A left Artinian ring is a left \(H\)-ring (Harada ring) [M. Harada, Ring Theory, Proc. 1978 Antwerp Conf., Lect. Notes Pure Appl. Math. 51, 669-689 (1979; Zbl 0449.16018), K. Oshiro, Math. J. Okayama Univ. 31, 161-178, 179-188 (1989; Zbl 0706.16009, Zbl 0734.16007), ibid. 32, 111-118 (1990; Zbl 0734.16008)] in case its basic ring \(R\) has the following property: If \(T=\{e_1,\dots,e_n\}\) is a complete set of orthogonal primitive idempotents of \(R\) and \(N=\text{rad}(R)\), then the \(e_i\) can be enumerated such that: \(T=\{e_{11},\dots,e_{1s_1},\dots,e_{m1},\dots,e_{ms_m}\}\), where (i) \(Re_{i1}\) is an injective \(R\)-module for \(i=1,\dots,m\), (ii) \(Ne_{i,j-1}\cong Re_{ij}\) for \(i=1,\dots,m\); \(j=2,\dots,s_i\). In particular, quasi-Frobenius rings and Nakayama rings are \(H\)-rings.
In the paper the question is studied, whether every \(H\)-ring has a selfduality, and it is shown that this question is equivalent to the question whether every Frobenius ring has a Nakayama automorphism. In more detail: Let \(E_i=E(Re_i/Ne_i)\) be the injective envelope of the simple module \(Re_i/Ne_i\) and \(E=\bigoplus_i E_i\). Let \(T=T(R)=\text{End}_R(E)\) and \(f_i\) is the unit element of the local ring \(\text{End}_R(E_i)\subset T\). If there exists an isomorphism \(\varphi\colon R\to T\), such that \(\varphi(e_i)=f_i\), \(\varphi\) is called a Nakayama isomorphism. In case \(R\) is a basic Frobenius ring, \(T\) can be identified with \(R\), hence \(\varphi\) is an automorphism of \(R\) if it exists, and is called a Nakayama automorphism of \(R\).
The result of the paper now is: The following conditions are equivalent: (i) For every basic left \(H\)-ring \(R\) there exists a Nakayama isomorphim \(\varphi\colon R\to T(R)\). (ii) Every (basic) Frobenius ring has a Nakayama automorphism. (iii) Every basic left \(H\)-ring has a selfduality. – The corresponding assertion holds, if in (i) and (iii) \(H\)-ring is replaced by Nakayama ring and in (iii) Frobenius ring by Frobenius and Nakayama ring.