Simple algebra

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In mathematics, specifically in ring theory, an algebra is simple if it contains no non-trivial ideals and the set {ab | a, b are elements of the algebra} ≠ {0}.

The second condition in the definition precludes the following situation: consider the algebra

$\{ \begin{bmatrix} 0 & \alpha \\ 0 & 0 \\ \end{bmatrix}\, | \, \alpha \in \mathbb{C} \}$

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 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 simple algebras up to isomorphism, i.e. any simple algebra is isomorphic to a matrix algebra over some division ring. This result was given in 1907 by 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.

[edit] Examples

A central simple algebra (sometimes called Brauer algebra) is a simple finite dimensional algebra over a field F whose center is F.

[edit] 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.