Studies in Rings generalised Unique Factorisation Rings

-23-
Definition 1 048.
A domain which satisfies the condition of
Corollary 1 047 is called a right Ore domain. Left Ore
domains are defined analogously.
Ore's theor~m, though proved in 1930, was only a
theoretical curiosity for a
long time until Alfred Goldie
proved some
results, nowadays known as Goldie's theorems,
in this direction in 1958. The importance of Goldie's
theorems is that it paved the way
to many new investigations
and answered many questions posed on
non-commutative ring
theory. We have seen that there are many non-commutative
domains whic~ do not possess a right or left division ring
of fractions and there are many rings which do not have
any factor rings
which are right or left Ore domains.
Instead of looking for Ore domains and
division rings of
fractions, we look for rings from which Simple Artinian
rings can be built using fractions.
Goldie's main result
states that if R is a Noetherian ring with 0 a prime ideal
(p a prime ideal), then R has (Rip has) a simple Artinian
ring of fractions. It turns out to be no extra work to
investigate rings from which semisimple ring of fractions
can be builto We begin with some definitions.