An isomorphism between rings and groups (Q1812829)

The author constructs bijective functors \(\varrho: {\mathcal G}\to{\mathcal R}\) and \(\sigma: {\mathcal R}\to{\mathcal G}\), where \(\mathcal G\) is a category of nilpotent groups and \(\mathcal R\) a category of commutative and not necessarily associative rings with identity. For all \(G\in \mathcal G\) and \(R\in{\mathcal R}\), \(\rho\sigma R\cong R\) and \(\sigma\rho G\cong G\). A subclass \(\mathcal L\) of \(\mathcal R\) has a decidable elementary theory if and only if \(\sigma{\mathcal L}\) has.