Studies in Rings generalised Unique Factorisation Rings

-.l(~il-
X is classically right localisable if C(X) is a right
Ore set and the localisation R., = H C(X)-l has the
A
following properties.
(1) For every P ~ X, the ring RX/PR X
is Artinian.
(2) The only right primitive ideals are PR X
for
p E. x.
(3) Every finitely generated RX-module which i~
an es~ential extension of a simple right
RX-module is Artinian.
Definition 4.21.
Let X S Spec R. Then X satisfies the right
intersection condition if for any r i qh t ideal I of R
such that I nCR(p) 1= ~ for every P £; X, the intersection
I () C (X) i s non- emp t Y• vV e say X sati s fie s right
second layer condition if every prime ideal in X
satisfies right second layer condition and we say X
satisfies the incomparability conditiol1 if there do
not
exist prime ideals P,Q € X with Q < P.
Proposition 4.22.
If R is a Noetherian ring and X is a
right stable
subset of Spec R satisfying the right intersection
condition and right second layer condition, then C(X)
is Cl right O'r-e set.