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Matrix rings with summand intersection property. (English) Zbl 1080.16503
Summary: A ring \(R\) has right SIP (SSP) if the intersection (sum) of two direct summands of \(R\) is also a direct summand. We show that the right SIP (SSP) is a Morita invariant property. We also prove that the trivial extension of \(R\) by \(M\) has SIP if and only if \(R\) has SIP and \((1-e)Me=0\) for every idempotent \(e\) in \(R\). Moreover, we give necessary and sufficient conditions for the generalized upper triangular matrix rings to have SIP.

MSC:
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16S50 Endomorphism rings; matrix rings
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