Modules over strongly semiprime ring

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.