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).
I found similar questions here and here but I think my question is (much) easier, though I don't manage to prove it.
What do you think about? Have you got any suggestions? Thanks in advance.