Studies in Rings generalised Unique Factorisation Rings

-87-
Let r E: I. • • 0 + r z
n n
where r. E R, for 1 S i S n.
1.
= rl(zl+u1x) + r 2(z2+
u2x) + ... + rn(zn+unx)
- (rlu l + r 2u 2 + 0 •• + rnun)x.
Thus r can be generated by [zi+uiX'x] 0 This completes
the proof.
Theorem 3 0 3.1 .
Let R be a Noetherian ring with many normal
elements.
Also assume
that for any pair of weakly I-invertible elements
x and y, either Rx ~ Ry or Ry ~ Rx. Then
A
= [1/ I is a left ideal of R containing a normal element]
is linearly ordered.
Proof:
Let I and J be two elements of A 0
Suppose if
possible that I $ J and J J I. Then, by theorem 3.30,
there exists at least one weakly I-invertible element b (say)