Studies in Rings generalised Unique Factorisation Rings

-68-
If S is a
finite normalising extension of Rand
S is right Noetherian,
then S satisfies essentiality.
Proof:
As
in [10, proposition 10.2.12J.
Theorem 3.50
Let R be a GUFR and S be a finite centralising
extension of Ro Also suppose that C 5 CS(O), where
C = ta ( R/aR=Ra is invertible). Then S is a GUFR.
Since S is a finitely generated left and right
R module, S is Noetherian. Now we prove C is an (left
and right) Ore set in S. Let a 6 C and s E S, then
+ r z )
n n
Here we are assuming that tZuZ2,.,zn1
set of generators of S over R and
is a centralising
we used the property
that for each r E R a rer ' a; for some r' c: R. Thus for
any s € S, there exists Si € S such that as = s I a, 1.e.
·
as ~ Sa. Similarly we have Sa ~ as, it follows that C is
an Ore set as in theorem 2 06.