Bicommutants and Arens regularity

TL;DR

This paper investigates the structure of biendomorphism rings of bimodules, explores their relation to dual modules, and characterizes Arens regularity of algebras, especially over fields, using duality and module theory.

Contribution

It provides new insights into the relationship between biendomorphism rings and dual modules, and characterizes Arens regularity for algebras over fields.

Findings

Identification of biendomorphism rings with subrings of duals

Conditions for inclusion and density of biendomorphism rings

Characterization of Arens regularity for algebras over fields

Abstract

Let $C$ and $R$ be unital rings and $Z$ an injective cogenerator for right $C$-modules. For an $R,C$-bimodule $U$ let $U^*=Hom_C(U,Z)$, $S=End_R(U)$ and $Biend_R(U)=End_S(U)$, the biendomorphism ring of $U$. Under suitable requirements on $U$ we show that $B:=Biend_R(U)$ can be identified with a subring of $\tilde{B}:=Biend_R(U^*)$,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 $Biend_R(R^*)$ in terms of the Arens products on $R^{**}$ and study Arens regularity of $R$ in the context of duality of modules. We characterize Arens regular algebras over fields.