Two Minimal Prime Ideals in k[X,Y]/<XY>

  • Thread starter Dragonfall
  • Start date
  • Tags
    Prime
In summary, the conversation discusses proving that there are exactly two minimal prime ideals in k[X,Y]/<XY>. The definition of a minimal prime ideal is modified to include the word 'non-trivial', meaning that every non-trivial subset of P that is a prime ideal is actually P. It is noted that non-trivial prime ideals of k[X,Y] are generated by irreducible elements. The idea of a correspondence between ideals in R and ideals in R/I is mentioned. The final question is about finding the factorization of (xn + ym) in R[x, y]/(xy).
  • #1
Dragonfall
1,030
4

Homework Statement


Show that there are exactly two minimal prime ideals in k[X,Y]/<XY>. P is a minimal prime ideal if it is prime and every subset of P that is a prime ideal is actually P. k is a field.


The Attempt at a Solution



Prime ideals of k[X,Y] are <0> and <f> for irreducibles f. But then doesn't every ideal contain <0>? So how can there be other prime ideals?
 
Physics news on Phys.org
  • #2
I think that that definition should be modified to include the word 'non-trivial' somewhere in there. How about:

P is a minimal prime ideal if it is prime and every non-trivial subset of P that is a prime ideal is actually P. k is a field.
 
  • #3
Alright, but even with that, I'm still not sure how to preceed. It probably has to do with the fact that nontrivial prime ideals of k[X,Y] are generated by irreducible elements. Somehow this translates to two nontrivial minimal prime ideals in k[X,Y]/<XY>
 
  • #4
I could be mistaken but I think that the idea of a correspondence between ideals in R and ideals in R/I might called for here.

Edit:

What's (xn + 1)(ym + 1) in R[x, y]/(xy)?
How does xn + ym factor?
 
Last edited:

FAQ: Two Minimal Prime Ideals in k[X,Y]/<XY>

What is a minimal prime ideal in k[X,Y]/?

A minimal prime ideal in k[X,Y]/ is a prime ideal that cannot be further reduced or decomposed into smaller prime ideals. It is the smallest prime ideal in the quotient ring k[X,Y]/, and it contains only irreducible elements.

How many minimal prime ideals are there in k[X,Y]/?

There are exactly two minimal prime ideals in k[X,Y]/. These are (X) and (Y), which correspond to the irreducible elements X and Y in the quotient ring.

Can you give an example of a minimal prime ideal in k[X,Y]/?

(X) is an example of a minimal prime ideal in k[X,Y]/. It contains all polynomials of the form f(X) where f(X) is irreducible.

How are minimal prime ideals related to irreducible elements in k[X,Y]/?

Every minimal prime ideal in k[X,Y]/ corresponds to an irreducible element in the quotient ring. This means that the minimal prime ideals (X) and (Y) correspond to the irreducible elements X and Y, respectively.

Why are minimal prime ideals important in k[X,Y]/?

Minimal prime ideals play an important role in the structure of the quotient ring k[X,Y]/. They provide a way to understand and classify the prime ideals in the quotient ring, and they also help to determine the properties of the quotient ring, such as its dimension and integral closure.

Similar threads

How Do Primary Ideals in a PID Relate to Their Product and Intersection?
Replies
1
Views
1K
Union of Prime Ideals Contains an Ideal
Replies
5
Views
2K
Strong Induction: Proving Every Int n>1 is a Product of Primes
Replies
5
Views
2K
Prove there are infinitely many primes using Mersenne Primes
Replies
1
Views
1K
A few questions about a ring of polynomials over a field K
Replies
6
Views
2K
Prove ##|\{ q\in\mathbb{Q}: q>0 \} |=|\mathbb{N}|##.
Replies
3
Views
1K
Rewriting ODE's into lower orders
Replies
2
Views
1K
Congruence and Primes: Proving the Order of an Integer Modulo a Prime
Replies
4
Views
1K
Prove that if p > 3 is a prime number, then p^2 = [....]
Replies
16
Views
2K
Repeated Roots and Being Relatively Prime w/Derivative
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top