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It is known that every proper ideal of a ring must be contained in a maximal ideal, by the Zorn's lemma. Is this true in general for proper homogeneous ideals in a graded ring?

Eric Wofsey
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Hang
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  • To be clear, by "homogeneous maximal ideal" do you mean an ideal that is both maximal and homogeneous or an ideal that is maximal among homgeneous ideals? – Eric Wofsey Jun 28 '17 at 00:30
  • I also feel confused about this. – Hang Jun 28 '17 at 00:30
  • Perhaps I mean the maximal ideal which corresponds to a point in a projective variety. – Hang Jun 28 '17 at 00:33
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    What kind of graded ring are you interested in? Just, say, $\mathbb{C}[x_0,\dotsc,x_n]$ with standard grading (by total degree)? More general (arbitrary grading group, etc.)? If you just want $\mathbb{C}[x_0,\dotsc,x_n]$ then, pretty trivially, every homogeneous ideal is contained in the "irrelevant ideal" $(x_0,\dotsc,x_n)$ which is maximal, homogeneous, and maximal homogeneous. Note that, if $P \in \mathbb{P}^n$ is a point in projective space then $I(P)$ is not a maximal ideal. – Zach Teitler Jun 28 '17 at 01:00

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