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I have an ideal in the polynomial ring $\mathbb{F}_2[x,y,z]$ given by $I=(1+ x + y + xy; 1+y + z + yz; 1 + x + z + xz )$. The annihilator of this ideal is generated by

$f_{xy}=\sum_{n,m\in \mathbb{Z}} x^n y^m$, $f_{xz}=\sum_{n,m\in \mathbb{Z}} x^n z^m$, $f_{zy}=\sum_{n,m\in \mathbb{Z}} z^n y^m$

because for example $f_{xy} I=((1+1+1+1)f_{xy},(1+1+z+z)f_{xy},(1+1+z+z)f_{xy})=(0,0,0)$.

Can I show this in sage? More generally, given an ideal like the above in $\mathbb{F_2}[x,y,z]$ can I find the generators of its annihilator?

I have an ideal in the polynomial ring $\mathbb{F}_2[x,y,z]$ given by $I=(1+ x + y + xy; 1+y + z + yz; 1 + x + z + xz )$. The annihilator of this ideal is generated by

$f_{xy}=\sum_{n,m\in \mathbb{Z}} x^n y^m$, $f_{xz}=\sum_{n,m\in \mathbb{Z}} x^n z^m$, $f_{zy}=\sum_{n,m\in \mathbb{Z}} z^n y^m$

because for example i.e. $f_{xy} I=((1+1+1+1)f_{xy},(1+1+z+z)f_{xy},(1+1+z+z)f_{xy})=(0,0,0)$. I=((1+1+1+1)f_{xy},(1+1+z+z)f_{xy},(1+1+z+z)f_{xy})=(0,0,0)$ and so on.

Can I show this find these generators in sage? More generally, given an ideal like the above in $\mathbb{F_2}[x,y,z]$ can I find the generators of its annihilator?