DING PROJECTIVE AND DING INJECTIVE MODULES OVER TRIVIAL RING EXTENSIONS

Let R ? M be a trivial extension of a ring R by an R-R-bimodule M such that MR, RM, (R, 0)(R?M) and (R? M)(R, 0) have finite flat dimensions. We prove that (X, a) is a Ding projective left R ? M-module if and only if the sequence M (?R) M (?R) X ?(M?a) M R-? X (a)? X is exact and coker(a) is a Ding projective left R-module. Analogously, we explicitly describe Ding injective R ? M-modules. As applications, we characterize Ding projective and Ding injective modules over Morita context rings with zero bimodule homomorphisms.