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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

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## 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?

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- #2

aPhilosopher

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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

Dragonfall

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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

aPhilosopher

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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 (x^{n} + 1)(y^{m} + 1) in R[x, y]/(xy)?

How does x^{n} + y^{m} factor?

Last edited:

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

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

A minimal prime ideal in k[X,Y]/

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

There are exactly two minimal prime ideals in k[X,Y]/

## 3. 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]/

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

Every minimal prime ideal in k[X,Y]/

## 5. 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]/

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