  We introduce a notion of duality for a Lie-Rinehart algebra giving certain
bilinear pairings in its cohomology generalizing the usual notions of
Poincar\'e duality in Lie algebra cohomology and de Rham cohomology. We show
that the duality isomorphisms can be given by a cap product with a suitable
fundamental class and hence may be taken natural in any reasonable sense.
Thereafter, for a Lie-Rinehart algebra satisfying duality, we introduce a
certain intrinsic module which provides a crucial ingredient for the
construction of the bilinear pairings. This module determines a certain class,
called modular class of the Lie-Rinehart algebra, which lies in a certain
Picard group generalizing the abelian group of flat line bundles on a smooth
manifold. Finally, we show that a Poisson algebra having suitable properties
determines a certain module for the corrresponding Lie-Rinehart algebra and
hence modular class whose square yields the module and characteristic class for
its Lie-Rinehart algebra mentioned before. In particular, this gives rise to
certain bilinear pairings in Poisson homology.
