A note on rings with the summand sum property

2011/07/02 by Liang Shen, Shen, Liang
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topics in Algebra #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #math.RA

paper · pdf · doi:10.48550/arxiv.1107.0384

7 pages

arxiv created 2011/07/02 · openalex publication_date 2011/07/02 · arxiv updated 2011/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A ring R is called right SSP (SIP) if the sum (intersection) of any two direct summands of R_R is also a direct summand. Left sides can be defined similarly. The following are equivalent: (1) R is right SSP. (2) R is right C3 and right SIP. (3) R is left C3 and left SIP. (4) R is left SSP. It is also shown that (1) R is a von-Neumann regular ring if and only if \mathbbM₂(R) is right SSP if and only if \mathbbM_n(R) is right SSP for some n>1; (2) R is a semisimple ring if and only if the column finite matrix ring ℂ\mathbbF\mathbbM_Λ(R) is right SSP for a countably infinite set Λ if and only if the column finite matrix ring ℂ\mathbbF\mathbbM_Λ(R) is right SSP for any infinite set Λ. Some known results are improved.

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