Semisimple Artinian ring: Difference between revisions

Revision as of 21:06, 5 May 2008

This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring
View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

Definition

Symbol-free definition

A commutative unital ring (or any commutative ring) is termed semisimple if it satisfies the following equivalent conditions:

The Jacobson radical (viz the intersection of its maximal ideals) is trivial
It is a subdirect product of fields
It is a direct product of fields

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-==Definition for commutative rings==
+{{curing property}}
+==Definition==
 ===Symbol-free definition===
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 * It is a [[subdirect product]] of fields
 * It is a [[direct product]] of fields
-==Definition for noncommutative rings==
-===Symbol-free definition===
-''Please check the content here''
-A [[unital ring]] (or any [[ring]]) is termed '''semisimple''' if it satisfies the following equivalent conditions:
-* The intersection of all its [[maximal left ideal]]s is trivial
-* The intersection of all its [[maximal right ideal]]s is trivial
-[[Category: Properties of noncommutative rings]]