On Semiperfect a-Rings

A ring is called a right a-ring if every right ideal is automorphism invariant. We describe some properties of a-rings over semiperfect rings. It is shown that an I-finite right a-ring is a direct sum of a semisimple Artinian ring and a basic ring. It is also demonstrated that if R is an indecomposable (as a ring) I-finite right a-ring not simple with nontrivial idempotents such that every minimal right ideal is a right annihilator and Soc(R_R) = Soc(_RR) is essential in R_R, then R is a quasi-Frobenius ring, which is also a right q-ring.