Artinian semisimple ring

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.

• $M_n(k)$ 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.