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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.

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  • $\begingroup$ This isn't true. $\endgroup$ – mweiss Mar 19 '18 at 2:23
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Consider $\Bbb Z/n\Bbb Z$ as a $\Bbb Z$-module. Then $n$ annihilates the whole module, but is nonzero.

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

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