# Prime ideal

In ring theory we say that an ideal of a ring is **prime** if and for any ideals with we have either or .

If is commutative then the following simpler definition holds: An ideal is prime iff and for any if then either or .

This second definition easily implies the the following important property of prime ideals in commutative rings with unity:

- Let be a commutative ring with unity, then an ideal is prime iff the quotient ring is an integral domain.