A note on deformations of finite dimensional modules over k-algebras

Let k be a field, and let Λ be a (not necessarily finite dimensional) k-algebra. Let V be an indecomposable left Λ-module which is finite dimensional over k and such that dimkExtΛ1(V,V)≤1. Assume further that V has a weak universal deformation ring Rw(Λ,V), which is a complete Noetherian commutative local k-algebra with residue field k. We prove in this note, under certain conditions on the Λ-module V, that Rw(Λ,V) is either isomorphic to k, or k[t], or to k[t]/(tN) for some integer N≥2.