Re: Integral closedness question

On Wed, 7 Dec 2005 11:05:16 -0500, "James" <James545@xxxxxxxxx> wrote:

>Let A be a (commutative with identity) ring that is integrally closed in its
>fraction field. If I is an ideal of A, then is A / I integrally closed in
>Frac(A / I)?
>
>James
>

Firstly, if you want to talk about fraction fields, you have to start
with an integral domain, so implicit in your question are two things
that you didn't state but are necessary for the question to even make
sense:

(1) A must be an integral domain.

(2) I must be a prime ideal of A.

With the above restrictions, the question can at least be considered.
but the answer to the question is no.

As a counterexample, let k be a field, A=k[x,y], I=(x^2-y^3).

Then I is a prime ideal of A but A/I is not a UFD, hence is not
integrally closed.

To see that A/I is not a UFD, note that x,y are irreducible in A/I,
but letting z=x^2=y^3 shows the factorization of z into irreducibles
is not unique.

quasi
.

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