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Let $A$ be an algebra with unity and $M$ be a left $A$-module. If $x$ is a nonzero element in $M$, then $Ax$ is a nonzero left $A$-module. If we define a map $f$ from the regular module $A$ onto $Av$ by $f(a)=av$, then $f$ is an onto module homomorphism. Is this map injective?

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Not necessarily. The kernel of the map $f$ is exactly the annihilator of $x\in M$. This can be non trivial.

An example of this is if you take $\mathbb{Z}/3\mathbb{Z}$ as a $\mathbb{Z}$-module. Then the annihilator of $1$ is the the ideal $3\mathbb{Z}$ and so the map $\mathbb{Z}\to \mathbb{Z}/3\mathbb{Z}$ given by $a \mapsto a\cdot 1$ is not injective.

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