Ring-induced localizations of nilpotent groups
arXiv:2606.15392
The paper investigates the localization of groups with respect to a commutative ring R and proves that if R is a binomial ring (such as the p‑adic integers), then the R‑localization of a nilpotent group is again nilpotent, using Hall‑Petresco R‑group structures.
Abstract
For a commutative ring , we study the -localization functor on the category of groups, defined as localization with respect to the homomorphism . Our main result is that, when is a binomial ring, the -localization of a nilpotent group is again nilpotent. Taking , the ring of -adic integers, yields a new example of a localization functor that preserves nilpotency. To prove this, we characterize -local groups in terms of -groups in the sense of Myasnikov-Remeslennikov. We call an -group a Hall-Petresco -group if it satisfies a version of the Hall-Petresco identity, and show that these form a quasivariety closed under quotients by the center. The crucial input to our main result is that every -local group carries a unique Hall-Petresco -group structure.