Characterization of Zero in a module

Question

Let $M$ be an $R-module$ over a ring $R$. Obviously, $0 \cdot m = 0, \forall m \in M$. Is it also true that $x \cdot m = 0, \forall m \in M$ if and only if $x = 0$?

Answer 1

Let $R = \mathbb Z_6$ and $M = \mathbb Z_2=\{0,3\}$. Then $2m = 0$ $\ \forall m \in M$, but $2 \ne 0$ in $R$.

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More generally, if $R$ is not an integral domain - say $xy = 0$ where $x \ne 0\ne y$, then if $M$ is the ideal generated by $y$ viewed as an $R$- module, then we will have $x\cdot m = 0$ $\ \forall m \in M$.

$R$ can even be an integral domain: $R = \mathbb Z$, $M = \mathbb Z_5$, then $x=5$ has this property.

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If $M$ is a free module over $R$, then this cannot happen, since no element $m \in M$ can be linearly independent, as $x\cdot m =0$.