Prime ideal: Difference between revisions

Revision as of 09:04, 7 August 2007

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

An ideal $I$ in a commutative unital ring $R$ is termed a prime ideal if whenever $a, b \in R$ are such that $a b \in I$ then either $a \in I$ or $b \in I$ .

Definition for non-commutative rings

The definition has many different forms for noncommutative rings:

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 ===Definition with symbols===
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+An [[ideal]] <math>I</math> in a [[commutative unital ring]] <math>R</math> is termed a '''prime ideal''' if whenever <math>a, b \in R</math> are such that <math>ab \in I</math> then either <math>a \in I</math> or <math>b \in I</math>.
 ==Definition for non-commutative rings==