  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.
