Re: Product of Prime Ideals

On 6 Mrz., 21:08, quasi <qu...@xxxxxxxx> wrote:
On Thu, 6 Mar 2008 19:53:10 +0000 (UTC), magi...@xxxxxxxxxxxxxxxxx

(Arturo Magidin) wrote:
In article <3gg0t3dq07fhdtdn73bmj7rm0u8l61m...@xxxxxxx>,
quasi <qu...@xxxxxxxx> wrote:

So then, how about an example of prime ideals P,Q in a commutative
unit ring R such that the product ideal P*Q is a proper subset of the
ideal (P intersect Q).

Probably there are easy examples, but thinking about it briefly, I
couldn't come up with one.

You mean, besides taking P=Q? (4) = (2)(2) =/= (2)/\(2) in Z.

Yes, of course, I meant to bar inclusion of either one into the other.

To restate the request ...

Give an example of prime ideals P,Q in a commutative ring R with 1,
where neither of P,Q is a subset of the other, and such that the
product ideal P*Q is a proper subset of the ideal (P intersect Q).

quasi

Since (P intersect Q) is in P, we must have P as a factor of (P
intersect Q).
Similarly, Q is a factor.
Since P and Q are different, they both appear in the prime ideal
decomposition
of (P intersect Q).
Hence P*Q subseteq (P intersect Q) = P * Q * ... subseteq P*Q

hagman.

.

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