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Let $\mathbb{k}$ be a field, and let $\Lambda$ be a (not necessarily finitedimensional) $\mathbb{k}$-algebra. Let $V$ be a left $\Lambda$-module such thatis finite dimensional over $\mathbb{k}$. Assume further that $V$ has a weakuniversal deformation ring $R^w(\Lambda,V)$, which is a complete Noetheriancommutative local $\mathbb{k}$-algebra with residue field $\mathbb{k}$. Weprove in this note that under certain conditions on the $\Lambda$-module $V$,that if $R^w(\Lambda,V)$ is a quotient of $\mathbb{k}[\![t]\!]$, then$R^w(\Lambda,V)$ is either isomorphic to $\mathbb{k}$, or$\mathbb{k}[\![t]\!]$, or to $\mathbb{k}[\![t]\!]/(t^N)$ for some integer$N\geq 2$.

A note on deformations of finite dimensional modules over $\mathbb{k}$-algebras