Prime element

Definition

Symbol-free definition

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 $p$ in an integral domain $R$ is said to be prime' if it satisfies the following:

• Whenever $p|ab$, then $p|a$ or $p|b$.
• The principal ideal $(p)$ is a prime ideal in $R$, or equivalently, $R/(p)$ 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.

Relation with other properties

Weaker properties

• Irreducible element
• Primary element