Let $R=k[x\_1,\ldots,x\_6]$ be a polynomial ring and $I=(x\_1x\_5-x\_2x\_4,x\_2x\_6-x\_3x\_5)$ be an ideal.



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> How to show that, $(I^2:x\_1x\_5-x\_2x\_4)=I$ ?
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Properties of colon ideal
Put $P=x\_1x\_5-x\_2x\_4$, $Q=x\_2x\_6-x\_3x\_5$. Assume a polynomial $A$ satisfies $AP\in I^2$. Then we have a relation $$(\*)\quad AP=UP^2+VQ^2+WPQ.$$ In particular, $P$ divides $VQ^2$. Since $P$ is prime and does not divide $Q$, it must divide $V$. Simplifying $(\*)$ by $P$ gives the result.


