2. (24 points) Let p be a positive prime integer and let I C Zp[x] be the set of all polynomials whose constant term is 0; that is, I= {f(x) EZ,[x] | f(x) = ajr +azr2 + ... + anr", (where n > 0) }. (a) (8 points) Prove that I is a two-sided ideal. You must test all properties that define "two-sided ideals" to receive full credit. (b) (8 points) Prove that I is the smallest ideal of Zo[r] that contains z. That is, prove that r E I and if J C Zp(x] is any ideal containing a, then I C J. (c) (8 points) Prove or disprove that Zy[x]/I = Ep