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In mathematics, the irrelevant ideal is the ideal of a graded ring generated by the homogeneous elements of degree greater than zero. More generally, a homogeneous ideal of a graded ring is called an irrelevant ideal if its radical contains the irrelevant ideal.

Say $R=k[x,y]$. Then irrelevant ideal is an ideal generated by $x,y,x^2,y^2,xy, \cdots$. Is this correct?

jk001
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    Boy, I do not think it was a good decision to define what “the” irrelevant ideal is, and then proceed to define what “other irrelevant ideals” are in terms of it. Seems like a job for a separate term. – rschwieb Jul 02 '21 at 02:52
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    I think what you wrote seems correct according to that definition. – rschwieb Jul 02 '21 at 02:54
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    One small thing - you have not defined a grading on your ring. If you're using a grading where $x$ and $y$ have positive degree, then you are correct, otherwise something else may happen. (I also agree with rschwieb's comment about the terminology.) – KReiser Jul 02 '21 at 03:34

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