Difference between revisions of "Classical semi-simple ring"
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An associative right Artinian (or, equivalently, left Artinian) ring with zero [[Jacobson radical|Jacobson radical]]. The [[Wedderburn–Artin theorem|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|Homological classification of rings]]). Every [[Group algebra|group algebra]] of a finite group over a field of coprime characteristic 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. | An associative right Artinian (or, equivalently, left Artinian) ring with zero [[Jacobson radical|Jacobson radical]]. The [[Wedderburn–Artin theorem|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|Homological classification of rings]]). Every [[Group algebra|group algebra]] of a finite group over a field of coprime characteristic 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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Revision as of 19:41, 19 October 2014
An associative right Artinian (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 coprime characteristic 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.
Classical semi-simple ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Classical_semi-simple_ring&oldid=18426