Recall that an ideal I ⊆ R is generated by x1 , . . . , xn if every y ∈ I can be written in the form y = r1x1 + · · · + rnxn for suitable elements ri ∈ R. (a) Show that K = { f (x) ∈ Z[x] : deg(f ) = 0 or f (x) = 0 } is a subring of Z[x], but is not an ideal. (b) Show that the ideal of all polynomials f (x) ∈ Z[x] with even constant term f0 is an ideal generated by 2 and x.

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