  In this paper, we study rings having the property that every right ideal is
automorphism-invariant. Such rings are called right $a$-rings. It is shown that
(1) a right $a$-ring is a direct sum of a square-full semisimple artinian ring
and a right square-free ring, (2) a ring $R$ is semisimple artinian if and only
if the matrix ring $\mathbb{M}_n(R)$ for some $n>1$ is a right $a$-ring, (3)
every right $a$-ring is stably-finite, (4) a right $a$-ring is von Neumann
regular if and only if it is semiprime, and (5) a prime right $a$-ring is
simple artinian. We also describe the structure of an indecomposable right
artinian right non-singular right $a$-ring as a triangular matrix ring of
certain block matrices.
