On Strongly 𝝅-regular Modules

Abstract

In this article, we introduce the notion of strongly 𝜋-regular module which is a generalization of von Neumann regular module in the sense [13]. Let 𝐴 be a commutative ring with 1 ≠ 0 and 𝑋 a multiplication 𝐴-module. 𝑋 is called a strongly 𝜋-regular module if for each 𝑥 ∈ 𝑋, (𝐴𝑥)m = 𝑐𝑋 = 𝑐2𝑋 for some 𝑐 ∈ 𝐴 and 𝑚 ∈ ℕ. In addition to give many properties and examples of strongly 𝜋-regular modules, we also characterize certain class of modules such as von Neumann regular modules and second modules in terms of this new class of modules. Also, we determine when the localization of any family of submodules at a prime ideal commutes with the intersection of this family.Keywords: von Neumann regular module, (𝑚, 𝑛)-closed ideal, strongly 𝜋-regular module, Krull dimension, (∗)-property, localization.