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 M-n(R) is a right a-ring for some n > 1, (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. (C) 2015 Elsevier B.V. All rights reserved.