Characterization of $\chi$-pure exact sequences

Abstract

In this paper, we introduce the notion of $\chi$-pure exact sequence. An exact sequence $0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0$ is said to be $\chi$-pure-exact if $0\longrightarrow A\bigotimes M\longrightarrow B\bigotimes M\longrightarrow C\bigotimes M\longrightarrow 0$ is again an exact sequence, where A, B, C are right R-modules and $M\simeq R/I$ is a left R-module for $I\in \chi$, where $\chi$ denotes the collection of left ideals of R. In this paper, we establish several equivalent conditions for a pure exact sequence to be $\chi$-pure exact. We further define a $\chi$-pure injective module and study the topological aspects of this module and introduce a condition under which a $\chi$-pure injective module coincides with an algebraically compact module.