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- Simple_algebra abstract "In mathematics, specifically in ring theory, an algebra is simple if it contains no non-trivial two-sided ideals and the multiplication operation is not uniformly zero (that is, there is some a and some b such that ab≠0).The second condition in the definition precludes the following situation; consider the algebra with the usual matrix operations:This is a one-dimensional algebra in which the product of any two elements is zero. This condition ensures that the algebra has a minimal nonzero left ideal, which simplifies certain arguments.An immediate example of simple algebras are division algebras, where every element has a multiplicative inverse, for instance, the real algebra of quaternions. Also, one can show that the algebra of n × n matrices with entries in a division ring is simple. In fact, this characterizes all finite-dimensional simple algebras up to isomorphism, i.e. any finite-dimensional simple algebra is isomorphic to a matrix algebra over some division ring. This result was given in 1907 by Joseph Wedderburn in his doctoral thesis, On hypercomplex numbers, which appeared in the Proceedings of the London Mathematical Society. Wedderburn's thesis classified simple and semisimple algebras. Simple algebras are building blocks of semi-simple algebras: any finite-dimensional semi-simple algebra is a Cartesian product, in the sense of algebras, of simple algebras.Wedderburn's result was later generalized to semisimple rings in the Artin–Wedderburn theorem.".
- Simple_algebra wikiPageID "577486".
- Simple_algebra wikiPageRevisionID "598772641".
- Simple_algebra hasPhotoCollection Simple_algebra.
- Simple_algebra subject Category:Algebras.
- Simple_algebra subject Category:Ring_theory.
- Simple_algebra type Abstraction100002137.
- Simple_algebra type Algebra106012726.
- Simple_algebra type Algebras.
- Simple_algebra type Cognition100023271.
- Simple_algebra type Content105809192.
- Simple_algebra type Discipline105996646.
- Simple_algebra type KnowledgeDomain105999266.
- Simple_algebra type Mathematics106000644.
- Simple_algebra type PsychologicalFeature100023100.
- Simple_algebra type PureMathematics106003682.
- Simple_algebra type Science105999797.
- Simple_algebra comment "In mathematics, specifically in ring theory, an algebra is simple if it contains no non-trivial two-sided ideals and the multiplication operation is not uniformly zero (that is, there is some a and some b such that ab≠0).The second condition in the definition precludes the following situation; consider the algebra with the usual matrix operations:This is a one-dimensional algebra in which the product of any two elements is zero.".
- Simple_algebra label "Algebra semplice".
- Simple_algebra label "Algèbre simple".
- Simple_algebra label "Enkelvoudige algebra".
- Simple_algebra label "Simple algebra".
- Simple_algebra label "単純環".
- Simple_algebra sameAs Algèbre_simple.
- Simple_algebra sameAs Algebra_semplice.
- Simple_algebra sameAs 単純環.
- Simple_algebra sameAs Enkelvoudige_algebra.
- Simple_algebra sameAs m.02rvks.
- Simple_algebra sameAs Q1405695.
- Simple_algebra sameAs Q1405695.
- Simple_algebra sameAs Simple_algebra.
- Simple_algebra wasDerivedFrom Simple_algebra?oldid=598772641.
- Simple_algebra isPrimaryTopicOf Simple_algebra.