Studies in Rings generalised Unique Factorisation Rings

Lemma 4.28.
Suppose P and Q are maximal ideals in a Noetherian
ring R. Then Q ~ P if and only if QP ~ Q 0 P.
Theorem 4 0 2 9 .
Let R be a GUFR and P,Q € Spec R with P € Min Spec R
and Q ~ Min (Spec R). Then Q is not linked to P.
Proof:
Suppose Q ~ P. Since Q is not minimal, there
exists a normal invertible ideal I of R such that I 5 Q.
By lemma 4 019, I has (right and left) AR property. Thus
we have a positive integer n such that rnn(pnQ) ~ I(pnQ)
by the left AR property of I. Because of the link from
Q to P, we have an ideal A of R with QP ~ A < Qn P such
tha t r ( Q ~P ) = P and I ( Q ~ P ) = Q. Thus,
(on p)In~ (Q(\ p)n In = Inn (on p) s I(Qn p) ~ Q(Qn p) ~ A.
i.e. (Qnp)I n ~ A and so In ~ r (Q~ P ) = P. Since P
is prime I ~ P, which violates the assumption that P is
minimal and contains no normal invertible ideals.
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Therefore
Q~ P.
Theorem 4 0 30 .
Let R be a GUFR and P,O e Min (Spec R)o Then
Q~ P if and only if QP F o n P.