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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. == Examples == * A central simple algebra (sometimes called Brauer algebra) is a simple finite-dimensional algebra over a field ''F'' whose center is ''F''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Simple algebra」の詳細全文を読む スポンサード リンク
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