Semisimple Artinian ring

Definition for commutative rings

Symbol-free definition

A commutative unital ring (or any commutative ring) is termed semisimple if it satisfies the following equivalent conditions:

• The Jacobson radical (viz the intersection of its maximal ideals) is trivial
• It is a subdirect product of fields
• It is a direct product of fields

Definition for noncommutative rings

Symbol-free definition

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A unital ring (or any ring) is termed semisimple if it satisfies the following equivalent conditions:

• The intersection of all its maximal left ideals is trivial
• The intersection of all its maximal right ideals is trivial