Studies in Rings generalised Unique Factorisation Rings

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( 1) RX- 1 is a right No e th e r i.an ring.
(2) For any ideal I of RX- 1, r C is an ideal of R.
(3) For any id ea.l I of R, .re is an ideal of RX- 1.
(4) For any ideal I of RX- 1 , I = (rc)e.
(5) An idea 1 I of RX- 1 is prime (semiprime) if
and orly if re is prime (semiprime) in R.
(6) Let P be a prime (semiprime) ideal of Ro
Then P = QC for some prime (semiprime) ideal
if and only if X ~ C(p).
Proof:
As in [19, theorem 9.20].
Remark 2 013.
We look at T, the partial quotient ring of R at c.
Since C S C R
(0),. it is obvious that T ~ Q(R), the Artinian
quotient ring ofR formed by localising R at CR(O)o
Now
T has the following properties.
Theorem 2.15.
Let R be a GUFR and T be the partial quotient ring
of Rat Co
Then
(1) T is a GUFR
(2) T has an Artinian quotient ring
(3) C(T) = '[t