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 algebrasalgebra]]s, where every nonzero 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 [[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]].
