On universal deformation rings and stable homogeneous tubes

Let k be a field of any characteristic and let Λ be a finite dimensional k-algebra. We prove that if V is a finite dimensional right Λ-module that lies in the mouth of a stable homogeneous tube T of the Auslander-Reiten quiver Λ with End_Λ (V) a division ring, then V has a versal deformation ring R (Λ, V) isomorphic to k\![t\!]. As consequence we obtain that if k is algebraically closed, Λ is a symmetric special biserial k-algebra and V is a band Λ-module with End_Λ (V) k that lies in the mouth of its homogeneous tube, then R (Λ, V) is universal and isomorphic to k\![t\!].