On dual-bimodules (Q1202793)

The following concept of dual-bimodules is introduced, a generalization of dual rings [\textit{C. R. Hajaranvis}, \textit{N. C. Norton}: J. Algebra 93, 253-266 (1985; Zbl 0595.16009)] and in particular of quasi-Frobenius rings: Let \(R\) and \(S\) be rings with identity. An \(R\)-\(S\) bimodule \(_ RQ_ S\) is called a left dual-bimodule, if \(\ell_ R(r_ Q(A)) = A\) and \(r_ Q(\ell_ R(Q')) = Q'\) for every left ideal \(A\) of \(R\) and every \(S\)-submodule \(Q'\) of \(Q\) resp. \(Q\) is a dual-bimodule, if it is a left and right dual-bimodule. Section 1 of the paper is concerned with basic properties of left dual- bimodules \(_ RQ_ S\). In particular it is shown that \(R\) is semilocal and that left dual-bimodules are invariant under Morita equivalence. Also a characterization of simple Artinian rings by means of left dual- bimodules is given. In Section 2 it is shown that for an \(R\)-\(S\) bimodule with \(_ RQ\) and \(Q_ S\) finitely generated, \(Q\) is a dual-bimodule iff \(_ RR\) and \(S_ S\) are \(Q\)-reflexive and every factor module of \(_ RR\), \(S_ S\), \(_ RQ\) and \(Q_ S\) is \(Q\)-torsionless. If \(Q_ S\) is finitely generated and \(\text{rad}(_ RQ) \leq \text{rad}(Q_ S)\), then \(R\) is semi-perfect. In Section 3 the question is investigated, when a duality, defined by a left dual-bimodule \(_ RQ_ S\), exists. Finally in Section 4 examples, illustrating the results, are given.