On Strongly ??-regular Modules

Abstract

In this article, we introduce the notion of strongly ??-regular module which is a generalization_x000D_ of von Neumann regular module in the sense [13]. Let ?? be a commutative ring with 1 ≠ 0 and_x000D_ ?? a multiplication ??-module. ?? is called a strongly ??-regular module if for each ?? ∈ ??,_x000D_ (????)m = ???? = ??2?? for some ?? ∈ ?? and ?? ∈ ℕ. In addition to give many properties and_x000D_ examples of strongly ??-regular modules, we also characterize certain class of modules such as_x000D_ von Neumann regular modules and second modules in terms of this new class of modules. Also,_x000D_ we determine when the localization of any family of submodules at a prime ideal commutes_x000D_ with the intersection of this family._x000D_ _x000D_ Keywords: von Neumann regular module, (??, ??)-closed ideal, strongly ??-regular module,_x000D_ Krull dimension, (∗)-property, localization.