Studies in Rings generalised Unique Factorisation Rings

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Theorem 4 023.
If R is a Noetherian ring and X ~ Spec R, then
X is classically right localisable if and only' if
(1) X is right stable,
(2) X sati s f i.es the right second layer condition,
(3) X satisfies the right intersection condition, and
(4) X sa tisfies the incomparability conditiono
Thus we have characterised the classically right
localisable subsets of Spec R in Noetherian ringso
The same can be done for classicully left localisable
subsets by defining the left second layer condition,
left intersection property and left stability etco
analogously.
We
conclude this section of preliminaries with two
theorems.
Theorem 4.24.
If R is a Noetherian ring and X is a right stable
subset of Spec R satisfying the right second layer condition
and the right intersection condition, then C(X) is a right
Ore set ..