Studies in Rings generalised Unique Factorisation Rings

-44-
Proof
Since R is a Noetherian ring and C is a right
and left Ore set, C is a right and left denominator
set by proposition 1.42. Now T = RC- l = C-1R is a
Noetherian ring by proposition 2 012(1).
By theorem 2.7, T has only a finite number of
maximal ideals. We prove that they are the minimal
prime ideals of T. Let M be a maximal ideal of T. If
possible assume P is a prime
ideal of T strictly contained
in Mo Then pC is strictly contained in MC, (otherwise
o c e
by (4) of proposition 2.12 JP
= (pc) ~ = (M ) = M ). But
M is the extension, in T, of some minimal prime ideal
P l
(say) of R. Since R has an Artinian quotient ring
n
CR(O) = n CR(P.), by proposition 1.62, where P1,P , ... ,P
i=l 1 2 n
are the minimal prime ideals of R. Thus we have
C-fCR(O) 5- CR(P l)
and so, by proposition 2.12(5), there
c
exists a prime ideal Q of T such that PI = Q. Therefore
e
e
M = p e l
= (Qc) , i.e., M = (Mc) = (Qc)e= Q. Consequently
we huve pC < MC = QC = Plo Also pC is a prime ideal of R
by proposition 2.12 (5).
This violates the minimality of Plo
Thus M does not contain any prime ideal properlyo Hence
the maximal ideals of T are the minimal prime ideals which
implies that T has no non minimal prime ideals and thus
T is obviously a GUFR.