Let $k$ be a field, and let $A$ be a finite-dimensional associative $k$-algebra. I assume that $A$ is basic (that is, in any decomposition of $A$ into a direct sum of indecomposable projective $A$-modules, these are pairwise non-isomorphic).
Wedderburn's principal theorem states that
If $A/rad(A)$ is separable, then there is a subalgebra $B$ of $A$ such that $B$ is semisimple and $A=B\oplus rad(A)$ as $k$-vector spaces.
In this post, I will say that Wedderburn's principal theorem holds for an algebra $A$ when the conclusion of the above theorem is true for $A$ (even if $A$ does not satisfy the assumptions of the theorem).
Here are some families of cases where the theorem does holds:
- if $k$ is algebraically closed, or even a perfect field;
- if $A$ is the path algebra of some finite quiver $Q$ modulo an admissible ideal (or, more generally, the tensor algebra of a $k$-species modulo an admissible ideal), without assumptions on the field $k$.
An example where it does not hold is the $\mathbb{F}_2(t)$-algebra $\mathbb{F}_2(t)[X]/(X^2-t)^2$ (Exercise 6.4 of Drozd and Kirichenko's book Finite Dimensional Algebras). Similar examples are obtained by replacing $\mathbb{F}_2$ by any field of characteristic $2$, and also in other positive characteristics.
The above examples are all local algebras. From these, we can build new examples by taking direct products or extensions.
My question is:
What are other families of examples where Wedderburn's principal theorem does not hold?
The bigger the family the better!
The above examples were all I could find in the literature. Without the assumption that $k$ is a field, there is also the case of endomorphism rings of finite abelian groups, mentionned in Auslander, Reiten and Smalø's book Representation Theory of Artin Algebras.