Triple cohomology of Lie-Rinehart algebras and the canonical class of associative algebras

We introduce a bicomplex which computes the triple cohomology of Lie-Rinehart algebras. We prove that the triple cohomology is isomorphic to the Rinehart cohomology provided the Lie-Rinehart algebra is projective over the corresponding commutative algebra. As an application we construct a canonical class in the third dimensional cohomology corresponding to an associative algebra and extend Sridharan's result on almost commutative algebras.