maximal ideal
Let $R$ be a ring with identity. A proper left (right, twosided) ideal $\U0001d52a\u228aR$ is said to be maximal if $\U0001d52a$ is not a proper subset^{} of any other proper left (right, twosided) ideal of $R$.
One can prove:

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A left ideal^{} $\U0001d52a$ is maximal if and only if $R/\U0001d52a$ is a simple left $R$module.

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A right ideal $\U0001d52a$ is maximal if and only if $R/\U0001d52a$ is a simple right $R$module.

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A twosided ideal $\U0001d52a$ is maximal if and only if $R/\U0001d52a$ is a simple ring^{}.
All maximal ideals^{} are prime ideals^{}. If $R$ is commutative^{}, an ideal $\U0001d52a\subset R$ is maximal if and only if the quotient ring^{} $R/\U0001d52a$ is a field.
Title  maximal ideal 
Canonical name  MaximalIdeal 
Date of creation  20130322 11:50:57 
Last modified on  20130322 11:50:57 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  8 
Author  djao (24) 
Entry type  Definition 
Classification  msc 13A15 
Classification  msc 16D25 
Classification  msc 81R50 
Classification  msc 46M20 
Classification  msc 18B40 
Classification  msc 22A22 
Classification  msc 46L05 
Related topic  ProperIdeal 
Related topic  Module 
Related topic  Comaximal 
Related topic  PrimeIdeal 
Related topic  EveryRingHasAMaximalIdeal 