Studies in Rings generalised Unique Factorisation Rings

Chapter-5
REMARKS
In this concluding chapter, we shall review some
of the results given in the previous chapters and discuss
the scope of further work.
In chapter 2, we have proved that every GUFR
has
an Artinian quotient ring, by proving that every nonminimal
prime ideal contains a
regular element, which
gives rise to a normal ideal. Thus the definition of a
GUFR can be reformed as·a Noetherian ring in,which every
non-minimal prime ideal contains a
4
normal regular element.
The theorem 2.30 that R is a commutative GUFR if
a nd only if t;,
has an Artinian quotient ring, leads to
the relevant question; is every commutative Noetherian
ring a GUFR?
Or does every commutative N6ethcrian ring
ha~e an Artinian quotient ring? In partjcular cases of
c omrnu t a t i ve Noetherian rings, I t vca n be proved t.hat they
have Artinian quotient ringso
For instance, if R is a
commutative Noetherian irreducible (i.e., for any ideal
A of R, A < A10 A 2
, whenever A < Al and A 2)
ring, then
R has an Artinian quotient ring.
In the general case,
we can say only upto the extent that a
commutative
Noethcrian ring R can be embedded in a commutativp