Studies in Rings generalised Unique Factorisation Rings

-89-
Definition 303.4
Let R be a 5ubring of Q. We say that R is classicaly
right (left) integrally closed in Q,
if R contains a nv
element y of Q for which there exiGt elements a
o
,a1,···,a
n-
1
n-l
of R such that yn = a + ya o l
+ + Y an-1
n-l)
a IY • n-
R is classically integrally
closed in Q if it is classically right and left integrally
closed in Q [14J.
Lemma 3 03.5.
Suppose the ring R is integrally closed and is an order
in a ring Q. Then the following assertions are true.
(1) bmb-1 € Rand b-1mb E R for all b e eR (0) and m E: R.
(2) If A is a finitely generated submodule of OR and
f e End A, then there exists d (; R such tria t f(a)=da
for all a € A.
Proof
As in [14, lemma 2 012J.
Theorem 3036.
Let R be a s ernipr i.me r i.qh t I\loetherian ring \vith the
~ ern i r; i mp 1 (; Arti n i a n Cl \J 0 t i (ln t .r i fl (1 lJ. ~-; u IJ}) 0 ~) e .il [;() t h