Why does $em=(1-e)m$ imply $m=0$

Question

Let $R$ be a ring and $M$ be a module over it. Suppose $e\in R$ is an idempotent and $m\in M$. Why does $em=(1-e)m$ imply $m=0$?

I can't see how I can use the property of idempotents other than noting that $1-e$ is also an idempotent. And the fact that two idempotents act in the same way on $m$ does not tell me anything.

Answer 1

Multiply both sides of the equation by $e$. This shows that $em = 0$. Then, you have that $em = 0 = m - em = m$.