Number Theory and Systems of Numbers!

HeY ppl! Can any1 help me wit eitha of these 2 questions plzzzzzzzzzz?!?!?! I'm in abit of a pickle! I'm rely rely lost! The question itself is confusing!! ><

1) Prove that a non-zero element in [Z]m is a zero divisor if and only if it is not relatively prime to m.

2) Let Zp[i] denote the ring {a + bi | a,b E [Z]p, i^2=-1}.

a) Show that if p is not prime, then Zp[i] is not an integral domain.

b) Suppose p is prime. Show that every non-zero element of Zp[i] is a unit iff x^2 + y^2 does not equal 0modp for any pairs x,y E [Z]p.

(Its a baby 'm' and 'p' next to the z's... couldn't find how to shrink letters! :S)

Hope all u smart ppl can help me!!!!!!!!!! Urgently!!!!!!!!!! :(:(:(