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Let m be the unique maximal ideal in $$\mathbb{C[x,y]}_{}:=\left\{\frac{f}{g}\mathrel{}\middle|\mathrel{}{f,g}\in\mathbb{C[x,y]}\text{ and } {g(0,0)}\neq 0\right\}$$ Show that m= $\langle x+7y+8x^2+9y^2, 2x-y-4x^3+8x^5y^5\rangle$. (Hint: Nakayama’s lemma)

I would very much appreciate some help on this question. I don't really understand how to use Nakayama's lemma to find a maximal ideal (if that is what I'm supposed to do here). I was thinking to choose a finitely generated module M:=$\langle x,y\rangle$ and then to find a basis of the quotient M\mM. Then by Nakayama's lemma, if m in the question really was the maximal ideal of the ring, then the basis would also generate M.

So in essence I'm trying to show a basis that generates M\mM, also generates M so that I can show the maximal ideal.

Is this the correct way of going about the question? Thank you in advance.

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    Solved [here](http://math.stackexchange.com/questions/2139904/finding-the-maximal-ideal-in-a-local-ring) in the comments. – 2017-02-20
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    Can you give me some hints please? – 2017-02-20
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    Another [removed duplicate](http://math.stackexchange.com/questions/2154486/unique-maximal-ideal-of-localisation-of-ring) since we're collecting them. Someone should answer this question sometime (and not in the comments) so that we could finally put the question to rest! – 2017-02-21

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