Studies in Rings generalised Unique Factorisation Rings

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expression. Then it can be interpreted that two fractions
are equal if and only if when they are brought to a common
denominator, their numerators agree. It follows from the
right common multiple property of 0 that any two expressions
can be brought to a
common denominator. So we'can define the
addition of two
fractions by the rule (a/s)+(b/s) = (a+b/s).
Here it can be easily verified that the expression in the
right depends only on a/s and b/s ond not on a,b and s.
To define the product of a/s and bit we determine b l
(
R
and SI E 0 such that bS 1
= sb 1
and then put (a/s) (b/t)=(ab l/tsl)
.
Again it is easy to check that this product is well defined.
A ring R is said to be a domain if, it is without zero
divisors.
It is obvious that the nonzero elements in a
domain form a multiplicative set and if 0 = R-o, 0 trivially
satisfies the right and
left reversibility conditions. From
this fact we get the following corollary of Ore's theorem.
Corollury 1 047.
A domain R has a
right division rinq of fractions
(right quotient division ring) if and only if D is a right
Ore set if and only if the intersection of any two
nonzero
right ideals is nonzero.