Prime ideal: Difference between revisions

Revision as of 17:05, 17 December 2007

This article defines a property of an ideal in a commutative unital ring |View other properties of ideals in commutative unital rings

This property of an ideal in a ring is equivalent to the property of the quotient ring being a/an: integral domain | View other quotient-determined properties of ideals in commutative unital rings

Definition for commutative rings

Symbol-free definition

An ideal in a commutative unital ringis 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 $b \in I$ .

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 ===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:
+An [[ideal]] in a [[commutative unital 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.
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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==
-The definition has many different forms for noncommutative rings:
-* [[Prime ideal (noncommutative rings)]]
-* [[Completely prime ideal]]