Studies in Rings generalised Unique Factorisation Rings

-55-
The next theorem characterises the GUFRs in
commutative case.
Theorem 2 030.
If R is a
commutative Noetherian ring, then R is
a GUFR if and only if R has an Artinian quotient ring.
Proof:
Follows from theorem 2.28, as, in every commutative
ring the principal left ideals are two
sided ideals.
POLYNOlv1IAL RINGS
Remark 2.31.
It is obvious that when P is a prime ideal of R[x],
P (\ R i saprime. i d ea 1 0 fR. I f Pis a pri IDe ide a 1 0 f R,
the map a o
+ alx + ... + anx n ~ (ao+P)·dal+p)~ + ... +(an+P)x n
is clearly a surjective homomorphism from R[x] to {R/p)[x] with
kernel PR[x]. Consequently ~ is isomorphic to (R/P)[x],
and thus (Rip) is isomorphic to a subring RI of (R[x]/pR[x.]).
First we consider the case when R is a
prime GUFR.
By a central element in a ring R, we mean any element x in R
such that xr = rx for all r E: R.