Automorphism-invariant non-singular rings and modules

$\mathbf{\text{Theorem}}\mathbf{\text{1.2.}}$ For a ring $A$, the following conditions are equivalent. $\mathbf{\text{1)}}$ $A$ is a right automorphism-invariant right non-singular ring. $\mathbf{\text{2)}}$ $A$ is a right automorphism-invariant regular ring. $\mathbf{\text{3)}}$ $A=S\times T$, 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. $\mathbf{\text{Theorem}}\mathbf{\text{1.5.}}$ For a ring $A$ with right Goldie radical $G(A_A)$, the following conditions are equivalent. $\mathbf{\text{1)}}$ $A\mathrm{/}G(A_A)$ is a semiprime right Goldie ring. $\mathbf{\text{2)}}$ Any direct sum of automorphism-invariant non-singular right $A$-modules is an automorphism-invariant module. $\mathbf{\text{3)}}$ Any direct sum of automorphism-invariant non-singular right $A$-modules is an injective module.