# Simple algebra

In mathematics, specifically in ring theory, an algebra is **simple** if it contains no non-trivial two-sided ideals and the multiplication operation is *not* 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:

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* × *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 algebras. 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 rings in the Artin–Wedderburn theorem.

## Examples

- A central simple algebra (sometimes called Brauer algebra) is a simple finite-dimensional algebra over a field
*F*whose 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 relations, 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

## References

- A. A. Albert,
*Structure of algebras*, Colloquium publications**24**, American Mathematical Society, 2003, ISBN 0-8218-1024-3. P.37.