  In this article we define and investigate a notion of parallel transport on
finite projective modules over finite matrix algebras. Given a derivation-based
differential calculus on the algebra and a connection on the module, we
construct for every derivation X a module parallel transport, which is a lift
to the module of the one-parameter group of algebra automorphisms generated by
X. This parallel transport morphism is determined uniquely by an ordinary
differential equation depending on the covariant derivative along X. Based on
these parallel transport morphisms, we define a basic set of gauge invariant
observables, i.e. functions from the space of connections to the complex
numbers. For modules equipped with a hermitian structure, we prove that this
set of observables is separating on the space of gauge equivalence classes of
hermitian connections. This solves the gauge copy problem for fuzzy gauge
theories.
