Studies in Rings generalised Unique Factorisation Rings

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Definition 2.2.
An element a in a ring R is said to be a normal
element, if aR = Ra = 10
In this case we call the
ideal I, a
normal, ideal.
Definition 2 03.
Let R be a Noetherian ring with an over-ring S.
Then R is called a Generalised Unique factorisation ring
(GUFR), if every non-minimal prime ideal of R contains
a normal~S-invertible ideal.
Examples 2o~.
(1) In any commutative Noetherian domain 0 every
nonzero prime ideal contains
Q-invertible principal
ideals, where Q is the quotient field of D.
Thus every
commutative i'Joetherian integral domain is a GUFR.
(2) A Noetherian unique factorisation ring, as defined
in [2J is a prime Noetherian ring R in which every non
zero prime ideal contains a normal prime ideal. Taking
S = Q(R), the simple Artinian quotient ring of R, it
can be
seen that every normal element in R is invertible
in S
and thus every normal prime ideal is S-invertible.
So R is a
prime GUFR.