MORPHIC RINGS AS TRIVIAL EXTENSIONS

A ring R is called left morphic if, for every a (element of) R, R/Ra (approximately equal to) 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 (proportional to) 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 (proportional to) R is morphic; for an integer n>1, Z(n) (proportional to) Z(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 (proportional to) Q/R is morphic; for a bimodule M over Z, Z (proportional to) M is morphic if and only if M (approximately equal to) Q/Z. Thus, Z (proportional to) Q/Z is a morphic ring which is not clean. This example settled two questions both in the negative raised by Nicholson and Sanchez Campos, and by Nicholson, respectively.