  $\textbf{Theorem 1.2.}$ For a ring $A$, the following conditions are
equivalent. $\textbf{1)}$ $A$ is a right automorphism-invariant right
non-singular ring. $\textbf{2)}$ $A$ is a right automorphism-invariant regular
ring. $\textbf{3)}$ $A=S\times T$, where $S$ is a right injective regular ring
and $T$ is a strongly regular ring which contains all invertible elements of
its maximal right ring of quotients. $\textbf{Theorem 1.5.}$ For a ring $A$
with right Goldie radical $G(A_A)$, the following conditions are equivalent.
$\textbf{1)}$ $A/G(A_A)$ is a semiprime right Goldie ring. $\textbf{2)}$ Any
direct sum of automorphism-invariant non-singular right $A$-modules is an
automorphism-invariant module. $\textbf{3)}$ Any direct sum of
automorphism-invariant non-singular right $A$-modules is an injective module.
