Re: On the primeness of the union of a chain of prime ideals.
- From: Marc Olschok <sa796ol@xxxxxxxxxxxxxxxxxxxxxx>
- Date: 21 Oct 2005 15:15:00 -0400
freedom641@xxxxxxxxx <freedom641@xxxxxxxxx> wrote:
>
> Dear All,
>
> Clearly in commutative case, the union any chain of prime ideals is
> prime. Is it true in non-commutative case!?
Yes.
Let V be such a union, and assume that I and J are two ideals,
such that IJ is a subset of V but I is not a subset of V.
Fix some i in I\V.
Then for every j in J there is some prime ideal P_j in the chain
with ij in P_j, and hence j in P_j subset V.
Therefore J is a subset of V.
My previous remarks about using the "usual proof" from the commutative
situation (using elements instead of ideals) were just wrong;
This would only work for "complete prime" ideals.
Marc
.
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- On the primeness of the union of a chain of prime ideals.
- From: freedom641@xxxxxxxxx
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