On strongly dccr* modules

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 (a(i)) of R, the descending chain of submodules a(1)N superset of a(1)a(2)N superset of ... superset of a(1)a(2 )...( )a(n) N superset of .... of M is stationary. We give many examples and properties of strongly (lax*. 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.