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Is there any example of a $P$-primary ideal $I$ in a noetherian domain $R$ such that $I^2=PI \not=P^2$?

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What a fun problem for the holiday season! Yes, an example is $R=\mathbb C[a,b,c,d]/(a^4-bd, b^3-cd, c^2-ad, a^3b^2c-d^3) \cong \mathbb C[t^9,t^{13},t^{16},t^{23}]$. Let $I=(a,b,c)$ and $P=(a,b,c,d)$.

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  • $\begingroup$ For another example see this answer. $\endgroup$
    – user26857
    Commented Jan 19, 2018 at 19:43

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