Let \(\sum \leq \prod\) be the direct sum and product of countably many copies of the integers, let \(\sum \leq D\leq \prod\) such that D/\(\sum\) is the divisible part of \(\prod /\sum\), and let \(e_ n=(\delta_{in})_{i\in {\mathbb{N}}}\in \prod\) be the generalized n-th unit vector. A subring R of the Z-endomorphism ring E(\(\prod)\) of \(\prod\) is called hyperpure if \(r\in R\), \(m\in {\mathbb{N}}\) and \(e_ nr\in m\prod\) for almost all n imply \(r\in mR\). The authors prove that, given any countable hyperpure unital subring R of E(\(\prod)\) such that DR\(\subseteq D\), there exists a pure subgroup A of D such that \(E(A)=R\oplus Fin(A)\) where Fin(A) consists of all endomorphisms of A with image of finite rank. Moreover, A may be chosen to contain any preassigned countable subset X of D. A similar realization theorem for uncountable subrings of E(\(\prod)\) is contained in a paper by \textit{M. Dugas} and \textit{R. Göbel} [Houston J. Math. 11, 471-483 (1985; Zbl 0597.20046)]. The authors give several interesting applications. One of them shows the existence of essentially indecomposable pure subgroups of \(\prod\).