Two Minimal Prime Ideals in k[X,Y]/<XY> (2024)

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).

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]/ 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.

2. 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.

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]/. It contains all polynomials of the form f(X) where f(X) is irreducible.

4. 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.

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]/. 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.