Summary: Let \(k\) be a field of characteristic 0 and let \(A\) be a supercommutative associative \(k\)-superalgebra. Let \({\mathcal L}\) be a \(k-A\)-Lie-Rinehart superalgebra. From these data, one can construct a superalgebra of differential operators \({\mathcal V}(A,{\mathcal L})\) (generalizing the enveloping superalgebra of a Lie superalgebra). We give a definition of Lie-Rinehart superalgebra morphisms allowing to generalize the notions of inverse image and direct image. We prove that a Lie-Rinehart superalgebra morphism decomposes into a closed imbedding and a projection. Furthermore, we see that, under some technical conditions, a closed imebedding decomposes into two closed imbeddings of different nature. The first one looks like a Lie superalgebra morphism. The second one looks like a supermanifold closed imbedding and satisfies a generalization of the Kashiwara’s theorem. Then, as in the \({\mathcal D}\)-module theory, we introduce a duality functor. Finally, we prove that, in the closed imbedding case, the direct image and the duality functor commute.