Studies in Rings generalised Unique Factorisation Rings

-105-
Proof:
----
If Q ~ P, it is obv i o us that there exists an
id ea 1 A with OP ~ A < Q n P and thus QP f:. Q n p •
Conversely, assume OP ~ o n r . Suppose if
possible that Q +> PS'
The set C of regular normal
elements is an Ore set in Rand P ne = Qnc = 91,
since
-1 -1
P and Q are minimal primes. So by r-erna r k 408)QC .~ PC
in RC-I. But QC- 1 and PC- 1 are maximal ideals of RC- 1
by theorem 2.7 and hence by lemma 4.28, QC- I PC-I=Qc-In PC-I:
Now let x E: Q n P, then T= xl- 1 E: PC- I n QC-I,
x EO (QC-I) (pC-I), thus there exist a. E Q, b. E P
1 J.
and c. ,d. E:
1 1
C for i=1,2, •.• ,n such that
-1
n
-1 -1
n a. b.
x 1. 1
X = xl = = I: (a.c. )(b.d. ) = ~
cr:- . But
a. b.
_.! 1
c. er:-
1 1
1
a .• b. '
1 1
= d. c. I
1. 1
i=l
1. 1 1 1
i=l
c.
1. 1
for each i = 1,2,o .• ,n, where b.' € R
1
and c . ' E: C such that b. c. , = c. b. , ( remark 1.46) .
1 1. 1 1 1
n a.b. ,
Therefore x
1 1
L
1 = d.e.
i=l I
1 1
c i
.Now b. E P, therefore
b ." = b v c ." c P, for each i = 1,2, •.• ,n. i.e,
1 1 1 1
1
R c v b ." 5. P. Since c. c C, R c. = c. R, which implies
1 J. 1 1 1
c. R b.' < .P for i = 1,2, ••• ,n. Hence b.' € P for each
1 1 - 1
i = 1,2, .•. ,n)as CnP =~. c. E
1
C for each i = 1,2, ... ,n