Studies in Rings generalised Unique Factorisation Rings

-106-
and
P is primeo
Cons equen t Lv a.b.' E OP for i = 1,2, ... n
1 1
x ( -1
and so I f Qp)C • Since C has only regular elements,
we have x € QP, it follows that Qnp ~ QP, wh i ch contradicts
the assumption tha t Q () P 1= QP and we have Q ~ P.
Remark 4 031.
Let P be a minimal prime ideal in a GUFR. Define,
x (p) = {Q E Spec R/Q ~PJ,
0
X1(P) = {Q e Spec R/O~Pl for some PI € Xo(p))
Xj+I(P)= {Q £ Spec R/Q~P. for some P.£x.(P)} for j > 1.
J J J
By theorems 4 029 and 4 0 30 we have
X o
(p) = {O E Min Spec R/OP 1= Q n p] and
x , l(P) = f O E Min Spec R/QP. 1= Q(lP. for some P. EX.(p)l
J+ t J J J J J
for j
= 0,1,2, .... Thus we have the right clique of
p = 00 u
j=o
x.(p)
J
= X(p).
Theorem 4.32.
Let R be a GUFR and P a minimal prime ideal of R.
Then} right clique of P = X(p) =
00
lJ
j=o
X. ( p), whe re
J
Xo(P) = {QE: Min spec H/QP 1= Q n p} and
X. J(P) ={"Q E Min Spec R/QP. 1= oo P., for some P.E X.(P)]
J+ . J J J J
for j = 0,1,2,....