Prime ideal
Definition for commutative rings
Symbol-free definition
An ideal in a commutative unital ring (or in any commutative ring) is termed a prime ideal if it satisfies the following equivalent conditions:
- Whenever the product of two elements in the ring lies inside that ideal, at least one of the elements must lie inside that ideal.
- It is an ideal whose complement is a saturated subset (that is, is clsoed with respect to the operation of multiplication).
- The quotient ring by that ideal is an integral domain
Definition with symbols
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Definition for non-commutative rings
The definition has many different forms for noncommutative rings: