  In this paper we define and study generalized Atiyah classes for quasi
coherent sheaves relative to arbitrary morphisms of schemes. We use derivations
and quasi coherent sheaves of left and right O-modules to define a generalized
first order jet bundle J(E) and a generalized Atiyah sequence for E. The
generalized jet bundle J(E) is a left and right module over a sheaf J of
associative rings on X. The sheaf J is an extension of O with a sheaf I of two
sided ideals of square zero. The Atiyah sequence gives rise to a generalized
Atiyah class c(E) with the property that c(E)=0 if and only if the left
structure on J(E) is O-isomorphic to the right structure on J(E). We give
examples where c(E)=0 and c(E)\neq 0 hence the class c(E) is a non trivial
characteristic class.
