  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 dimensional cohomology groups.
