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

Let $\mathbb{k}$ be a field, and let $\mathrm{\Lambda }$ be a (not necessarily finite dimensional) $\mathbb{k}$-algebra. Let $V$ be a left $\mathrm{\Lambda }$-module such that is finite dimensional over $\mathbb{k}$. Assume further that $V$ has a weak universal deformation ring $R^w(\mathrm{\Lambda },V)$, which is a complete Noetherian commutative local $\mathbb{k}$-algebra with residue field $\mathbb{k}$. We prove in this note that under certain conditions on the $\mathrm{\Lambda }$-module $V$, that if $R^w(\mathrm{\Lambda },V)$ is a quotient of $\mathbb{k}[\text{\hspace{0.17em}⁣}[t]\text{\hspace{0.17em}⁣}]$, then $R^w(\mathrm{\Lambda },V)$ is either isomorphic to $\mathbb{k}$, or $\mathbb{k}[\text{\hspace{0.17em}⁣}[t]\text{\hspace{0.17em}⁣}]$, or to $\mathbb{k}[\text{\hspace{0.17em}⁣}[t]\text{\hspace{0.17em}⁣}]\mathrm{/}(t^N)$ for some integer $N\ge 2$.