Studies in Rings generalised Unique Factorisation Rings

-39-
aR and Rb are S-invertible ideals, there are R-bimodules
-1 -1 ( ) -1 ( )-1
I and J such that aR I = bR J = Ro If we write
Thus a,b f C implies ab E C, i.e. C is a multiplicative
set. To prove that C is an Ore set, let a E C and r ~ R,
then raf Ra ::-:: aR and so ra = art for some r t e R. Thus
C satisfies right Ore condition.
Similarly C satisfies
left Ore condition.
Theorem 2 07.
Let R be a GUFR with the over-ring S. Let
T = RC- 1 = C-IR be the localised ring of R at C. Then
T has atmost a
finite number of maximal ideals o
Since C is. a right and left Ore set, by proposition
1.42 and theorem 1.45, T = RC- 1 = C- 1R exists and the
homomorphism from R to RC- 1 (r __ rI-I) is a monomorphism,
since C has only regular elements.
Thus T is an over-ring
of S.
To
prove that T has only finite number of maximal
ideals, we use the correspondence P ~ PT which is a
bijection from [p ~ Spec Rip n C = ~] to Spec T. Let
PI, •.•,P n
be the minimal prime ideals of R such that