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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 oftopological inverse semigroup actions to connect simplicity of the associatedskew inverse semigroup ring with topological properties of the action.Furthermore, we use our result to present a new proof of the simplicitycriterion for a Steinberg algebra $A_R(\mathcal{G})$ associated with aHausdorff and ample groupoid $\mathcal{G}$.

Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics