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 uniformly 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 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.

Simple (abstract algebra)

In mathematics, the term simple is used to describe an algebraic structure which in some sense cannot be divided by a smaller structure of the same type. Put another way, an algebraic structure is simple if the kernel of every homomorphism is either the whole structure or a single element. Some examples are:

The general pattern is that the structure admits no non-trivial congruence relations.

The term is used differently in semigroup theory. A semigroup is said to be simple if it has no nontrivial ideals, or equivalently, if Green's relation J is the universal relation. Not every congruence on a semigroup is associated with an ideal, so a simple semigroup may have nontrivial congruences. A semigroup with no nontrivial congruences is called congruence simple.