Bousfield localization as an algebraic closure of groups (Q1825472)

The notions of HR-local group and the HR-localization of a group arose in A. K. Bousfield's work on homology localization of spaces. (Here R is a subring of the rationals.) \textit{A. K. Bousfield} [Mem. Am. Math. Soc. 186 (1977; Zbl 0364.20058)] constructs the HR-localization of a group via certain natural transfinite towers which stabilize appropriately and then proceeds to show, for example, that HR-localization coincides with R- completion for certain groups (e.g., polycyclic-by-finite). The present paper provides an alternative characterization of HR-localness in terms of the unique solvability of certain (\(\Gamma\)-)systems of equations defined over the group in question. Furthermore, adjoining the roots of all such \(\Gamma\)-systems to the group (with appropriate relations provided by the \(\Gamma\)-systems) essentially constructs the HR- localization of the group. The authors apply their result, in particular, to derive bounds on the size of the HZ-localization.