Universal Localization

Localization at a prime ideal is a powerful tool
in the study of commutative rings. Several
approaches have been studied in an attempt to
extend this to noncommutative rings. I have been
working on Cohn's notion of universal localization,
in which he inverted all square matrices that are
not zero divisors modulo the prime ideal (rather
than just attempting to invert individual elements).
This localization always exists, but has proved
to be difficult to compute. I will discuss this
construction and its relationship to several other
approaches to localization. In particular, Goldie's
localization, defined in his 1967 paper, is a factor
ring of the universal localization.