Hide/Show Apps

Noncommutative Localizations of Lie-Complete Rings

2016-01-01
Dosi, Anar
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 is an element of 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 I-1-adic topology, where I-1 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.