1. Prove that $\displaystyle R = \{a+bi: a,b \in \mathbb{Z} \} $ is a subring of $\displaystyle \mathbb{C} $ and that $\displaystyle M = \{a+bi: 3|a \ \text{and} \ 3|b \} $ is a maximal ideal in $\displaystyle R $.

Showing it's a subring is obvious. Just show closure under addition and multiplication and that $\displaystyle 0_{\mathbb{C}} \in R $. Also show that the solution of the equation $\displaystyle a+x = 0_{\mathbb{C}} $ is in $\displaystyle R $.

To show that $\displaystyle M $ is maximal consider the following: $\displaystyle M \subset J \subset R $ means that there is some $\displaystyle j \in J $ such that $\displaystyle j \notin M $ where $\displaystyle j = r+si $. So $\displaystyle 3 \not| \ |j| = r^2+s^2 = (r+si)(r-si) $. And any ideal that contains $\displaystyle r+si $ must contain $\displaystyle 1 $ so that $\displaystyle J = R $.

Is this correct?

Not as far as I can see. The element $\displaystyle j= r+si$ is an arbitrary element in $\displaystyle J$; how does containing this element makes an ideal the whole ring?! I think you must show that the norm of such an element j is coprime with the norm of any element in $\displaystyle M$ and thus... Read here, in particular 10.2: it is exactly what you need and want for this and the next question: http://www.fen.bilkent.edu.tr/~franz/nt/ch10.pdf Tonio
2. To show that $\displaystyle R/M $ is a field we know this from the fact that $\displaystyle M $ is maximal. How do you show it has 9 elements?