Modules over strongly semiprime ring

TL;DR

This paper characterizes modules over strongly semiprime rings, establishing equivalences involving injectivity conditions and the structure of the ring related to the right Goldie radical.

Contribution

It provides new equivalences characterizing strongly semiprime rings via module injectivity properties and the structure of the quotient by the Goldie radical.

Findings

Equivalence between module injectivity conditions and ring being strongly semiprime.

Characterization of strongly semiprime rings via module properties and non-singularity.

Conditions under which modules are injective over such rings.

Abstract

$\textbf{Theorem 1.3.}$ For a given ring $A$ with right Goldie radical $G(A_A)$, the following conditions are equivalent. $\textbf{1)}$ Every non-singular right $A$-module $X$ which is is injective with respect to some essential right ideal of the ring $A$ is an injective module. $\textbf{2)}$ $A/G(A_A)$ is a right strongly semiprime ring. $\textbf{Theorem 1.4.}$ For a given ring $A$, the following conditions are equivalent. $\textbf{1)}$ $A$ is a right strongly semiprime ring. $\textbf{2)}$ Every right $A$-module which is injective with respect to some essential right ideal of the ring $A$, is an injective module and $A$ is right non-singular.