Displaying similar documents to “Matrix rings with summand intersection property”

⊕ -cofinitely supplemented modules

H. Çalışıcı, A. Pancar (2004)

Czechoslovak Mathematical Journal

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Let R be a ring and M a right R -module. M is called ⊕ -cofinitely supplemented if every submodule N of M with M N finitely generated has a supplement that is a direct summand of M . In this paper various properties of the ⊕ -cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of ⊕ -cofinitely supplemented modules is ⊕ -cofinitely supplemented. (2) A ring R is semiperfect if and only if every free R -module is ⊕ -cofinitely supplemented. In addition, if M has the...

Strong separativity over exchange rings

Huanyin Chen (2008)

Czechoslovak Mathematical Journal

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An exchange ring R is strongly separative provided that for all finitely generated projective right R -modules A and B , A ⊕ A ≅ A ⊕ B ⇒ A ≅ B . We prove that an exchange ring R is strongly separative if and only if for any corner S of R , a S + b S = S implies that there exist u , v ∈ S such that a u = b v and S u + S v = S if and only if for any corner S of R , a S + b S = S implies that there exists a right invertible matrix a b * ∈ M 2 ( S ) . The dual assertions are also proved.

On rings all of whose modules are retractable

Şule Ecevit, Muhammet Tamer Koşan (2009)

Archivum Mathematicum

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Let R be a ring. A right R -module M is said to be retractable if 𝕋 H o m R ( M , N ) ≠ 0 whenever N is a non-zero submodule of M . The goal of this article is to investigate a ring R for which every right R-module is retractable. Such a ring will be called right mod-retractable. We proved that ( 1 ) The ring ∏ i ∈ ℐ R i is right mod-retractable if and only if each R i is a right mod-retractable ring for each i ∈ ℐ , where ℐ is an arbitrary finite set. ( 2 ) If R [ x ] is a mod-retractable ring then R is a mod-retractable ring.