Given a right Noetherian ring R
and a prime ideal P of R, the injective hull of the right R-module R∕P is a finite
power of a uniquely determined indecomposable injective IP. One forms the ring of
right quotients RP of R relative to IP and the right ideal M = PRP of RP
generated by P. The M-adic and IP-adic topologies are compared; they turn out
to coincide on every finitely generated RP-module when RP is a classical
quasi-local ring with maximal ideal M. This condition also implies that
R satisfies the right Ore condition with respect to the multiplicative set
𝒞(P) introduced by Goldie, that the M-adic completion RP of RP is the
bicommutator of IP, and that RP is an n by n matrix ring over a complete local
ring.
