Rings all of whose right ideals are automorphism-invariant are called right a-rings. In the present paper, we study rings having the property that every right cyclic module is dual automorphism-invariant. Such rings are called right a*-rings. We obtain some of the relationships a-rings and a*-rings. We also prove that; (i) A semiperfect ring R is a right a*-ring if and only if any right ideal in J(R) is a left T-module, where T is a subring of R generated by its units, (ii) R is semisimple artinian if and only if R is semiperfect and the matrix ring M-n(R) is a right a*-ring for all n > 1, (iii) Quasi-Frobenius right a*-rings are Frobenius.