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Matrix rings with summand intersection property

F. Karabacak, Adnan Tercan (2003)

Czechoslovak Mathematical Journal

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 the 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 ) M e = 0 for every idempotent e in R . Moreover, we give necessary and sufficient conditions for the generalized upper triangular matrix rings to have SIP.

Modules commuting (via Hom) with some limits

Robert El Bashir, Tomáš Kepka (1998)

Fundamenta Mathematicae

For every module M we have a natural monomorphism   Φ : ∐ i ∈ I H o m R ( A i , M ) → H o m R ( ∏ i ∈ I A i , M ) and we focus attention on the case when Φ is also an epimorphism. The corresponding modules M depend on thickness of the cardinal number card(I). Some other limits are also considered.

Modules with the direct summand sum property

Dumitru Vălcan (2003)

Czechoslovak Mathematical Journal

The present work gives some characterizations of R -modules with the direct summand sum property (in short DSSP), that is of those R -modules for which the sum of any two direct summands, so the submodule generated by their union, is a direct summand, too. General results and results concerning certain classes of R -modules (injective or projective) with this property, over several rings, are presented.

Notes on slender prime rings

Robert El Bashir, Tomáš Kepka (1996)

Commentationes Mathematicae Universitatis Carolinae

If R is a prime ring such that R is not completely reducible and the additive group R ( + ) is not complete, then R is slender.

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