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| In [[mathematics]], specifically in [[ring theory]], an [[algebra (ring theory)|algebra]] is '''simple''' if it contains no non-trivial two-sided [[ideal (ring theory)|ideal]]s and the multiplication operation is ''not'' uniformly zero (that is, there is some ''a'' and some ''b'' such that ''ab''≠0).
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| The second condition in the definition precludes the following situation; consider the algebra with the usual matrix operations:
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| :<math> | | [http://search.Un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=Roberto&Submit=Go Roberto] is what's written in his birth certificate fortunately he never really appreciated that name. Managing people is undoubtedly where his primary income comes from. Base jumping is something that bigger been doing for several years. Massachusetts has always been his located place and his [https://Www.google.com/search?hl=en&gl=us&tbm=nws&q=people+loves&btnI=lucky people loves] it. Go of his website to hit upon out more: http://circuspartypanama.com<br><br>Feel free to surf to my site - [http://circuspartypanama.com clash of clans hack no survey] |
| \left\{\left.
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| \begin{bmatrix}
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| 0 & \alpha \\
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| 0 & 0 \\
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| \end{bmatrix}\,
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| \right| \,
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| \alpha \in \mathbb{C}
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| \right\}
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| </math>
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| 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.
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| 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 algebra]]s. 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.
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| Wedderburn's result was later generalized to [[semisimple ring]]s in the [[Artin–Wedderburn theorem]].
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| == Examples == | |
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| * A [[central simple algebra]] (sometimes called Brauer algebra) is a simple finite dimensional algebra over a [[field (mathematics)|field]] ''F'' whose [[center of an algebra|center]] is ''F''.
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| == Simple universal algebras == | |
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| In [[universal algebra]], an abstract algebra ''A'' is called "simple" [[if and only if]] it has no nontrivial [[congruence relation]]s, or equivalently, if every homomorphism with domain ''A'' is either [[injective]] or constant.
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| As congruences on rings are characterized by their ideals, this notion is a straightforward generalization of the notion from ring theory: a ring is simple in the sense that it has no nontrivial ideals if and only if it is simple in the sense of universal algebra.
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| == See also == | |
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| * [[simple group]]
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| * [[simple ring]]
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| * [[central simple algebra]]
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| ==References==
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| * [[A. A. Albert]], ''Structure of algebras'', Colloquium publications '''24''', [[American Mathematical Society]], 2003, ISBN 0-8218-1024-3. P.37.
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| [[Category:Algebras]]
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| [[Category:Ring theory]]
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