Semisimple Artinian ring

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

Stronger properties

 * Field

Weaker properties

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