I solved the following problem:
Show that for any monomial order $>$ it is true that $\mathrm{in}_{>}(I)\mathrm{in}_{>}(J)\subseteq \mathrm{in}_{>}(IJ)$ for any two ideals $I,J$ of $k[x_{1},...,x_{n}]$, where $\mathrm{in}_{>}(I)$ is the initial or leading-term ideal of $I$ with respect to the monomial order $>$.
In addition to the above, I am asked to give two ideals $I,J$ such that the inclusion is proper, but I have not been able to find them. Could you please give me a hint that I need in my ideals for strict inclusion to be fulfilled?
Thanks!
Take an ideal, in general, it is very likely that you will get $(\mathrm{in}_{>}(I))^2\subsetneq\mathrm{in}_{>}(I^2)$. For example, take the ideal of twisted cubic, $I=(y-z^2,x-z^3)$ then (by Macaulay2 or you can just compute the Groebner basis) $\mathrm{in}_{>}(I)= (y^2,yz,z^2)$ and $\mathrm{in}_{>}(I^2)=(y^4,y^3z,y^2z^2,xz^3,yz^3,z^4)$ where $>$ is the degree reverse lexicographic ordering.