A note on rings with the summand sum property

الأصل · EN

A ring R is called right SSP (SIP) if the sum (intersection) of any two direct summands of Rᵣ 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 M₂(R) is right SSP if and only if Mₙ(R) is right SSP for some n>1; (2) R is a semisimple ring if and only if the column finite matrix ring CFMΛ(R) is right SSP for a countably infinite set Λ if and only if the column finite matrix ring CFMΛ(R) is right SSP for any infinite set Λ. Some known results are improved.