Prime Ideal

A proper ideal P of a commutative ring R such that the quotient ring R/P is an integral domain; equivalently, P is proper and whenever a product ab lies in P, at least one of a or b lies in P (the complement R\P is multiplicatively closed).

A proper ideal P of a commutative ring R such that whenever a·b ∈ P then either a ∈ P or b ∈ P; equivalently the quotient R/P is an integral domain.