An associative $K$-algebra A is called reduced 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 in $char(K)=p>0$ it is shown that for group algebras reduced=soluble holds.
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.