Semisimple Artinian ring
From Commalg
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
Contents
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)
