Noncommutative Localizations of Lie-Complete Rings

In this paper we investigate the topological localizations of Lie-complete rings. It has been proved that a topological localization of a Lie-complete ring is commutative modulo its topological nilradical. Based on the topological localizations we define a noncommutative affine scheme X = Spf(A) for a Lie-complete ring A. The main result of the paper asserts that the topological localization A_(f) of A at f ∈ A is embedded into the ring 𝒪_A(X_f) of all sections of the structure sheaf 𝒪_A on the principal open set X_f as a dense subring with respect to the weak I₁-adic topology, where I₁ is the two-sided ideal generated by all commutators in A. The equality A_(f) = 𝒪_A(X_f) can only be achieved in the case of an NC-complete ring A.