Studies in Rings generalised Unique Factorisation Rings

-50-
Remark 2.22.
Prime GUFRs are the prime Noetherian rings in
which every non zero (non minimal) prime ideal contains
a normal Q(R)- invertible ideal, where Q(R) is the
simple Artinian quotient ring of R. Thus every NUFR is
a prime GUFR. Examples of prime GUFRs which are not
NUFRs are g~ven in Chapter 3. As in corollary 2.8, it
can be seen that, if R is a prime GUFR, T is a simple
Noetherian ring.
Noetherian rings in which every prime ideal contains
a normal invertible ideal are a generalisation of GUFRs.
But we show that there is nothing to be gained by
this
extension as such rings turn out to be
prime GUFRs.
Theorem 2.23.
Let R be a GUFI-\
in which ever)' prime ideal c o n t.a i.ns
normal invertible ideals. Then R is a prime GUFR.
Proof
First we prove, in GUFRs with every prime ideal
contains normal invertible ideals, every non zero ideal
contains a normal invertible ideal.