PlanetMath: prime ideal

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prime ideal

(Definition)

Let be a ring. A two-sided proper ideal $\mathfrak{p}$ of a ring is called a prime ideal if the following equivalent conditions are met:

If and are left ideals and the product of ideals satisfies $IJ \subset \mathfrak{p}$ , then $I \subset \mathfrak{p}$ or $J \subset \mathfrak{p}$ .
If and are right ideals with $IJ \subset \mathfrak{p}$ , then $I \subset \mathfrak{p}$ or $J \subset \mathfrak{p}$ .
If and are two-sided ideals with $IJ \subset \mathfrak{p}$ , then $I \subset \mathfrak{p}$ or $J\subset \mathfrak{p}$ .
If and are elements of with $xRy \subset \mathfrak{p}$ , then $x \in \mathfrak{p}$ or $y \in \mathfrak{p}$ .

$R/\mathfrak{p}$ is a prime ring if and only if $\mathfrak{p}$ is a prime ideal. When is commutative with identity, a proper ideal $\mathfrak{p}$ of is prime if and only if for any $a,b \in R$ , if $a\cdot b \in \mathfrak{p}$ then either $a \in \mathfrak{p}$ or $b \in \mathfrak{p}$ . One also has in this case that $\mathfrak{p} \subset R$ is prime if and only if the quotient ring $R/\mathfrak{p}$ is an integral domain.

"prime ideal" is owned by djao.

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See Also: maximal ideal, ideal, prime element

Attachments:: characterization of prime ideals (Result) by GrafZahl
ideal included in union of prime ideals (Result) by polarbear
quotient ring modulo prime ideal (Theorem) by pahio

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Cross-references: integral domain, quotient ring, prime, identity, commutative, prime ring, two-sided ideals, right ideals, product of ideals, left ideals, equivalent, proper ideal, ring
There are 70 references to this entry.

This is version 10 of prime ideal, born on 2001-10-20, modified 2005-07-24.
Object id is 409, canonical name is PrimeIdeal.
Accessed 9497 times total.

Classification:

AMS MSC:	13C99 (Commutative rings and algebras :: Theory of modules and ideals :: Miscellaneous)
	16D99 (Associative rings and algebras :: Modules, bimodules and ideals :: Miscellaneous)

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