A nonzero element in an integral domain is said to be a prime element if it satisfies the following equivalent conditions:
- Whenever it divides the product of two elements, it must divide at least one of them.
- The principal ideal generated by it is a prime ideal, or equivalently, the quotient ring by this principal ideal is an integral domain.
Definition with symbols
A nonzero element in an integral domain is said to be prime' if it satisfies the following:
- Whenever , then or .
- The principal ideal is a prime ideal in , or equivalently, is an integral domain.
Invariance up to associates
Further information: Prime element property is invariant upto associates
Given two associate elements, one of them is prime if and only if the other one is.