  $\textbf{Theorem 1.3.}$ 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.
