Maximal ideal: Difference between revisions
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==Definition | {{curing-ideal property}} | ||
{{quotient is a|field}} | |||
==Definition== | |||
===Symbol-free definition=== | ===Symbol-free definition=== | ||
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* For any ideal <math>J</math> such that <math>M</math> ≤ <math>J</math> ≤ <math>R</math>, <math>J</math> is equal either to <math>M</math> or to <math>R</math>. | * For any ideal <math>J</math> such that <math>M</math> ≤ <math>J</math> ≤ <math>R</math>, <math>J</math> is equal either to <math>M</math> or to <math>R</math>. | ||
* The quotient ring <math>R/M</math> is a [[field]]. | * The quotient ring <math>R/M</math> is a [[field]]. | ||
Revision as of 09:35, 7 August 2007
This article defines a property of an ideal in a commutative unital ring |View other properties of ideals in commutative unital rings
This property of an ideal in a ring is equivalent to the property of the quotient ring being a/an: field | View other quotient-determined properties of ideals in commutative unital rings
Definition
Symbol-free definition
An ideal in a commutative unital ring (or more generally, in any commutative ring) is termed maximal if it is proper (not the whole ring) and it satisfies the following equivalent conditions:
- There is no ideal of the ring properly in between this ideal and the whole ring
- The quotient of the ring by this ideal is a field
- The natural action of on makes into a simple -module.
Definition with symbols
An ideal in a commutative ring is termed maximal if it satisfies the following equivalent conditions:
- For any ideal such that ≤ ≤ , is equal either to or to .
- The quotient ring is a field.
