Let $R$ be a commutative unitary ring and suppose that the abelian group $(R,+)$ is finitely generated. Let's also $P$ be a maximal ideal of $R$.
Then $R/P$ is a finite field.
Well, the fact that the quotient is a field is obvious. The problem is that I have to show it is a finite field. I do not know how to start: I think that we have to use some tools from the classification of modules over PID (the hypotesis about the additive group is quite strong).
What do you think about? Have you got any suggestions? Thanks in advance.