Abstract
We introduce a notion of n-Lie-Rinehart algebras as a generalization of Lie-Rinehart algebras to n-ary case. This notion is also an algebraic analogue of n-Lie algebroids. We develop representation theory and describe a cohomology complex of n-Lie-Rinehart algebras. Furthermore, we investigate extension theory of n-Lie-Rinehart algebras by means of 2-cocycles. Finally, we introduce crossed modules of n-Lie-Rinehart algebras to gain a better understanding of their third cohomology groups.