Semisimple Artinian ring

From Commalg
Jump to: navigation, search
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


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