In a module X over R, if $a\in R$ such that $a.x=0$ for all $x\in X$, then $a=0$?

Question

I've been trying to prove the statement:

In a module X over R, if $a\in R$ such that $a.x=0$ for all $x\in X$, then $a=0$.

I don't know if it is true, but it seems reasonable, I'm trying to use the definition of a module over R, but I can't get to the result, any hints?

Detail: R is a commutative ring.

Answer 1

Consider $\Bbb Z/n\Bbb Z$ as a $\Bbb Z$-module. Then $n$ annihilates the whole module, but is nonzero.

Answer 2

It is false. Such elements form an ideal called the annihilator of the module $X$.