Studies in Rings generalised Unique Factorisation Rings

-37-
entries 'at is in M 2
( Q) = N. Put I = X M 2 ( R) = M 2
( R) X,
-1 (' -1
then I .s N. Furthe rm0 re X € M 2
RpI, since Xis
the scalar matrix/with non zero entries a~l
which is
-1 -1 () () -1
in Rp and thus I = X M 2
R = M 2
R X is contained
() -1 -1 ( ) ( )
in M 2
Rp and 11 = I I = M 2
R. Therefore M 2
R is
a GUFR
which is not prime.
Remark 2 05.
(1) The principal ideal theorem for a right Noetherian
ring asserts that the minimal prime ideals over any normal
ideal has height atmost 10 Thus in a GUFR even though
every non-mi~imal
prime ideal contains normal ideals, each
normal ideal is contained in either a minimal prime ideal
or in a prime ideal of height 1.
(2) If R is Noetherian ring satisfying descending
chain condition on prime ideals, then R is a GUFR with
the over ring 5
if and only if every height 1 prime ideal
of R contains an S-invertible principal ideal.
(3) By Proposition 1 0 16 , if R is a GUFR with over ring S,
then every prime ideal contains a normal S-invertible ideal
if and only if every minimal prime ideal contains a normal
S-invertib1e ideal.