In this paper, the authors introduce a class of Monge-Ampère equations (defined intrinsically on a contact manifold as exterior systems and called Monge-Ampère systems [see T. Morimoto, C. R. Acad. Sci., Paris, Sér. A 289, 25-28 (1979; Zbl 0425.35023)]) on contact manifolds of arbitrary odd dimension which they call decomposable Monge-Ampère systems. They show that one can canonically associate with a decomposable Monge-Ampère system a linear object, called characteristic system, that has nice properties, and show that most of the results that hold in the case of two independent variables, as discussed in [T. Morimoto, Budzynski, Robert (ed.) et al., Symplectic singularities and geometry of gauge fields. Proceedings of the Banach Center symposium on differential geometry and mathematical physics in Spring 1995, Warsaw: Polish Academy of Sciences, Banach Cent. Publ. 39, 105-121 (1997; Zbl 0879.35008)], can be naturally generalized to this class of decomposable Monge-Ampère systems.