 
Summary: Dual Entwining Structures and Dual Entwined
Modules
Jawad Y. Abuhlail
Mathematics Department
Birzeit University
P.O. Box 14, Birzeit  Palestine
Abstract
In this note we introduce and investigate the concepts of dual entwining structures
and dual entwined modules. This generalizes the concepts of dual DoiKoppinen
structures and dual DoiKoppinen modules introduced (in the infinite case over rings)
by the author is his dissertation.
Introduction
This note deals with the following problem: let (A, C, ) be a given entwining structure over
a commutative base ring R. Find an Rsubalgebra A C
and an Rcoalgebra C A
,
such that (A, C,
) is an entwining structure.
For general entwining structures, it's not clear if such a dual entwining structure exists.
However, once it's found, we have the expected duality theorems between the corresponding
