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**Siknature** Let R= ℚ[X,Y] and let I=<X + 1>

I am trying to prove that I is a prime ideal of R.

I have tried arguing the following way (but i am unsure whether it is correct because the ring has two variables X,Y):

Suppose that f,g ∈ℚ[X,Y] and that fg∈ <X + 1>

That is, (x+1)|fg.

Hence, fg(-1)=0, that is f(-1)g(-1)=0 .

Therefore either f(-1)=0 , that is (X+1)|f, or g(-1)=0, that is (x+1)|g.

Thus either f or g is in the ideal <X+1>.

Therefore the ideal is prime.

Is this a correct argument?

How would i show that I is not a maximal ideal of R?

Thanks