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Let 
$X$ be a nonempty set and 
${\mathcal{P}}(X)$ the power set of 
$X$. The aim of this paper is to identify the unital subrings of 
${\mathcal{P}}(X)$ and to compute its cardinality when it is finite. It is proved that any topology 
$\unicode[STIX]{x1D70F}$ on 
$X$ such that 
$\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$, where 
$\unicode[STIX]{x1D70F}^{c}=\{U^{c}\mid U\in \unicode[STIX]{x1D70F}\}$, is a unital subring of 
${\mathcal{P}}(X)$. It is also shown that 
$X$ is finite if and only if any unital subring of 
${\mathcal{P}}(X)$ is a topology 
$\unicode[STIX]{x1D70F}$ on 
$X$ such that 
$\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}^{c}$ if and only if the set of unital subrings of 
${\mathcal{P}}(X)$ is finite. As a consequence, if 
$X$ is finite with cardinality 
$n\geq 2$, then the number of unital subrings of 
${\mathcal{P}}(X)$ is equal to the 
$n$th Bell number and the supremum of the lengths of chains of unital subalgebras of 
${\mathcal{P}}(X)$ is equal to 
$n-1$.
