Studies in Rings generalised Unique Factorisation Rings

-93-
Noetherian ring is right localisable if CR(P)
is a
right Ore seta
It is obvious that in commutative rings
C R
(p) ,coincides with the complement of P in R. But
Unlike in commutative rings, CR(P) need not be a right
Ore set in many
cases.
First we look at the obstacles to
the localisation
at a prime ideal in .Noetherian rings and then discuss
a newly developed t.e chn Lqu e of Lo c a Li s a t i.on at a
collection of prime ideals in which the elements are
related in a
special manner.
We b eq i n with an example. rAost of the material
in the preliminaries of this chapter is taken from
[25J and [26].
[16J,
Example 4 0 1 0
Let k be a
field and let R be the 2 x 2 upper
triangular matrices over k. Then R is an Artinian (and
thus Noetherian) ring with two prime ideals, the ideal Q
of rnatrices in R who s e
upper left corner is zero and the
ideal P
of matrices in R whose lower right corner is zeroo
NO'N Rip and R/Q are both isomorphic to k, and QP = O.
Also PO = pn Q = J, the Jacobson radical of R. Note that
Q and P are completely prime ideals and thus C(Q)
= R-Q
and C(p) = R- P.