Simple ring

In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field.

The center of a simple ring is necessarily a field. It follows that a simple ring is an associative algebra over this field. So, simple algebra and simple ring are synonyms.

Several references (e.g., Lang (2002) or Bourbaki (2012)) require in addition that a simple ring be left or right Artinian (or equivalently semi-simple). Under such terminology a non-zero ring with no non-trivial two-sided ideals is called quasi-simple.

According to the Artin–Wedderburn theorem, every simple ring that is left or right Artinian is a matrix ring over a division ring. In particular, the only simple rings that are a finite-dimensional vector space over the real numbers are rings of matrices over either the real numbers, the complex numbers, or the quaternions.

An example of a simple ring that is not a matrix ring over a division ring is the Weyl algebra.

A ring is a simple algebra if it contains no non-trivial two-sided ideals.

Wedderburn's result was later generalized to semisimple rings in the Artin–Wedderburn theorem.