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maximal ideal (Definition)

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.




"maximal ideal" is owned by djao.
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See Also: proper ideal, module, comaximal ideals, prime ideal, existence of maximal ideals


Attachments:
existence of maximal ideals (Theorem) by yark
maximal ideal is prime (Theorem) by pahio
maximal ideal is prime (general case) (Theorem) by mclase
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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 MSC13A15 (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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