  Given a partial action $\pi$ of an inverse semigroup $S$ on a ring
$\mathcal{A}$ one may construct its associated skew inverse semigroup ring
$\mathcal{A} \rtimes_\pi S$. Our main result asserts that, when $\mathcal{A}$
is commutative, the ring $\mathcal{A} \rtimes_\pi S$ is simple if, and only if,
$\mathcal{A}$ is a maximal commutative subring of $\mathcal{A} \rtimes_\pi S$
and $\mathcal{A}$ is $S$-simple. We apply this result in the context of
topological inverse semigroup actions to connect simplicity of the associated
skew inverse semigroup ring with topological properties of the action.
Furthermore, we use our result to present a new proof of the simplicity
criterion for a Steinberg algebra $A_R(\mathcal{G})$ associated with a
Hausdorff and ample groupoid $\mathcal{G}$.
