On strongly dccr∗ modules

Abstract

In this paper, we introduce and study the concept of strongly dccr∗ modules. Strongly dccr∗ condition generalizes the class of Artinian modules and it is stronger than dccr∗ condition. Let R be a commutative ring with nonzero identity and M a unital R-module. A module M is said to be strongly dccr∗ if for every submodule N of M and every sequence of elements (ai) of R, the descending chain of submodules a1N ⊇ a1a2N ⊇ ⋯ ⊇ a1a2⋯anN ⊇⋯ of M is stationary. We give many examples and properties of strongly dccr∗. Moreover, we characterize strongly dccr∗ in terms of some known class of rings and modules, for instance in perfect rings, strongly special modules and principally cogenerately modules. Finally, we give a version of Union Theorem and Nakayama's Lemma in light of strongly dccr∗ concept. © 2022 World Scientific Publishing Company.