Abstract
We construct a finitely generated monoid S with a zero element such that for every field K the Jacobson radical of the monoid algebra K[S] is a sum of nilpotent ideals but is not nilpotent. Moreover, the contracted monoid algebra K 0[S] 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 K[H p ] is a sum of nilpotent ideals, but is not nilpotent. Moreover, K[H p ]/J is a domain.
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