  We introduce the new notion of epsilon-graded associative algebras which
takes its root into the notion of commutation factors introduced in the context
of Lie algebras. We define and study the associated notion of
epsilon-derivation-based differential calculus, which generalizes the
derivation-based differential calculus on associative algebras. A corresponding
notion of noncommutative connection is also defined. We illustrate these
considerations with various examples of epsilon-graded algebras, in particular
some graded matrix algebras and the Moyal algebra. This last example permits
also to interpret mathematically a noncommutative gauge field theory.
