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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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##### References:
 [1] F. W. Anderson and K. R. Fuller: Rings and Categories of Modules. Springer-Verlag, 1974. · Zbl 0301.16001 [2] G. F. Birkenmeier, J. Y. Kim and J. K. Park: When is the CS condition hereditary. Comm. Algebra 27 (1999), 3875-3885. · Zbl 0946.16004 [3] J. L. Garcia: Properties of direct summands of modules. Comm. Algebra 17 (1989), 73-92. · Zbl 0659.16016 [4] K. R. Goodearl: Ring Theory. Marcel Dekker, 1976. · Zbl 0336.16001 [5] J. Hausen: Modules with the summand intersection property. Comm. Algebra 17 (1989), 135-148. · Zbl 0667.16020 [6] I. Kaplansky: Infinite Abelian Groups. University of Michigan Press, 1969. · Zbl 0194.04402 [7] G. V. Wilson: Modules with the summand intersection property. Comm. Algebra 14 (1986), 21-38. · Zbl 0592.13008
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