  We show that cluster algebras do not contain non-trivial units and that all
cluster variables are irreducible elements. Both statements follow from Fomin
and Zelevinsky's Laurent phenomenon. As an application we give a criterion for
a cluster algebra to be a factorial algebra. This can be used to construct
cluster algebras, which are isomorphic to polynomial rings. We also study
various kinds of upper bounds for cluster algebras, and we prove that factorial
cluster algebras coincide with their upper bounds.
