Semisimple Artinian ring: Difference between revisions

Revision as of 01:37, 6 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 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

Relation with other properties

Stronger properties

Weaker properties

Zero-dimensional ring: For full proof, refer: Semisimple implies zero-dimensional

@@ Line 4: / Line 4: @@
 ===Symbol-free definition===
-A [[commutative unital ring]] (or any [[commutative ring]]) is termed '''semisimple''' if it satisfies the following equivalent conditions:
+A [[commutative unital ring]] is termed '''semisimple''' if it satisfies the following equivalent conditions:
-* The [[Jacobson radical]] (viz the intersection of its [[maximal ideal]]s) is trivial
+* Every module over it is [[semisimple module|semisimple]]
-* It is a [[subdirect product]] of fields
+* Every module over it is [[projective module|projective]]
-* It is a [[direct product]] of fields
+* Every module over it is [[injective module|injective]]
+* Every short exact sequence of modules over it, splits
+* Its [[global dimension]] is zero
+==Relation with other properties==
+===Stronger properties===
+* [[Field]]
+* [[Finite direct product of fields]]
+===Weaker properties===
+* [[Zero-dimensional ring]]: {{proofat|[[Semisimple implies zero-dimensional]]}}