A note on rings with the summand sum property

TL;DR

This paper investigates rings with the summand sum property (SSP), establishing their equivalences, characterizations, and connections to von-Neumann regular and semisimple rings, along with improvements on existing results.

Contribution

It characterizes rings with SSP, explores their properties, and links SSP to well-known classes like von-Neumann regular and semisimple rings, extending prior results.

Findings

Rings with SSP are equivalent to being right C3 and SIP.

Von-Neumann regular rings are characterized by matrix ring SSP properties.

Semisimple rings are characterized via column finite matrix ring SSP.

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 $\mathbb{M}_{2}(R)$ is right SSP if and only if $\mathbb{M}_{n}(R)$ is right SSP for some $n>1$; (2) $R$ is a semisimple ring if and only if the column finite matrix ring $\mathbb{C}\mathbb{F}\mathbb{M}_{\Lambda}(R)$ is right SSP for a countably infinite set $\Lambda$ if and only if the column finite matrix ring $\mathbb{C}\mathbb{F}\mathbb{M}_{\Lambda}(R)$ is right SSP for any infinite set $\Lambda$. Some known results are improved.