Prime Ideals of Z/3Z X Z/9Z

[shadowstalker]

Having worked on prime ideals recently and finding them for Z_n I was wondering how you can to find all the prime ideals of a multiplication of Z/3Z X Z/9Z or Z/2Z X Z/4Z for example. I'm mostly having trouble starting this problem and feel that if I could get an idea where to start that I could finish off the proof myself.

Thanks

 

[Deveno]

if R is a commutative ring, and P is a prime ideal, then R/P is an integral domain (some authors require R to have unity for both concepts, however Z3 x Z9 has unity, namely (1,1), so does Z2 x Z4).

what's another name for a finite integral domain?

(hint #2: is Z9 a field? what about Z3 x Z3? think in terms of possible ring homomorphisms).