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.

  • $\begingroup$ I posted this at mathoverflow as well. $\endgroup$ – Sven Wirsing Sep 4 '14 at 13:49

The question was answered in mathoverflow:


  • $\begingroup$ Add a link to the question there, please. =) $\endgroup$ – Pedro Tamaroff Sep 4 '14 at 18:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.