How do I prove that this ideal is

Devin Dougherty

Devin Dougherty

Answered question

2022-04-12

How do I prove that this ideal is not a ' ideal?
Let K be a field and R=K[X,Y](XY).F or PK[X,Y] we denote [P] its class in R. Show that the Ideal (XY) is not a ' ideal.

Answer & Explanation

Alonso Christian

Alonso Christian

Beginner2022-04-13Added 11 answers

Step 1
Here is one possible proof:
Let φ:K[X,Y]K
be the unique K-algebra homomorphism such that φ(X)=1 adn φ(Y)=0. Then
φ(XY)=φ(X)φ(Y)=10=0 so this factors through the quotient
K[X,Y]K[X,Y](XY)=R to give a homomorphism
φ~:RK such that widetilde{φ}([P])=φ(P) for all PK[X,Y]
Since K is a field, φ~([X])=φ(X)=1, so [X]0
In other words, Y(XY)
Symmetrically, we must have X(XY), showing that (XY) is not '.

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