Nonzerodivisor on a module

Definition

Suppose $M\neq 0$ 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 $0\neq m\in M$ 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 $M$. For full proof, refer: Degree of Hilbert polynomial drops on quotienting by nonzerodivisor