On strongly dccr⋆ modules

Abstract

In this paper, we introduce and study the concept of strongly dccr[Formula: see text] modules. Strongly dccr[Formula: see text] condition generalizes the class of Artinian modules and it is stronger than dccr[Formula: see text] condition. Let [Formula: see text] be a commutative ring with nonzero identity and [Formula: see text] a unital [Formula: see text]-module. A module [Formula: see text] is said to be strongly [Formula: see text] if for every submodule [Formula: see text] of [Formula: see text] and every sequence of elements [Formula: see text] of [Formula: see text], the descending chain of submodules [Formula: see text] of [Formula: see text] is stationary. We give many examples and properties of strongly dccr[Formula: see text]. Moreover, we characterize strongly dccr[Formula: see text] 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[Formula: see text] concept.