MORPHIC RINGS AS TRIVIAL EXTENSIONS

Abstract

A ring R is called left morphic if, for every a R, R/Ra = l(a) where l(a) denotes the left annihilator of a in R. Right morphic rings are defined analogously. In this paper, we investigate when the trivial extension R M of a ring R and a bimodule M over R is (left) morphic. Several new families of (left) morphic rings are identified through the construction of trivial extensions. For example, it is shown here that if R is strongly regular or semisimple, then R R is morphic; for an integer n > 1, n n is morphic if and only if n is a product of distinct prime numbers; if R is a principal ideal domain with classical quotient ring Q, then the trivial extension R Q/R is morphic; for a bimodule M over , M is morphic if and only if M = ./ Thus, / is a morphic ring which is not clean. This example settled two questions both in the negative raised by Nicholson and Snchez Campos, and by Nicholson, respectively.