Examples of QF Rings without Nakayama Automorphism and H-Rings without Self-Duality

J. Kado and K. Oshiro (J. Algebra211 (1999), 384–408) proved the equivalence of (A) the existence of a Nakayama isomorphism for every basic left H-ring (Harada ring) and (B) the existence of a Nakayama automorphism for every basic QF ring. In this paper, by using examples of J. Kraemer (“Characterizations of the Existence of (Quasi-) Self-Duality for Complete Tensor Rings,” Algebra Berichte 56, Verlag Reinhard Fischer, Munich, 1987), we give examples of QF rings without Nakayama automorphism and H-rings without self-duality.