Let $R$ be a ring. Recall that an $R$-module $M$ is said to be $C2$-module if it satisfies the following property: Whenever $A,B$ are submodules of $M$ such that $A$ is isomorphic to $B$, and $B$ is a direct summand of $M$, then $A$ is a direct summand of $M$ as well.
Recall that an $R$-module $M$ is said to be a $C3$-module if whenever $A,B$ are direct summands of $M$ with $A\cap B = \lbrace 0 \rbrace$ , then $A+B$ is a summand of $M$.
It is well known that every $C2$-module is $C3$. How can I prove this?. I am stuck with this proof for more than a week.
I really appreciate any help.
