  In this paper, we introduce the concept of a Rota-Baxter paired module to
study Rota-Baxter modules without necessarily a Rota-Baxter operator. We obtain
two characterizations of Rota-Baxter paired modules, and give some basic
properties of Rota-Baxter paired modules. Beginning with the connection between
the notion of integrals in the representations of Hopf algebra and of the
notion of an integral algebra as a motivation and special case of Rota-Baxter
algebra of weight zero, we obtain a large number of Rota-Baxter paired modules
from Hopf related algebras and modules, including semisimple Hopf algebras,
weak Hopf algebras, Long bialgebras, quasitriangular Hopf algebras, weak Hopf
modules, dimodules and Doi-Hopf modules. Some of them give new examples of
Rota-Baxter algebras.
