 
Summary: DUALITY AND RATIONAL MODULES IN HOPF ALGEBRAS OVER
COMMUTATIVE RINGS.
J. Y. ABUHLAIL, J. G´OMEZTORRECILLAS, AND F. J. LOBILLO
Abstract. Let A be an algebra over a commutative ring R. If R is noetherian and A
is pure in RA
, then the categories of rational left Amodules and right A
comodules are
isomorphic. In the Hopf algebra case, we can also strengthen the BlattnerMontgomery
duality theorem. Finally, we give sufficient conditions to get the purity of A
is RA
.
Introduction
It is well known that the theory of Hopf algebras over a field cannot be trivially passed
to Hopf algebras over a commutative ring. For instance let us consider Z[x] as Hopf algebra
and let a be the Hopf ideal generated by 4, 2x . Let H be the Hopf Zalgebra H = Z[x]/a.
The finite dual is zero in this situation. However H = Z4[x]/ 2x , so we can view H as a
Hopf Z4algebra. If I is a Z4cofinite ideal of H then every element nonzero in H/I has
order 2, which implies every element in H
has order 2. In this situation H
is not pure
