Simple algebra

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An algebra, consisting of more than one element, without two-sided ideals different from 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.

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Simple algebra. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article