Studies in Rings generalised Unique Factorisation Rings

-21-
( a)
( b) The map r ----1
-1
R to RD •
-1
= 1 in RD for all d E D.
-1
rl is a ring homomorphism from
(c) For r,s ~ Rand d c 0
-1 -1
rd = sd if and only if
re = se for some c E Do The c occurs because 0
may contin zero divisors 0 If 0 cons r.s ts of non
1 -1
zero divisors, then rd- = sd if and only if r=s.
It can he easily seen that the definitions of a right
quotient ring in 1 043 and the right localisation in remark 1.44
are equivalent.
Now we state Ore's theorem.
Theorem 1 045.
Suppose 0 is a multiplicative set in a ring R.
A right localisation of R relative to D exists if and only
if 0 is a
right Ore right reversible set.
Remark 1 046.
Let us write an element of RO- l as a/s where a ~ R,
5 ~ 0 and call a' the numerator and' s'the denominator of this