An associative $K$-algebra A is called reduced (or often basic) if $A/rad(A)$ has no nilpotent elements. It can be shown that this is equivalent to that $A/rad(A)$ is a isomorphic to a direct sum of division algebras. Here $rad(A)$ is the Jacobson radical of $A$.


In the article "Reduced group algebras" and a related one (<https://math.stackexchange.com/questions/819466/the-division-algebras-arising-in-the-wedderburn-decomposition-of-a-finite-group>) in $char(K)=p>0$ it is shown that for group algebras reduced=soluble holds. (An associatve algebra is called soluble if the factor $A/rad(A)$ is commutative.)


If we take a soluble associative algebra and the direct sum with a division algebra (e.g. lower diagonal matices togehther with real quaternion) we obtain an example of that kind.


My question is whether there are some natural examples of such reduced associative algebras which are not soluble and arising from direct products of division algebras, soluble (or abelian) algebras.


Is it possible to obtain a reduced algebra from every associative algebra?


One source is local algebras and their direct sum. But not all reduced algebras are a direct sum of local algebras.



Examples of reduced associative algebras
Let $Q$ be the quaternion group of order $8$ and let $P(Q)$ be the power set of $Q$ viewed as a monoid by $AB=\{ab\mid a\in A,b\in B\}$. Let $\mathbb QP(Q)$ be the contracted monoid algebra of $P(Q)$ over $\mathbb Q$ (so identify the zero of the monoid with the zero of the algebra). I claim that $\mathbb QP(Q)$ is reduced but not solvable. 


The algebra $\mathbb QP(Q)$ is not semisimple because $P(Q)$ contains elements which are not von Neumann regular. Its semisimple quotient is the direct product $\prod\_{H\leq Q} \mathbb Q[Q/H]$ of the group algebras of the quotients of $Q$. Since it is well known that $\mathbb QQ$ has a quaternion Wedderburn component, and the other direct factors are commutative this is an example of a reduced algebra which is not semisimple.


There might be a clever argument to show it is a direct product of a quaternion division algebra and a soluble algebra using Mobius inversion on the subgroup lattice. I will have to think about it.


