Studies in Rings generalised Unique Factorisation Rings
Studies in Rings generalised Unique Factorisation Rings
Studies in Rings generalised Unique Factorisation Rings
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-34-<br />
universal envelop<strong>in</strong>g algebra- U(L)<br />
of any solvable or<br />
s emi s i.mpLe Lie algebra <strong>in</strong> the non-cc ommut a t i.v e case.<br />
Several basic facts about commutative UFDs are<br />
extended to NUFDs by Chatters <strong>in</strong> [1]. MoPoGillchrist<br />
and MoKo Smith have proved that NUFDs<br />
are often<br />
pr<strong>in</strong>cipal ideal doma<strong>in</strong>s<br />
(<strong>in</strong> one of their papers).<br />
In 1986 Chatters and Jordan [2J<br />
<strong>in</strong>vestigated<br />
unique factorisation <strong>in</strong> prime Noetherian r<strong>in</strong>gso<br />
They<br />
def<strong>in</strong>ed a Noetherian<br />
unique factorisation r<strong>in</strong>g by analogy<br />
with the characterisation of UFDs by Kaplansky.<br />
They<br />
called a<br />
prime Noetherian r<strong>in</strong>g a Noetherian unique<br />
factorisation r<strong>in</strong>g (NUFR)- if every non zero prime ideal<br />
conta<strong>in</strong>s a<br />
pr<strong>in</strong>cipal prime ideal.<br />
In this chapter we def<strong>in</strong>e <strong>generalised</strong> unique<br />
factorisation r<strong>in</strong>gs and<br />
study the properties of these<br />
r<strong>in</strong>gs.<br />
BASIC DEFINITION AND<br />
EXAMPLES.<br />
Def<strong>in</strong> i t ion 2., J~ •<br />
Let R be any r<strong>in</strong>g and S an over-r<strong>in</strong>g of R.<br />
An<br />
ideal I<br />
of R is said to be S-<strong>in</strong>vertible, if the R-bimodu1e<br />
-1 -1 -1<br />
S conta<strong>in</strong>s an R-subbimodule I such that 11 =1 I=R.
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