The reciprocal complements of classes of integral domains

Given an integral domain $D$ with quotient field $\mathcal{Q}(D)$, the reciprocal complement of $D$ is the subring $R(D)$ of $\mathcal{Q}(D)$ whose elements are all the sums $\frac{1}{d_1}+\dots +\frac{1}{d_n}$ for $d_1,\dots ,d_n$ nonzero elements of $D$. In this article we study problems related with prime ideals, localizations and Krull dimension of rings of the form $R(D)$ and we describe the reciprocal complements of classes of domains, including semigroup algebras and $D+\mathfrak{m}$ constructions. We also characterize when $R(D)$ is a DVR.