Dual coalgebras of algebras over commutative rings

In the study of algebraic groups the representative functions related to monoid algebras over fields provide an important tool which also yields the finite dual coalgebra of any algebra over a field. The purpose of this note is to transfer this basic construction to monoid algebras over commutative rings R. As an application we obtain a bialgebra (Hopf algebra) structure on the finite dual of the polynomial ring R[x] over a noetherian ring R. Moreover, we give a sufficient condition for the finite dual of any R-algebra A to become a coalgebra. In particular, this condition is satisfied provided R is noetherian and hereditary.