Studies in Rings generalised Unique Factorisation Rings

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in the generating set of I such tha t b 9- J and
similarly there is a weakly I-invertible element
c E J such tha t c ~ I. Since Rb 5 I and Rc ! I,
we have Rc ~ Rb and similarly Rb ~ Rc, which contradicts
the hypothe3is~ Thus either I $ J or J ~ I, and the
proof is complete.
Theorem 3.32.
Let R be a Noetherian ring with many
normal
elementsn Also assume that for any pair of weakly
I-invertible elements x and y either xR ~ yR or
yR s xR. Then = {Ill is a right ideal of R }
'. conta ining a normal e leme n t
is linearly ordered.
Proof:
As in theorem 3.31.
INTEGRALLY eraSED RINGS
Q~finition 3 033.
Let R be any ring and M be an R-module. Then M
is said to be integrally closed if any endomorphism of
any finitely generated submodule extends to an endomorphism
of M. A ring R is said to be right (left) integrally closed
if RR(RR) is integrally closed [14J.