Artinian semisimple ring - Imported Knowlpedia Content

An Artinian semisimple ring is a ring $R$ that is semisimple and Artinian (i.e. it satisfies the descending chain condition on ideals, equivalently on left ideals).

Remarks

By the Artin–Wedderburn theorem, such rings are precisely finite direct products of matrix rings over division rings, and these rings are the basic building blocks for finite-length module categories.

Examples

$M_n(k)$ is Artinian semisimple for any field $k$ .
A finite product of fields, e.g. $\mathbb{Q}\times \mathbb{Q}\times \mathbb{F}_5$ , is Artinian semisimple.
An infinite product $\prod_{i\in \mathbb{N}} k$ (with $k$ a field) is not Artinian, hence not Artinian semisimple.