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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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