Bicommutants and Arens regularity

Abstract

Let $C$ and $R$ be unital rings and $Z$ an injective cogenerator for right $C$-modules. For an $R,C$-bimodule $U$ let $U^{\ast }=Ho{m}_C(U,Z)$, $S=En{d}_R(U)$ and $Bien{d}_R(U)=En{d}_S(U)$, the biendomorphism ring of $U$. Under suitable requirements on $U$ we show that $B:=Bien{d}_R(U)$ can be identified with a subring of $\tilde{B}:=Bien{d}_R(U^{\ast })$,study conditions for the reverse inclusion and density of $B$ in $\tilde{B}$. In the case $C$ is contained in the center of $R$ we describe $Bien{d}_R(R^{\ast })$ in terms of the Arens products on $R^{\ast \ast }$ and study Arens regularity of $R$ in the context of duality of modules. We characterize Arens regular algebras over fields.

21 pages