Weakly Continuous and C2-Rings

Nicholson, W. K.; Yousif, Mohamed

doi:10.1081/agb-100002399

Cited by 38 publications

(31 citation statements)

References 18 publications

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“…If M is semiregular and F is a submodule of M such that RadðMÞ F then M is F-semiregular. For M ¼ R and an ideal F ¼ I, I-semiregularity of rings is defined by Nicholson and Yousif (2001). Now we consider the module theoretic version of some results of Nicholson and Yousif.…”

Section: F-semiregular Modulesmentioning

confidence: 99%

“…M is called continous if M is CS and has (C 2 ) (Mohamed and Müller, 1990). A module M is said to be an ACS-module if for every element a 2 M, Ra ¼ P È S where P is projective and S is singular (Nicholson and Yousif, 2001).…”

Section: The Singular Submodule Z(m)mentioning

confidence: 99%

“…Zhou (2000) defines d-semiregular and d-semiperfect rings as a generalization of semiregular and semiperfect rings. On the other hand, Nicholson and Yousif (2001) consider I-semiregular rings for an ideal I of a ring R and study Zð R RÞ-semiregular rings. Now in this paper, we define F-semiregular modules M for a submodule F of a module M and consider some certain fully invariant submodules such as ZðMÞ; SocðMÞ; dðMÞ (is defined in Zhou, 2000).…”

Section: Introductionmentioning

confidence: 99%

“…Let M ¼ R ¼ Z 8 . Then R is a self-injective ring, JðRÞ ¼ ZðRÞ ¼ 2R and SocðRÞ ¼ 4R: Hence R is a ZðRÞ-semiregular ring by Nicholson and Yousif (2001) but not SocðRÞ-semiregular since JðRÞ-semiregular is not contained in SocðRÞ.…”

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Semiregular Modules with Respect to a Fully Invariant Submodule

Alkan

Özcan

2004

Communications in Algebra

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Let M be a left R-module and F a submodule of M for any ring R. We call M F-semiregular if for every x 2 M, there exists a decomposition M ¼ A È B such that A is projective, A Rx and Rx \ B F. This definition extends several notions in the literature. We investigate some equivalent conditions to F-semiregular modules and consider some certain fully invariant submodules such as ZðMÞ; SocðMÞ; dðMÞ. We prove, among others, that if M is a finitely generated projective module, then M is quasi-injective if and only if M is ZðMÞ-semiregular and M È M is CS. If M is projective SocðMÞ-semiregular module, then M is semiregular. We also characterize QF-rings R with JðRÞ 2 ¼ 0.

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Section: F-semiregular Modulesmentioning

confidence: 99%

Section: The Singular Submodule Z(m)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Semiregular Modules with Respect to a Fully Invariant Submodule

Alkan

Özcan

2004

Communications in Algebra

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“…Following [14], Module M is said to be an ACS-module if for every element a ∈ M, aR = P ⊕ S, where P is projective and S is singular. ACS stands for annihilator-CS and this property is named so because (by Lemma 2.9 of [14]) the condition is equivalent to saying that for every a ∈ M, the right annihilator ideal r(a) is essential in a direct summand of M. Following [19], module M is called a weak CS-module provided that for each semisimple submodule S of M there exists a direct summand K of M such that S is essential in K. We generalize this definition slightly; an R-module M is called SCS if every closed simple module is a direct summand of M. Then, clearly a weak CS-module is SCS and any summand of an SCS module is SCS.…”

Section: Theorem 52 If a Right Artinian Ring R Has Only One Simple mentioning

confidence: 99%