# Classical semi-simple ring

An associative right (or, equivalently, left) Artinian ring with zero Jacobson radical. The Wedderburn–Artin theorem describes the structure of the classical semi-simple rings. The class of classical semi-simple rings can also be characterized by homological properties (see Homological classification of rings). Every group algebra of a finite group over a field of characteristic coprime with the order of the group is a classical semi-simple ring. Commutative classical semi-simple rings are finite direct sums of fields. Connected with classical semi-simple rings is Goldie's theorem, which states that a ring has a left classical ring of fractions that is a classical semi-simple ring if and only if it satisfies the maximum condition for left annihilators and does not contain any infinite direct sums of left ideals.

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Classical semi-simple ring.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Classical_semi-simple_ring&oldid=33976