$PS$-Injective Modules, $PS$-Flat Modules and $PS$-Coherent Rings

Abstract

A left ideal $I$ of a ring $R$ is small in case for every proper left ideal $K$ of $R, K +I ≠ R$. A ring $R$ is called left $PS$-coherent if every principally small left ideal $Ra$ is finitely presented. We develop, in this paper, $PS$-coherent rings as a generalization of $P$-coherent rings and $J$-coherent rings. To characterize $PS$-coherent rings, we first introduce $PS$-injective and $PS$-flat modules, and discuss the relation between them over some spacial rings. Some properties of left $PS$-coherent rings are also studied.