ZASSENHAUS RINGS AS IDEALIZATION OF MODULES

A ring R is called a Zassenhaus ring if any homomorphism phi of the additive group of R that leaves all left ideals of R invariant, is a left multiplication by some element a of R, i.e., phi(x) = ax for all x epsilon R. Let M be an R-R-bimodule. Then the direct sum R circle plus M turns naturally into a ring R(+)M by defining M M = {0}. This ring is called the idealization of the module M, which is an ideal of R(+)M. We will investigate conditions under which R(+)M is a Zassenhaus ring.