Prime element

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 defining ingredient::principal ideal generated by it is a defining ingredient::prime ideal, or equivalently, the quotient ring by this principal ideal is an defining ingredient::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
Given two associate elements, one of them is prime if and only if the other one is.

Weaker properties

 * Irreducible element
 * Primary element