Rings for which every cyclic module is dual automorphism-invariant

Rings all of whose right ideals are automorphism-invariant are called right [Formula: see text]-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 [Formula: see text]-rings. We obtain some of the relationships [Formula: see text]-rings and [Formula: see text]-rings. We also prove that; (i) A semiperfect ring [Formula: see text] is a right [Formula: see text]-ring if and only if any right ideal in [Formula: see text] is a left [Formula: see text]-module, where [Formula: see text] is a subring of [Formula: see text] generated by its units, (ii) [Formula: see text] is semisimple artinian if and only if [Formula: see text] is semiperfect and the matrix ring [Formula: see text] is a right [Formula: see text]-ring for all [Formula: see text], (iii) Quasi-Frobenius right [Formula: see text]-rings are Frobenius.