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We give an example of a cluster-tilted algebra A with quiver Q, such that theassociated cluster algebra has a denominator vector which is not the dimensionvector of any indecomposable A-module. This answers a question posed by T.Nakanishi. The relevant example is a cluster-tilted algebra associated with atame hereditary algebra. We show that for such a cluster-tilted algebra A, wecan write any denominator vector as a sum of the dimension vectors of at mostthree indecomposable rigid A-modules. In order to do this it is necessary, andof independent interest, to first classify the indecomposable rigid A-modulesin this case.

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