Some Examples of Affine Monoid Algebras

We construct a finitely generated monoid S with a zero element such that for every field K the Jacobson radical of the monoid algebra KS] is a sum of nilpotent ideals but is not nilpotent. Moreover, the contracted monoid algebra K ₀S] is a monomial algebra.

If K is a field of characteristic p > 0, then we construct a finitely presented group H _p such that the Jacobson radical J of the group algebra KH _p ] is a sum of nilpotent ideals, but is not nilpotent. Moreover, KH _p ]/J is a domain.