Studies in Rings generalised Unique Factorisation Rings

-59-
Let PI be a non-minimal prime ideal of R[x].
Then,
Case-I.
P 1 C\ R .i_~) a non minimal prime ideal of R. Then,
from the definition of GUFR,
Pln R contains an element a
, J
such that aR = Ra 5 PIn Rand aR = Ra is invertibleo
Hence aR[x] = R[x]a $ PI and it is easy to see that
aR[x] = R[x]a is Q(R[x])-invertible.
Case-lIe
PI tl R is a minimal prime ideal of R. Let P = PIn R.
Then PR[x] is a minimal prime ideal of R[x] and so
PR[x] ~ PI ioe., PR[x] < PI' By lemma 2.19, P contains
no normal invertible idea~ and as in the proof of theorem 2 026
(Rip) is a prime GUFR.
Since (Rip) is a prime GUFH, (R/P)[x] is a prime
GUFR by theorem 2.33 and hence so is (R[x]/PR[x]). Since
PR [ x] < PIthere i sag e PIsueh
tha t
(g+PR[xJ) (R[x]/PR[x]) = (R[x]/PR[x]) (g+PR[x]) is contained
in Pl' (where PI' is the copy of P l
in R[x]/PR[x]), ioeo,
9 € E p $ CR[x] (0) n Cl and thus g is regular in R[x] and
gR[x] = R[xJgo Consequently gR[x] = R[x]g is ·contained
in PI and is Q(R[x])-invertible.
Thus in both cases PI contains
Q(R[x])-invertible,
normal ideals. Hence R[x] is a GUFR.