Studies in Rings generalised Unique Factorisation Rings

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so that Sg = gS $ P. Since S is prime 9 € CS(O) and
hence gS = 5g is Q(S)-invertible. Thus when S is prime
every non zero prime ideal P contains a normal invertible
ideal and so S is a GUFR.
Remark 3 014.
In the proof of the above
theorem, we have not used
the non-minimality of P. Thus in S every (non zero)
prime ideal contains a normal invertible ideal.
Hence
S is a prime GUFR by theorem 2.2~ and R is an a-prime ring
by lemma 3011. However, in this case we have the following
cha r ac ter t s e t.Lo n ,
Theorem 3.15.
Let R be a Noetherian ring with an Artinian quotient
ring and let a be an automorphism on R.
Then S = R[x,a]
is a prime GUFR
if and only if R is an a-prime ring in
which every non-zero a-prime ideal contains a normal
invertible a-ideal.
Proof:
Sufficient part of the theorem follows from theorem 3.13.