Artinian semisimple ring

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).

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.