TITLE: Square of primary ideals
QUESTION [12 upvotes]: Is there any example of  a $P$-primary ideal $I$ in a noetherian domain $R$ such that $I^2=PI \not=P^2$?

REPLY [7 votes]: 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 $P=(a,b,c,d)$.