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Difference between revisions of "Simple algebra"

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An algebra, consisting of more than one element, without two-sided ideals different from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085190/s0851901.png" /> and the entire algebra. A simple algebra without a unit element need not be a [[Simple ring|simple ring]], since in this case not every ideal in the ring is an ideal in the algebra. The classification of the finite-dimensional simple algebras is known for certain classes of algebras (cf. [[Alternative rings and algebras|Alternative rings and algebras]]; [[Jordan algebra|Jordan algebra]]; [[Lie algebra|Lie algebra]]). Every associative algebra over a field possessing a unit element is imbeddable in a simple algebra with the same unit element.
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An algebra, consisting of more than one element, without two-sided ideals different from $0$ and the entire algebra. A simple algebra without a unit element need not be a [[simple ring]], since in this case not every ideal in the ring is an ideal in the algebra. The classification of the finite-dimensional simple algebras is known for certain classes of algebras (cf. [[Alternative rings and algebras]]; [[Jordan algebra]]; [[Lie algebra]]). Every associative algebra over a field possessing a unit element is imbeddable in a simple algebra with the same unit element.
  
For references, see [[Simple ring|Simple ring]].
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For references, see [[Simple ring]].

Latest revision as of 20:33, 28 December 2014

An algebra, consisting of more than one element, without two-sided ideals different from $0$ and the entire algebra. A simple algebra without a unit element need not be a simple ring, since in this case not every ideal in the ring is an ideal in the algebra. The classification of the finite-dimensional simple algebras is known for certain classes of algebras (cf. Alternative rings and algebras; Jordan algebra; Lie algebra). Every associative algebra over a field possessing a unit element is imbeddable in a simple algebra with the same unit element.

For references, see Simple ring.

How to Cite This Entry:
Simple algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simple_algebra&oldid=35929
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article