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I'm working on my problem which proves that every minimal primary ideal in $\mathbb{Z}$-graded ring is also graded. My strategy was to build an ideal from all homogeneous elements of the given ideal. And then reasoning such an ideal would be graded using the primary property. And then I would try to say those two ideals are the same kind.

However, this proof only works if the built ideal is not trivial. In other words, in a graded integral ring, every minimal primary ideal has non-zero homogeneous element. But I also wonder if this is true in a general sense, i.e in a graded ring does every ideal contain some non-zero homogeneous elements?

How should I approach this? I'm quite stuck at the moment. Thank you for reading.

Minh Khôi
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  • By "graded integral ring" do you mean $\mathbb{Z}$-graded ring? – Eric Wofsey Oct 29 '19 at 06:38
  • What about the ideal generated by x+1 in k[x]? – user26857 Oct 29 '19 at 06:45
  • Yes, sorry for not specifying that out. Let me edit my question. It was Z-graded – Minh Khôi Oct 29 '19 at 06:45
  • Probably a duplicate of https://math.stackexchange.com/questions/32595/minimal-primes-of-a-homogeneous-ideal-are-homogeneous – user26857 Oct 29 '19 at 06:47
  • Sorry, I tried to read the thread but I couldn't understand what it was about. The link that was given by the OP was dead. So I couldn't say if my question was really a duplicated to it. Can you explain it to me? – Minh Khôi Oct 29 '19 at 06:58
  • I did not check the link which was supposed to contain the result you need. In this case maybe another link can be helpful: https://stacks.math.columbia.edu/tag/00JU – user26857 Oct 29 '19 at 12:24

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