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Let $R$ be a ring. A two-sided proper ideal $\mathfrak{p}$ of a ring $R$ is called a prime ideal if the following equivalent conditions are met:
- If $I$ and $J$ are left ideals and the product of ideals $IJ$ satisfies $IJ \subset \mathfrak{p}$ , then $I \subset \mathfrak{p}$ or $J \subset \mathfrak{p}$ .
- If $I$ and $J$ are right ideals with $IJ \subset \mathfrak{p}$ , then $I \subset \mathfrak{p}$ or $J \subset \mathfrak{p}$ .
- If $I$ and $J$ are two-sided ideals with $IJ \subset \mathfrak{p}$ , then $I \subset \mathfrak{p}$ or $J\subset \mathfrak{p}$ .
- If $x$ and $y$ are elements of $R$ 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 $R$ is commutative with identity, a proper ideal $\mathfrak{p}$ of $R$ 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.
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"prime ideal" is owned by djao.
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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 90 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 12172 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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Pending Errata and Addenda
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