Examples of Semi-Simple Rings

Question

I am studying Semi-Simple Rings from "A first course in Non-Commutative Rings " by T.Y.Lam .

But hardly I am finding any examples to the definitions that are being taught.There is no example of a semisimple ring only the definition is provided i.e "A ring is semisimple if it is simple as a module over itself"

When I read rings ,fields there were so many examples but here I get none.Why ??

Also how to verify whether the popular rings such as $\mathbb Z$ ,$M_n(\mathbb R)$ are simple or not?

Answer 1

You mean semisimple as a module over itself. Examples include

• Any field. More generally, any division ring.
• If $R$ is semisimple, then so is $M_n(R)$ (exercise). Hence, for example, $M_n(\mathbb{R})$ is semisimple.
• If $R$ and $S$ are semisimple, then so is $R \times S$ (exercise).

It follows that any ring which is a finite product of matrix rings over division rings is semisimple. By the Artin-Wedderburn theorem, this exhausts all possibilities. In particular, $\mathbb{Z}$ is not semisimple: it is not semisimple as a module over itself because it has nontrivial submodules, such as $2 \mathbb{Z}$, which don't have complements.

In practice, a common source of examples of semisimple rings come from finite groups: if $k$ is a field and $G$ is a finite group of order not divisible by the characteristic of $k$, then the group ring $k[G]$ is semisimple by Maschke's theorem: equivalently, the category of $G$-representations over $k$ is semisimple.

See this blog post for a longer discussion with proofs.