Studies in Rings generalised Unique Factorisation Rings

-36-
(3) We give an example of a GUFR which is neither a
cornmu t a t i.ve Noetherian domain nor a
f'JUFR.
k 2
Let k be a field and T = k[x1... x ]. Let 1= L x. T
n .11
J.=
k
where k $ n. Set R = T/l, then P = ~ x.R is the unique
.1 1 l=
minimal prime ideal of R, where ~.
= x.+I for i=1,2,o •• ,k.
1 1
Localise R at P and let the localised ring be Rp. Now
it is easy to see that Rp is an over-ring of R and
that
P contains no f~p-invertible principal .idea l s , B\Jt every
non-minimal prime ideal of R strictly contains P and thus
contains elements of the complement of P, i .n. ~ units
in Rp' which
in turn lead to Rp-invertible principal
ideals in non-minimal prime ideal. Thus R is a commutative
GUFR.
Since R can be embedded in Rp' M 2(R)
~an be embedded
in M 2(R p)' Because of the order preserving bijection
between the nrime ideals of R and that of M 2(R),
M 2(P)
is the unique minimal prime ideal of M 2(R).
None of the
elements of ~(P) is invertibl.e in M 2(R p)' therefore
M 2(P)
contains no M 2(R p)-invertible normal ideah.
Let N be a non-minimal prime ideal of M 2(R),
then
N ~ M 2(P). Let N = N2(Q), where Q is a prime ideal of R.
Then Q I=.P a nd hence there exists at least one element fa·
-in Q such that a ~ ·P. Then the scalar matrix X with non zero