On matrix rings with the SIP and the Ads

Abstract

In this paper, matrix rings with the summand intersection property (SIP) and the absolute direct summand (ads) property (briefly, SA) are studied. A ring R has the right SIP if the intersection of two direct summands of R is also a direct summand. A right R-module M has the ads property if for every decomposition M = A circle plus B of M and every complement C of A in M, we have M = A circle plus C. It is shown that the trivial extension of R by M has the SA if and only if R has the SA, M has the ads, and (1 - e)Me = 0 for each idempotent e in R. It is also shown with an example that the SA is not a Morita invariant property.