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.
- Prime ideals are primary and radical.
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