Exercise
Consider the ring R = Q[X, Y]. Which of the following ideals is equal to R?
({X, Y})
({X2-Y, Y-1, X2+1})
({X2-Y, Y-1, X2-1})
No, the degree (maximum sum of X-exponent and Y-exponent of any monomial)
of a nonzero polynomial in this ideal is always positive.
Correct.
For
(X2-Y) + (Y-1) - (X2+1) = -2
is invertible.
Wrong.
The third polynomial is not needed to generate the ideal. Polynomials of degree 1 in X
do not occur in the ideal.