Studies in Rings generalised Unique Factorisation Rings

-19-
Ore has proved a more general result by classifying
the multiplicative sets in a ring R, at which the right
(left) ring of quotients (fractions) of R exists.
Definition 1 039.
Let R be any ring.
A multiplicative set D in R
is said to s2tisfy the right (left) Ore condition if
given r G· R, s E. 0 there exist r' E- Rand s' ~ S such
that r s ' = s r ' (59 r = r' s ) 0 In this case 0 is said to
be a right (left) Ore set. If D satisfies both right
and left conditions, 0 is simply called an Ore set.
Property 1 040.
We have a very useful property in a right Ore set
known
as the right common multiple property.
If 0 is a right Ore set in R, then given any
d 1,d2,
.•• ,d n
E 0, there exist d E D and r 1,r2,
... ,r n
in R
such that d = dlr l
= d 2r2
= ... = dnr n.
The left common
multiple property is defined like wise.
Definition 1 041.
A multiplicative set 0 in a ring R is said to be
right reversible in R, if for any d ~ 0, rt R with dr = 0,