On ϕ-P-flat modules and ϕ-von Neumann regular rings

Abstract

Let [Formula: see text] be a commutative ring with a nonzero identity and [Formula: see text] an [Formula: see text]-module. Set [Formula: see text], if [Formula: see text]-[Formula: see text] then [Formula: see text] is called a [Formula: see text]-torsion module. An [Formula: see text]-module [Formula: see text] is said to be [Formula: see text]-flat, if [Formula: see text] is an exact [Formula: see text]-sequence, for any exact sequence of [Formula: see text]-modules [Formula: see text], where [Formula: see text] is [Formula: see text]-torsion. In this paper, we study some new properties of [Formula: see text]-flat modules. Then we introduce and study the class of [Formula: see text]-[Formula: see text]-flat modules which is a generalization of [Formula: see text]-flat modules and [Formula: see text]-flat modules. Finally, we give some new characterizations of the [Formula: see text]-von Neumann regular ring and its transfer to various contexts of constructions such as the amalgamation of rings along an ideal and trivial ring extension.