Recall that a right $R$-module $M_R$ is called square-free if whenever $A,B\leq M$ are submodules with $A\cap B=0$ and $A\cong B=0$ then $A=B=0$.
A submodule $N$ of a module $M$ is called a summand (written $N \leq^{\oplus}M$) if $M$ is the internal direct sum $M=N\oplus K$ for some submodule $K\leq M$.
$M_R$ is called summand-square-free if whenever $A,B$ are summands with $A\cap B=0$ and $A\cong B$ then $A=B=0$.
If $M$ is a module, by a factor module of $M$ we mean a quotient $M/N$ for some $N\leq M$.
Indeed, any square-free module is summand-square-free but the converse is not necessarily true.
We have the following characterization:
$M_R$ is square-free $\iff$ Every factor module of $M$ is summand-square-free.
How can I prove this statement?!.
I appreciate any help. Thanks in advance.