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:

[math]\displaystyle{ \left\{\left. \begin{bmatrix} 0 & \alpha \\ 0 & 0 \\ \end{bmatrix}\, \right| \, \alpha \in \mathbb{C} \right\} }[/math]

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 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 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. The same remark applies with respect to groups and normal subgroups; hence the universal notion is also a generalization of a simple group (it is a matter of convention whether a one-element algebra should be or should not be considered simple, hence only in this special case the notions might not match).

A theorem by Roberto Magari in 1969 asserts that every variety contains a simple algebra.

See also

• simple group
• simple ring
• central simple algebra

References

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