Studies in Rings generalised Unique Factorisation Rings

-56-
Lemma 2 032.
Let R be a prirne GlJFr{ a nd T be the partial quotient
ring of R at Co Then every non zero prime ideal of T[x]
can be gen~rated by a central element in T[x].
Proof:
Although the proof is similar to the proof given
in [2J, "'le give it. Le t P be a non ZC1'O p r i me ideal of
T[x]. Since R is prime GUFR, as in corollary 2 08, it can
be seen that T is a simple Noetherian ring. Let f be a nonzero
polynomial of P of least degree, deg f = n (say).
The subset of T consists of the leading coefficients of
the polynomials of P of degree n,
together with zero,
is a non zero ideal of T and .this equal to T, since T
is simple. Thus 1 is an element of that ideal and hence
wi t ho u t 10S5 of generality we can a s s urne that f i.s a manic
polynomial. Let 9 € P, using division algorithm 9 = fq+r,
where q and ~ are in T[x] and deg r < deg f or r = 0,
but r = 9-fq € P and f is a polynomial of least degree
in P, whi.c h implies r = O. Hen c e 9 = fq ~ fT[x]. i.e.,
P S fT[x] and so P = fT[x]. Furthermore x s f = fox and sf-fs (P
for all s f T, and its degree < degree f. Thus sf-fs - 0
and we get sf = fs for 511 5 € T and consequently
fT [ x]
= T [ x] f •