A ring $R$ has right SIP (SSP) if the intersection (sum) of two direct summands of $R$ is also a direct summand. We show that the right SIP (SSP) is the Morita invariant property. We also prove that the trivial extension of $R$ by $M$ has SIP if and only if $R$ has SIP and $(1-e)Me=0$ for every idempotent $e$ in $R$. Moreover, we give necessary and sufficient conditions for the generalized upper triangular matrix rings to have SIP.

For every module $M$ we have a natural monomorphism $$\Psi :\coprod _{i\in I}{\mathrm{H}om}_R(M,A_i)\to {\mathrm{H}om}_R\left(M,\coprod _{i\in I}{A}_i\right)$$ and we focus our attention on the case when $\Psi$ is also an epimorphism. Some other colimits are also considered.

The present work gives some characterizations of $R$-modules with the direct summand sum property (in short DSSP), that is of those $R$-modules for which the sum of any two direct summands, so the submodule generated by their union, is a direct summand, too. General results and results concerning certain classes of $R$-modules (injective or projective) with this property, over several rings, are presented.