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 $M=A\oplus B$ of $M$ and every complement $C$ of $A$ in $M$, we have $M=A\oplus C$. 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.