Nonzerodivisor on a module

Definition
Suppose $$M \ne 0$$ is a module over a commutative unital ring $$R$$ and $$x \in R$$ is an element. We say that $$x$$ is a nonzerodivisor on $$M$$ if the following equivalent conditions hold:


 * The mapping $$M \to M$$ given by $$m \mapsto xm$$ is injective.
 * There does not exist $$0 \ne m \in M$$ such that $$xm = 0$$

Facts

 * If $$M$$ is a graded module over a graded algebra over a field, that occurs as a quotient of a multivariate polynomial ring, and $$x$$ is a nonzerodivisor on $$M$$, then the degree of the Hilbert polynomial for $$M/xM$$ is less than the degree of the Hilbert polynomial on $$M$$.