On matrix rings with the SIP and the Ads

Authors : Figen Takil MUTLU

Pages : 2657-2663

View : 10 | Download : 6

Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :In this paper, matrix rings with the summand intersection property insert ignore into journalissuearticles values(SIP); and the absolute direct summand insert ignore into journalissuearticles values(ads); property insert ignore into journalissuearticles values(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 $insert ignore into journalissuearticles values(1-e);Me=0$ for each idempotent $e$ in $R$. It is also shown with an example that the SA is not a Morita invariant property.
Keywords : Ads property, summand intersection property, trivial extension