Studies in Rings generalised Unique Factorisation Rings

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Definition 4 015.
An ideal I of R is said to have the right
Artin-Rees (Arrl-2£Qper~y if for any finitely generated
right R module M containing an essential submodule L
with LI = 0, there is a positive integer n such that
n
MI = O. In this case we call I a right AR Ldea l ,
Left AR
property is defined analogously.
Re ma r k Lt 4 1 () .
( 1) A prime idea 1 P wi th the ri gh t AR property
always satisfies the right second layer
conditiono
( 2 ) Ani(j E: a 1 I 0 f R i srigh tAR i fandon 1y i f
for every right ideal K of R, there is a
positive integer n such that KC\I n < KI.
Theorem 4 017.
If R is a Noetherian ring and P is a prime ideal
with the right AR
property, then P is classically
localisable if and only if there is no
prime ideal Q
of R with P < Q and Q ~ P.
Lemma 4.18.
Suppose an ideal in a
right Noetherian ring R has
the right AR p rop e r t v , If Q ~ P in Spec R and if I .$ P,
then I S Q.