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?