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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.

In this article, we study modules with the weak $FI$-extending property. We prove that if $M$ satisfies weak $FI$-extending, pseudo duo, $C_3$ properties and $M/\mathrm{Soc}M$ has finite uniform dimension then $M$ decomposes into a direct sum of a semisimple submodule and a submodule of finite uniform dimension. In particular, if $M$ satisfies the weak $FI$-extending, pseudo duo, $C_3$ properties and ascending (or descending) chain condition on essential submodules then $M=M_1\oplus M_2$ for some semisimple submodule $M_1$ and Noetherian (or Artinian, respectively)...