Cyclic Lie–Rinehart algebras

We study Lie–Rinehart algebra structures in a framework provided by duality pairings of modules over unital commutative associative algebras. Thus, we construct new examples of Lie brackets corresponding to a fixed anchor map whose image is a cyclic submodule of the derivation module, and therefore we call them cyclic Lie–Rinehart algebras. Special cases of our construction include Lie algebroid structures on cotangent bundles of differential manifolds and also certain differential operators that occur in mathematical physics.