zbMATH — the first resource for mathematics

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.

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
PDF BibTeX Cite
Full Text: DOI EuDML
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.