Studies in Rings generalised Unique Factorisation Rings

-98-
Remark 4 012.
Let P be a classically right localisable prime
idea 1 in R, Q be a pri IDe idea1 0 f R wi t h Q < P, and
there exist a f,·9 uniform R-module lvi, with ann(M) = Q,
containing a copy U of a non zero right ideal of Rip.
By passing to R/Q, we assume Q = O. We can localise at
CR(P) and get the simple Rp/PR p- module U ® Rp insid e
M @R p, so there is. an n with (M ® Rp)
pn R p = o. This
implies tha t Mpn is CR(P)-torsion, so Mpn () U = o.
n
Thus 1vlP = 0 and hence P n = o. This contradiction
shows
that apart from the links between prime ideals
we have another obstruction to localisation at a
prime
id ea 1.
Definition 4 0 13 .
A prime ideal P, in a ring R, satisfies the right
second layer condition (sol.c) if the situation of the
above remark does not occur, i.e., no such Q exists.
Left second layer condition is defined analogously.
Theorem 4.14.
Let R be a Noetherian ring and let P be a prime
ideal of Ro
Then P is classically right localisable
if and only if [pJ
is right stable and P satisfies
the right second layer condition.