Let $k$ be a field and $S=k[x,y,z]$. Let $m=(x,y,z)$ and $I\subseteq m$ a proper homogeneous ideal in $S$. Is this true that we always have:


$$[Im:(x)][Im:(y,z)]\subseteq Im \ ?$$


In a paper we needed this statement for monomial ideals and it is easy to prove, but I can not see either way for general ideals. Has anyone seen this kind of statements before?



Is $[Im:(x)][Im:(y,z)]\subseteq Im$ in $k[x,y,z]$?
Perhaps it is not so bad once we are willing to get our hand dirty a little. Let $f\in Im:(x)$. Then $xf = xf\_1+yf\_2+zf\_3$ with $f\_i \in I$. Rewriting, we have $x(f-f\_1) = yf\_2+zf\_3$. Since $x,y,z$ form a regular sequence we must have $f=f\_1+h$, with $h\in (y,z)$. 


Now let $g\in Im:(y,z)$. Since $I$ is proper, $g\in m$. Then $fg = f\_1g + hg \in Im$, as desired. 


