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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.

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    You are correct that your case is easier than both of the linked questions, and that you should use the classification of finitely generated abelian groups.2012-08-13

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As abelian groups, both $\,R\,,\,P\,$ are f.g. and thus the abelian group $\,R/P\,$ is f.g....but this is also a field so if it had an element of additive infinite order then it'd contain an isomorphic copy of $\,\Bbb Z\,$ and thus also of $\,\Bbb Q\,$, which of course is impossible as the last one is not a f.g. abelian group. (of course, if an abelian group is f.g. then so is any subgroup)

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    @ThomasAndrews , I indeed see your point, yet I think anyone dealing with the OP's question's stuff must have already studied basic group theory, so f.g. abelian groups must be already well-known. Anyway, if the OP has any doubt I'll be glad to add a little explanation about this.2012-08-13