Re: Noetherian??

On Tue, 26 Jul 2005 16:54:34 -0700, quasi <quasi@xxxxxxxx> wrote:

>On Tue, 26 Jul 2005 16:44:56 -0700, quasi <quasi@xxxxxxxx> wrote:
>
>>Let T be the ring R+xR[x,y] and let J be xR[x,y] so J is an ideal of
>>T. Does the ideal (x) in T contain the element xy? If so, show it. If
>>not then (x) is a proper ideal of (x,xy) which is a subideal of J,
>>where all ideals are viewed as ideals of T.
>
>To correct the terminology, change the sentence:
>
>>If not then (x) is a proper ideal of (x,xy) which is a subideal of J,
>>where all ideals are viewed as ideals of T.
>
>to this:
>
>>If not, then the ideal (x) is a proper subset of the ideal (x,xy) which is a subset of the ideal J,
>>where all ideals are viewed as ideals of T
>
>To say (x) is an ideal of (x,xy) is confusing since (x,xy) is just an
>ideal of T.
>
>quasi

In fact, the part of this problem where you need to be most careful is
keeping track of which ring an ideal relates to.

For example, (x) as an ideal of R[x,y] is the same (as a set) as the
ideal xR[x,y] of T.

But you can also talk about the ideal (x) of T which can be specified
more precisely as xT.

Clearly xT is a subset of xR[x,y], but are they equal?

This question is at the heart of the problem.

quasi
.

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