BICOMMUTANTS AND ARENS REGULARITY

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), R' = End(R)(U) and Biend(R)(U) = End(R')(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 (B) over tilde := Biend(R)(U*), and study conditions for the reverse inclusion and density of B in (B) over tilde. In the case C is contained in the center of R we describe Biend(R)(R*) in terms of the Arens products in R** and study Arens regularity of R in the context of duality of modules. We characterize Arens regular algebras over fields.