Semisimple Artinian ring: Difference between revisions
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==Definition | {{curing property}} | ||
==Definition== | |||
===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[commutative unital ring]] | 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 [[projective module|projective]] | ||
* | * Every module over it is [[injective module|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=== | ||
* [[Field]] | |||
===Weaker properties=== | |||
[[ | * [[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)