Nonzerodivisor on a module
Definition
Suppose is a module over a commutative unital ring and is an element. We say that is a nonzerodivisor on if the following equivalent conditions hold:
- The mapping given by is injective.
- There does not exist such that
Facts
- If is a graded module over a graded algebra over a field, that occurs as a quotient of a multivariate polynomial ring, and is a nonzerodivisor on , then the degree of the Hilbert polynomial for is less than the degree of the Hilbert polynomial on . For full proof, refer: Degree of Hilbert polynomial drops on quotienting by nonzerodivisor