Math Is Fun Forum

laipou

Member

Registered: 2009-09-19

Posts: 24

polynomial ring and Noetherian ring

1.
Let K be a field.
K[x,y]:polynomial ring in two variables.
Is there exists a method to determine whether (x),(x^2,y),(x^2,x+y),(x^2,xy,y^2)...etc
are maximal or prime or primary ideals in K[x,y]?

2.
Let R be a commutative Noetherian ring.
Let I and J be two ideals in R.
Show that if J is contained in Rad(I),then J^n is contained in I for some n.

I think I have to use the fact that J is finitely generated.
But I dont know how to find a generating set for J^n.

Thanks for any help.

Last edited by laipou (2009-12-19 19:35:59)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: polynomial ring and Noetherian ring

1. For maximality, typically the trick is to mod out by the ideal then prove you get a field.  Remember this works equally well for ideals in quotients of a polynomial ring because of the third isomorphism theorem.

If you have a suspicion that your ring is local, remember to just look at the set of all nonunits.

For rings, it seems that proving something is primary is always done by this proposition:

2. Don't worry about generators for J^n, they exist but they aren't important.  Remember what it means to be a generating set: every element is expressible as a "linear combination" with coefficients from R.  It's obvious that you want to take your n as something like the highest n from the definition of rad(I).  But this won't work.  Why? (And think pigeon hole principle)

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."