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Is there an easy example that shows that the initial ideal of a radical ideal is not necessarily a radical ideal itself?

This is the converse of if the initial ideal of an ideal is radical, then the ideal is radical.

Also, how can I create more examples?

user26857
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Mark
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  • Related: http://math.stackexchange.com/questions/1169096/showing-that-if-the-initial-ideal-of-i-is-radical-then-i-is-radical – user26857 Apr 03 '16 at 15:37
  • @user26857 Yes, that is the result for which I am looking for an example that shows its converse is false. – Mark Apr 03 '16 at 15:39

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Let $R=K[X,Y,Z]$ and $I=(X^2-YZ)$. If one considers the lexicographical order on $R$ with $X>Y>Z$ then $\operatorname{in}(I)=(X^2)$.

user26857
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  • There is an interesting result of Migliore and Nagel which says that any monomial ideal is the initial ideal of a radical ideal. – user26857 Apr 03 '16 at 16:46
  • You can create more examples by using irreducible polynomials as I did. – user26857 Apr 03 '16 at 16:50
  • I found an example using the CAS Singular. Your example is much simpler, however. And I will look for that result. Thanks. – Mark Apr 03 '16 at 18:05