Liu Zhongkui

Abstract
Let $\Cal X$ be a class of left $R$-modules. It is proved that if type 2 $\Cal X$-extending left $R$-modules 
$M_1$ and $M_2$ are relatively essentially $\Cal X^e$-injective and $M_1$ is pseudoly $M_2$-$\Cal X^e$-injective (or $M_2$ is pseudoly $M_1$-$\Cal X^e$-injective) then $M_1\oplus M_2$ is type 2 $\Cal X$-extending. As applications, we characterize when the direct sum of two extending left $R$-modules is 
extending, when the direct sum of two CESS-modules is CESS, and when the direct sum of two uniform-extending left $R$-modules  is uniform-extending.