To show that we need to show and .

Note this means by Euclidean algorithm. Thus, by relative primeness there exists such that . Since is an ideal it means . All ideals which contain have to be the improper ideal, i.e. .2. Suppose that I is an ideal of Q[X] which contains both X^{2} + 2X + 4 and X^{3} - 3. Show that I = Q[X].

Find two elements so that .3. Prove that in the ring Z[X], <2>U<X> is not an ideal