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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).


The second condition in the definition precludes the following situation; consider the algebra with the usual matrix operations:


:<math>
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\left\{\left.
\begin{bmatrix}
0 & \alpha \\
0 & 0 \\
\end{bmatrix}\,
\right| \,
\alpha \in \mathbb{C}
\right\}
</math>  
 
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'' &times; ''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.
 
Wedderburn's result was later generalized to [[semisimple ring]]s in the [[Artin–Wedderburn theorem]].
 
== Examples ==
 
* 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''.
 
== Simple universal algebras ==
 
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.
 
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.
 
== See also ==
 
* [[simple group]]
* [[simple ring]]
* [[central simple algebra]]
 
==References==
* [[A. A. Albert]], ''Structure of algebras'', Colloquium publications '''24''', [[American Mathematical Society]], 2003, ISBN 0-8218-1024-3.  P.37.
 
[[Category:Algebras]]
[[Category:Ring theory]]

Revision as of 06:47, 27 February 2014


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