Studies in Rings generalised Unique Factorisation Rings

-47-
theorem 2 0 6 . Since R has an Artinian quotient ring,
n
CR(O) = o CH 0\) , where Pl,P2'··· 'Pn are t h e d i s t i n c t
i=l
minimal prime ideals of R, so tha t P =
P. for some i ,
1
1 5 i S n. Thu s a f CR(O) 5 C R
(pi) = CR(P) contradicting
the fact tha~ aR = Ra S P.
Remark 2.20.
if'le consid.er a s pecia I case of GUFRs , i.e. GUFRs
wi t h all. height 1 primes are of the form pR == Rp , Then)
by lemma 2.1~each p is a regular element in R and so
each pIt = Rp is invertible ( in Q(R) ) and thus p € c.
Further it can be seen that, each c E: C can be written as
, uPl ... P n
, [01" some unit u in R and for some positive
integer n, and p.s are such that p.R = Rp. is a height 1
1 1 1
prime ideal of R for i = 1,2, •.. ,ne Thus the ring T ,
localised ring of R at C,
coincides with the partial
quotient ring of R with respect to the multiplicative
set generated by the elements p of R such that pR = Rp
is a
height 1 prime.
Theorem 2.21.
Let R be a GUFR and every height one prime ideal
is of the form pR = Rp for some p £ R. Then the following
are e qu i va Len t .