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I have a submodule of the module $\mathbb{Z}_2[x,y,z]^3$ given by the 6 generators which are the columns of the matrix

\begin{align} \left(\begin{array}{cccccc} 0 & 0 & 0 & 1+x+y+xy & 1+y+z+yz & 1+x+z+xz\newline 1+z & 1+x & 0 & 0 & y+z & 0\newline 0 & 1+x& 1+y & x+y & 0 & z^2 \end{array}\right) \end{align}

How can I implement the sub-module membership test in sage? For example, I want to check whether \begin{align} \left(\begin{array}{c} x+z \newline x+y\newline y+z \end{array}\right) \end{align}

belongs to the above submodule or not?

I have a submodule of the module $\mathbb{Z}_2[x,y,z]^3$ given by the 6 generators which are the columns of the matrix

below \begin{align} \left(\begin{array}{cccccc} 0 & 0 & 0 & 1+x+y+xy & 1+y+z+yz & 1+x+z+xz\newline 1+z & 1+x & 0 & 0 & y+z & 0\newline 0 & 1+x& 1+y & x+y & 0 & z^2 \end{array}\right) \end{array} \right) \end{align}

How can I implement the sub-module membership test in sage? For example, I want to check whether \begin{align} \left(\begin{array}{c} x+z \newline x+y\newline y+z \end{array}\right) \end{align}

belongs to the above submodule or not? not?

I have a submodule of the module $\mathbb{Z}_2[x,y,z]^3$ given by the 6 generators which are the columns below \begin{align} \left(\begin{array}{cccccc} \begin{array}{cccccc} 0 & 0 & 0 & 1+x+y+xy & 1+y+z+yz & 1+x+z+xz\newline 1+z & 1+x & 0 & 0 & y+z & 0\newline 0 & 1+x& 1+y & x+y & 0 & z^2 \end{array} \right) \end{align}\end{array}

How can I implement the sub-module membership test in sage? For example, I want to check whether \begin{align} \left(\begin{array}{c} \begin{array}{c} x+z \newline x+y\newline y+z \end{array}\right) \end{align}

\end{array} belongs to the above submodule or not?

I have a submodule of the module $\mathbb{Z}_2[x,y,z]^3$ given $\mathbb{Z}_2[x,y,z]^3$, which can be specified by the its 6 generators which that are the columns below \begin{array}{cccccc} 0 & 0 & 0 & 1+x+y+xy & 1+y+z+yz & 1+x+z+xz\newline 1+z & 1+x & 0 & 0 & y+z & 0\newline 0 & 1+x& 1+y & x+y & 0 & z^2 \end{array}

How can I implement the sub-module membership test in sage? For example, I want to check whether \begin{array}{c} x+z \newline x+y\newline y+z \end{array} belongs to the above submodule or not?