Studies in Rings generalised Unique Factorisation Rings

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and so a I = R and a a = 1. Thus by the lemma given
above,
in a right bounded prime GUFR, every essential
right ideal is projective, which is a partial converse
of theorem ~~44. We do not know whether every right
ideal of a right bounded prime GUFR is projective.
Ano ther question that arose in chapter 2 is about
the integrally closed rings. We proved that the semiprime
GUFRs are integrally closed, if every right and
left endomorphisms of Q over R takes the identity
element of Q to R itself. The relevant question is:
If R is a commutative Noetherian UFO, then, is every R
endomorp~ism of Q takes the identity element of Q to R?
(Here Q is the quotient field of R).
The question is
important because in the case when R is a commutative
Noetherian UFO,
it is always integrally closed.
In chapter 3, we proved that the finite centralising
extension of a GUFR is a GUFR. The case of finite normalising
extension of a GUFR is yet to be proved. Ihe obstacle in this
case is that we cannot connect the prime ideals of R with
the prime ideals of a finite normalising extension directly.
(lemma 303)