Modules Satisfying Extension Conditions Under Monomorphism of Their Closed Submodules

A right [Formula: see text]-module [Formula: see text] is called to satisfy condition [Formula: see text] if, for every [Formula: see text] and [Formula: see text], there exists [Formula: see text] such that [Formula: see text] and if [Formula: see text] is a direct summand of [Formula: see text], then [Formula: see text] is a direct summand of [Formula: see text]. In this paper, we give some properties of rings and modules to satisfy condition [Formula: see text]. Moreover, their connections with von Neumann regular rings, Hereditary rings, Noetherian rings and (semi)artinian rings are addressed.

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New characterizations of quasi-Frobenius rings

Thuyet

Diallo

Diop

et al. 2023

Asian-European J. Math.

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In this paper, we firstly provide several new characterizations of quasi-Frobenius rings by using some generalized injectivity of rings with certain chain conditions. For example, [Formula: see text] a ring [Formula: see text] is quasi-Frobenius if and only if [Formula: see text] is right [Formula: see text], right minfull with ACC on right annihilators; [Formula: see text] a ring [Formula: see text] is quasi-Frobenius if and only if [Formula: see text] is two-sided min-CS with ACC on right annihilators in which [Formula: see text]; [Formula: see text] a ring [Formula: see text] is quasi-Frobenius if and only if [Formula: see text] is right Johns left [Formula: see text]; [Formula: see text] a ring [Formula: see text] is quasi-Frobenius if and only if [Formula: see text] is quasi-dual two-sided [Formula: see text] with ACC on right annihilators. Moreover, it is shown that a ring [Formula: see text] is quasi-Frobenius if and only if [Formula: see text] is a left [Formula: see text]-injective left IN-ring with right RMC and [Formula: see text]. Also, we prove that if [Formula: see text] is a right duo, right QF-[Formula: see text] left quasi-duo ring satisfying ACC on right annihilators, then [Formula: see text] is quasi-Frobenius. In this paper, several known results on quasi-Frobenius rings are reproved as corollaries.

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Modules Satisfying Extension Conditions Under Monomorphism of Their Closed Submodules

Cited by 3 publications

References 8 publications

On generalizations of ADS modules and rings

On generalizations of ADS modules and rings

On modules and rings satisfy condition (𝒞)

New characterizations of quasi-Frobenius rings

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