Exercise
Which of the following ideals in R[X,Y] is prime?
(X2 - Y2)
({X2,Y2})
({X2 + 1,Y})
No. The product of X - Y and X + Y belongs to
the ideal (X2 - Y2).
No.
X does not belong to the ideal, but its square does.
Yes.
"Modding out Y", we see the ideal (X2 + 1) in R[X], which is prime as the polynomial involved is irreducible.
In the next section, the modding out procedure will be made more precise.