Semisimple Artinian ring

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

• Field

Weaker properties

• Artinian ring
• Zero-dimensional ring
• Semiprimitive ring: (some people use the term semisimple for such a ring)