Semisimple Artinian ring: Difference between revisions
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[commutative unital ring]] is termed '''semisimple''' if it satisfies the following equivalent conditions: | A [[commutative unital ring]] is termed '''semisimple Artinian''' if it satisfies the following equivalent conditions: | ||
* Every module over it is [[semisimple module|semisimple]] | * Every module over it is [[semisimple module|semisimple]] | ||
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* Every short exact sequence of modules over it, splits | * Every short exact sequence of modules over it, splits | ||
* Its [[global dimension]] is zero | * Its [[global dimension]] is zero | ||
* The ring is semisimple as a module over itself | |||
* The ring is a direct product of finitely many fields | |||
==Relation with other properties== | ==Relation with other properties== | ||
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* [[Field]] | * [[Field]] | ||
===Weaker properties=== | ===Weaker properties=== | ||
* [[Zero-dimensional ring]] | * [[Artinian ring]] | ||
* [[Zero-dimensional ring]] | |||
* [[Semiprimitive ring]]: (some people use the term '''semisimple''' for such a ring) | |||
Latest revision as of 16:34, 12 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 is termed semisimple Artinian if it satisfies the following equivalent conditions:
- Every module over it is semisimple
- Every module over it is projective
- Every module over it is injective
- Every short exact sequence of modules over it, splits
- Its global dimension is zero
- The ring is semisimple as a module over itself
- The ring is a direct product of finitely many fields
Relation with other properties
Stronger properties
Weaker properties
- Artinian ring
- Zero-dimensional ring
- Semiprimitive ring: (some people use the term semisimple for such a ring)