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On matrix rings with the SIP and the ads. (English) Zbl 1424.16060
Summary: 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.
MSC:
 16S50 Endomorphism rings; matrix rings 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
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References:
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