  Let $\mathbb{k}$ be a field, and let $\Lambda$ be a (not necessarily finite
dimensional) $\mathbb{k}$-algebra. Let $V$ be a left $\Lambda$-module such that
is finite dimensional over $\mathbb{k}$. Assume further that $V$ has a weak
universal deformation ring $R^w(\Lambda,V)$, 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 $\Lambda$-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
$N\geq 2$.
