 
Summary: Dual coalgebras of algebras over
commutative rings
Jawad Y. Abuhlail, Jos´e G´omezTorrecillas
and Robert Wisbauer
Abstract
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 coal
gebra 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 fi
nite dual of any Ralgebra A to become a coalgebra. In particular this condition is
satisfied provided R is noetherian and hereditary.
Introduction
Let k be a field and consider a group G. The commutative Hopf algebra Rk(G) of all
kvalued representative functions over G plays a prominent role in the finite dimensional
representation theory of G (e.g., [4]). From the point of view of the algebraic theory
of Hopf algebras, Rk(G) can be considered as the dual Hopf algebra k[G]
of the group
algebra k[G]. In fact, the construction of the dual coalgebra A
