Abstract

It is shown that if R is a right automorphism-invariant ring and satisfies ACC on right annihilators, then R is a semiprimary ring. By this useful fact, we study finiteness conditions which ensure an automorphism-invariant ring is quasi-Frobenius (QF). Thus, we prove, among other results, that: (1) R is QF if and only if R is right automorphism-invariant, right min-CS and satisfies ACC on right annihilators; (2) R is QF if and only if R is left Noetherian, right automorphism-invariant and every complement right ideal of R is a right annihilator; (3) If R is right CPA, right automorphism-invariant and every complement right ideal of R is a right annihilator, then R is QF.