Rigid and Schurian modules over cluster-tilted algebras of tame type

We give an example of a cluster-tilted algebra Λ with quiver Q , such that the associated cluster algebra A ( Q ) has a denominator vector which is not the dimension vector of any indecomposable Λ -module. This answers a question posed by T. Nakanishi. The relevant example is a cluster-tilted algebra associated with a tame hereditary algebra. We show that for such a cluster-tilted algebra Λ , we can write any denominator vector as a sum of the dimension vectors of at most three indecomposable rigid Λ -modules. In order to do this it is necessary, and of independent interest, to first classify the indecomposable rigid Λ -modules in this case.