On Small Injective Rings and Modules

Thuyet, Le Van; Quynh, Truong Cong

doi:10.1142/s0219498809003436

Cited by 7 publications

(11 citation statements)

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“…Note that R is right minsymmetric. So, in view of [11,Lemma 2.2], R is right perfect. Then R is a right GP F ring by Theorem 2.15.…”

Section: Resultsmentioning

confidence: 99%

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Principally Small Injective Rings

Xiang¹

2011

Kyungpook mathematical journal

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A right ideal I of a ring R is small in case for every proper right ideal K of R, K + I = R. A right R-module M is called P S-injective if every R-homomorphism f : aR → M for every principally small right ideal aR can be extended to R → M . A ring R is called right P S-injective if R is P S-injective as a right R-module. We develop, in this article, P S-injectivity as a generalization of P -injectivity and small injectivity. Many characterizations of right P S-injective rings are studied. In light of these facts, we get several new properties of a right GP F ring and a semiprimitive ring in terms of right P S-injectivity. Related examples are given as well.

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“…Note that R is right minsymmetric. So, in view of [11,Lemma 2.2], R is right perfect. Then R is a right GP F ring by Theorem 2.15.…”

Section: Resultsmentioning

confidence: 99%

“…In recent decades, the generalizations of injective rings are extensively studied by many authors (see [3][4][7][8][9][10][11][12]). Let R be a ring.…”

Section: Introductionmentioning

confidence: 99%

Principally Small Injective Rings

Xiang¹

2011

Kyungpook mathematical journal

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“…(3) Every Z-module is ss-injective. In fact, if M is a Z-module, then M is small injective (by [19,Theorem 2.8] and hence it is ss-injective. (4) The two classes of principally small injective rings and ss-injective rings are different (see [15,Example 5.2], Example 4.4 and Example 5.6).…”

Section: Ss-injective Modulesmentioning

confidence: 99%

SS-Injective Modules and Rings

Tayyah,

Mehdi

2016

Preprint

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We introduce and investigate ss-injectivity as a generalization of both soc-injectivity and small injectivity. A module M is said to be ss-N-injective (where N is a module) if every R-homomorphism from a semisimple small submodule of N into M extends to N. A module M is said to be ss-injective (resp. strongly ss-injective), if M is ss-Rinjective (resp. ss-N-injective for every right R-module N). Some characterizations and properties of (strongly) ss-injective modules and rings are given. Some results of Amin, Yuosif and Zeyada on soc-injectivity are extended to ss-injectivity. Also, we provide some new characterizations of universally mininjective rings, quasi-Frobenius rings, Artinian rings and semisimple rings.

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“…(2) All finitely generated -modules are not injective and this follows from ( 1) and the fact that every non-trivial finitely generatedmodule is not injective [7, p.31]. Also, we have from [17,Theorem 2.8] that any -module is small injective.…”

Section: Introductionmentioning

confidence: 99%