Studies in Rings generalised Unique Factorisation Rings

-45-
.(2) Follows immediately from theorem 2.9.
(3) Follows from [1, theorem 2.7J.
vve sketch it. Let t c:: C(T) < T.
still for completion
-1
Then t = ac , for
s 0 me a f Rand c € C. Thusa = t c , \IV [1e 1'"e c i 5 a unit
of To Since c is a unit of T, we have T = eT = Te and
so C c C(T), sothat a € C(1'). N0 \V a € C f 0 11 0 W 5 from
the fact that a f R. Thus 'a' is also a unit in T.
-1
Consequently t = ac is a unit in T.
De fin i t·i 0 n 2. 15 .
An idea: P in a ring R is said to be right
localisable, if C(p) = (Xf R/x+p is regular in RIP}
is a right reversible set in R.
Definition 2 016.
A ring R ~s said to have a right Quotient ring,
if CH(O) .is a right reversible set. R is said to ha ve
quotient rinQ t if CR(O) is a right and left reversible set.
For instance, every GUFR
has a quotient ring.
Lemma 2.17.
Let R be a" No e the r i e n ring VJi th a quot.ient ring Q.
Let P = pl~ = Rp be a normal prime ideal of R wi t h p
regular.
Then P is localisable.