Studies in Rings generalised Unique Factorisation Rings

-72-
Lemma 3.110
Let P be a prime ideal of S, such that a(P) = P;
then pn R is a
prime ideal of Rand a(pnR) = P "R.
Lemma 3.12.
Let P be a prime ideal of S with a(P) = P; then
(PO R)S is a prime ideal of S.
Theorem 3.13.
Let R be a Noe therian ring wi th an Artinia 11 quotient
ring and let a be an automorphism on R. If every non zero
a-prime ideal of R contains a normal invertible a-ideal;
then S = R[x,~]
is a GUFR.
Proof:
By lemma 3 0 9 , S is a NoethGrian ring. Suppose that
every non zero a-prime ideal of R contains a
normal
invertible a-idealo We shall show that S has an Artinian
quotient ring Q(5)
~
conta ins a "norma 1
and every non-minimal prime ideal of S
invertible idea1.
Assume t.h a t S is not prime. Let P be a non-minimal
prime ideal of S. If x € P, then x € Cs(O) by remark 308.
If x i P, then x ~ CS(P) and a(P) = P by lemma 3 010.