On dual-Bimodules

Abstract

A ring $R$ with identity in which $l=r_{R}l_{R}(l)$ for every right ideal 1 and $J=$ $l_{R}r_{R}(J)$ for every left ideal $J$ of $R$ is called a dual ring.This ring has been investigated by many authors.As is well-known, an Artinian dual ring is a QF-ring and, recently, Hajarnavis and Norton [4] have studied dual rings and pointed out that certain properties well-known for QF-rings are also seen to hold without the Artinian assumption.In this paper, we shall introduce the notion of dual-bimodules and try to give a module-theoretic characterization of dual rings.Let $R$ and $S$ be rings with identity and $RQ_{s}$ an $(R, S)$ -bimodule.We shall call $Q$ a left dual-bimodule if(1) $l_{R}t_{Q}(A)=A$ for every left ideal $A$ of $R$ , and(2) $r_{Q}l_{R}(Q^{\prime})=Q^{\prime}$ for every S-submo\'oule $Q^{\prime}$ of $Q$ .A right dual-bimodule is similarly defined and we shall call $Q$ a dual- bimodule if it is a left dual-bimodule and is a right dual-bimodule as well.A left dual-bimodule need not be a right dual-bimodule in general (see Example 4.2).Trivially a dual ring is a dual-bimodule.A bimodule which defines a Morita duality is a dual-bimodule [1, Exercise 24.7].Furthermore, a dual-bimodule is a quasi-Frobenius bimodule in the sense of Azumaya [2] (cf.also [5, Theo- rem 4]).In Section 1, we shall study basic properties of left dual-bimodules and show that, among other things, an $(R, S)$ -bimodule $Q$ such that the mapping $\lambda:R-End(Q_{S})$ given by $a\rightarrow a_{L}$ , the left multiplication by $a$ , is surjective is a left dual-bimodule if and only if every factor module of $RR$ and $Q_{S}$ is Q-torsionless (Theorem 1.4), for a left dual-bimodule $RQ_{s}$ the ring $R$ is semilocal (Theorem 1.10) and that for every R-module $RQ\neq 0,$ $RQ_{s}$ is a left dual-bimodule with $S=End(_{R}Q)$ if and

Locations

• Tsukuba Journal of Mathematics - View - PDF