Studies in Rings generalised Unique Factorisation Rings

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In the proof of le~ma 2.17 we have to make use
of the well known AR
property and some other localisation
techniques which we have not yet discussed in this thesis.
When we discuss the localisation at a
prime ideal in
chapter 4, Wb
will give a proof of this lemma.
Lemma 2.18.
Let R be a GUFR and P = pR = Rp be a non-minimal
prime ideal of R. Then p is regular and P is localisable.
Proof
Since P is a non minimal prime ideal of H, from the
definition of GUFR and by'lb:oran 2 0 6 , P contains a regular
normal element e(say). Therefore e = pr l
= r 2P
for some
r l,r2
€ R. Now the regularity of p follows from the
regularity of e. The second part of the .lemma follows
from lemma 2 017 and from theorem 2.9.
Lemma 2.19.
Let R be a GUFR and P be minimal prime ideal of R.
Then P cannot contain any normal invertible ideal.
Proof
Suppose if possible that P contains a normal
invertible ideal aR = Ra (say). Then a E: C R
(0) by