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maximal ideal
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(Definition)
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Let $R$ be a ring with identity. A proper left (right, two-sided) ideal $\mathfrak{m} \subsetneq R$ is said to be maximal if $\mathfrak{m}$ is not a proper subset of any other proper left (right, two-sided) ideal of $R$ .
One can prove:
- A left ideal $\mathfrak{m}$ is maximal if and only if $R/\mathfrak{m}$ is a simple left $R$ -module.
- A right ideal $\mathfrak{m}$ is maximal if and only if $R/\mathfrak{m}$ is a simple right $R$ -module.
- A two-sided ideal $\mathfrak{m}$ is maximal if and only if $R/\mathfrak{m}$ is a simple ring.
All maximal ideals are prime ideals. If $R$ is commutative, an ideal $\mathfrak{m} \subset R$ is maximal if and only if the quotient ring $R/\mathfrak{m}$ is a field.
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"maximal ideal" is owned by djao.
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Cross-references: field, quotient ring, commutative, prime ideals, simple ring, two-sided ideal, right ideal, simple, left ideal, proper subset, ideal, right, identity, ring
There are 64 references to this entry.
This is version 3 of maximal ideal, born on 2001-10-20, modified 2002-04-20.
Object id is 410, canonical name is MaximalIdeal.
Accessed 11535 times total.
Classification:
| AMS MSC: | 13A15 (Commutative rings and algebras :: General commutative ring theory :: Ideals; multiplicative ideal theory) | | | 16D25 (Associative rings and algebras :: Modules, bimodules and ideals :: Ideals) |
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Pending Errata and Addenda
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