Studies in Rings generalised Unique Factorisation Rings

-75-
Lo t P be a non-minimal prime ideal of S. If x E' P,
then xS = Sx s P and is QCS)-invertibleo O'therwi s e ,
cons ider E = [a £ R/aR = Ra is an invertible a-ideal of R}.
It is easy to ,see that E is an Ore set in R. Let T be the
localised ring of R at E. Then a can be extended to an
automorphism ~ T such ( -1
= a:(a)c -1
on that ~ ac ) for all
-1 E T. Thus T* T[x,~J is an Ore extension of T and
ac =
T is ~-simple, ioeo, T and 0 are the only ~-ideals of T.
As in the proof of lemma 2.32, it can be seen
that for any prime ideal P of T* with x ~
P, there exists
a central element f € T* such that P = fT* = T*f.
If P n R ~
0, the proof is as in the general case.
If Pl1 R = 0, ~~ €/ P, it can be seen that x 4 PT* and PT*
is a non zero prime ideal of T* and thus PT* = fT* = T*f
for some f E T*, by the above ob s e r va t i cn , It is obvious
that xf = fx and Rf = fR. By the common multiple property
-1
of E, we have f = gd where q € P and d € E. Using the
fact a(d) = du (since a(dR) = dR) for some unit u in R
and dR = Rd, we get
Rg = Rfd = fRd = fdR = gR and
xg = xfd = fxd = fa:(d)x = fdux = fdxu- 1 = gxu
-1