The reciprocal complements of classes of integral domains

Given an integral domain D with quotient field $Q\left(D\right)$, the reciprocal complement of D is the subring $R\left(D\right)$ of $Q\left(D\right)$ 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 $R\left(D\right)$ and we describe the reciprocal complements of classes of domains, including semigroup algebras and $D+m$ constructions. We also characterize when $R\left(D\right)$ is a DVR.