula: see text] of [Formula: see text] such that [Formula: see text] is finitely generated ([Formula: see text] is simple), there exists a direct summand [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is finitely generated. In this paper, we show that free modules are coseparable. We also investigate whether or not the ([Formula: see text]-)coseparability is stable under taking submodules, factor modules, direct summands, direct sums and direct products. We show that a finite direct sum of coseparable modules is not, in general, coseparable. But the class of [Formula: see text]-coseparable modules is closed under finite direct sums. Moreover, it is shown that the class of coseparable modules over noetherian rings is closed under finite direct sums. A characterization of coseparable modules over noetherian rings is provided. It is also shown that every lifting (H-supplemented) module is coseparable ([Formula: see text]-coseparable).

On a class of separable modules

2021 ◽  
pp. 2250034
Author(s):  
Rachid Ech-chaouy ◽  
Abdelouahab Idelhadj ◽  
Rachid Tribak
Keyword(s):  
Direct Summand ◽  
Finitely Generated ◽  
Direct Sums ◽  
Simple Modules ◽  
Direct Sum ◽  
The Stability

A module [Formula: see text] is called [Formula: see text]-separable if every proper finitely generated submodule of [Formula: see text] is contained in a proper finitely generated direct summand of [Formula: see text]. Indecomposable [Formula: see text]-separable modules are shown to be exactly the simple modules. While direct summands of an [Formula: see text]-separable module do not inherit the property, in general, the question of the stability under direct sums is unanswered. But we obtain some partial answers. It is shown that any infinite direct sum of [Formula: see text]-separable modules is [Formula: see text]-separable. Also, we prove that if [Formula: see text] and [Formula: see text] are [Formula: see text]-separable modules such that [Formula: see text] is [Formula: see text]-projective, then [Formula: see text] is [Formula: see text]-separable. We conclude the paper by providing some characterizations of several classes of rings in terms of [Formula: see text]-separable modules. Among others, we prove that the class of rings [Formula: see text] for which every (injective) [Formula: see text]-module is [Formula: see text]-separable is exactly that of semisimple rings.

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 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).

scholarly journals On Rad-D12 Modules

2013 ◽  
Vol 21 (1) ◽  
pp. 201-208
Author(s):  
Yahya Talebi ◽  
Ali Reza Moniri Hamzekolaee ◽  
Derya Keskin Tütüncü
Keyword(s):  
Direct Summand ◽  
Direct Sums ◽  
Direct Sum

Abstract Let M be a right R-module. We call M Rad-D12, if for every sub- module N of M, there exist a direct summand K of M and an epimor- phism α : K → M/N such that Kererα ⊆ Rad(K). We show that a direct summand of a Rad-D12 module need not be a Rad-D12 module. We investigate completely Rad-D12 modules (modules for which every direct summand is a Rad-D12 module). We also show that a direct sum of Rad-D12 modules need not be a Rad-D12 module. Then we deal with some cases of direct sums of Rad-D12 modules.

scholarly journals On strongly coseparable modules

2021 ◽  
Author(s):  
Rachid Ech-chaouy ◽  
Abdelouahab Idelhadj ◽  
Rachid Tribak
Keyword(s):  
Direct Summand ◽  
Noetherian Ring ◽  
Commutative Rings ◽  
Free Module ◽  
Finitely Generated ◽  
Direct Sums ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Regular Rings

AbstractA module M is called $$\mathfrak {s}$$ s -coseparable if for every nonzero submodule U of M such that M/U is finitely generated, there exists a nonzero direct summand V of M such that $$V \subseteq U$$ V ⊆ U and M/V is finitely generated. It is shown that every non-finitely generated free module is $$\mathfrak {s}$$ s -coseparable but a finitely generated free module is not, in general, $$\mathfrak {s}$$ s -coseparable. We prove that the class of $$\mathfrak {s}$$ s -coseparable modules over a right noetherian ring is closed under finite direct sums. We show that the class of commutative rings R for which every cyclic R-module is $$\mathfrak {s}$$ s -coseparable is exactly that of von Neumann regular rings. Some examples of modules M for which every direct summand of M is $$\mathfrak {s}$$ s -coseparable are provided.

Endoprime modules and their direct sums

2018 ◽  
Vol 17 (08) ◽  
pp. 1850155 ◽  
Author(s):  
Gangyong Lee ◽  
S. Tariq Rizvi
Keyword(s):  
Direct Summand ◽  
Prime Ring ◽  
Endomorphism Ring ◽  
Matrix Ring ◽  
Direct Sums ◽  
Direct Sum ◽  
Finite Matrix ◽  
Reduced Ring ◽  
Baer Modules ◽  
Special Classes

The purpose of this paper is to further study the endoprime modules as one of the special classes of quasi-Baer modules. As a module theoretic analogue of a prime ring, we characterize an endoprime module via its endomorphism ring and a weak retractability condition. It is shown that any direct summand of an endoprime module is an endoprime module. A characterization is obtained when a direct sum of endoprime modules is an endoprime module. It is well known that every prime ring is semicentral reduced. We prove that a column (and row) finite matrix ring over a semicentral reduced ring is also a semicentral reduced ring. Consequently, it is shown that a column (and row) finite matrix ring over a prime ring is prime. Applications and examples illustrating our results are provided.

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