Uintell Mathematics Encyclopedia

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.