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### vector space basis for a quotient module

For my question, let's say I have the following quotient module as an example, \begin{align} & \frac{\left(\mathbb{Z}_{2}\left[x,y,z\right]\right)^{3}}{\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 & 0 & 0\newline 0 & 1+x& 1+y & 0 & 0 & 0 \end{array}\right)} \end{align}

where $\mathbb{Z}_{2}\left[x,y,z\right]^{3}$ is a polynomial ring in variables $x,y,z$ over field $\mathbb{F}_2$. I am interested in calculating the Groebner basis of the submodule in the denominator using Sage and I can do the rest. I am finally interested in finding the vector space basis of the quotient module or its dimension. If that is also possible directly using Sage, it will be great.

### vector space basis for a quotient module

For my question, let's say I have the following quotient module as an example, \begin{align} & \frac{\left(\mathbb{Z}_{2}\left[x,y,z\right]\right)^{3}}{\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 & 0 & 0\newline 0 & 1+x& 1+y & 0 & 0 & 0 \end{array}\right)} \end{align}

where $\mathbb{Z}_{2}\left[x,y,z\right]^{3}$ is a polynomial ring in variables $x,y,z$ over field $\mathbb{F}_2$. I am interested in calculating the Groebner basis of the submodule in the denominator using Sage and I can do the rest. I am finally interested in finding the vector space basis of the quotient module or its dimension. If that is also possible directly using Sage, it will be great. great.

### vector space basis for a quotient module

For my question, let's say I have the following quotient module as an example, \begin{align} & \frac{\left(\mathbb{Z}_{2}\left[x,y,z\right]\right)^{3}}{\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 & 0 & 0\newline 0 & 1+x& 1+y & 0 & 0 & 0 \end{array}\right)} \end{align}

where $\mathbb{Z}_{2}\left[x,y,z\right]^{3}$ is a polynomial ring in variables $x,y,z$ over field $\mathbb{F}_2$. I am interested in calculating the Groebner basis of the submodule in the denominator using Sage and I can do the rest. I am finally interested in finding the vector space basis of the quotient module or its dimension. If that is also possible directly using Sage, it will be great.

### vector space basis for a quotient module

For my question, let's say I have the following quotient module as an example, \begin{align} & \frac{\left(\mathbb{Z}_{2}\left[x,y,z\right]\right)^{3}}{\left(\begin{array}{cccccc} \frac{\left(\mathbb{Z}_{2}\left[x,y,z\right]\right)^{6}}{\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 & 0 & 0\newline 0 & 1+x& 1+y & 0 & 0 & 0 \end{array}\right)} \end{align}

where $\mathbb{Z}_{2}\left[x,y,z\right]^{3}$ $\mathbb{Z}_{2}\left[x,y,z\right]^{6}$ is a polynomial ring in variables $x,y,z$ over field $\mathbb{F}_2$. I am interested in calculating the Groebner basis of the submodule in the denominator using Sage and I can do the rest. I am finally interested in finding the vector space basis of the quotient module or its dimension. If that is also possible directly using Sage, it will be great.

### vector space basis for a quotient module

For my question, let's say I have the following quotient module as an example, \begin{align} & \frac{\left(\mathbb{Z}_{2}\left[x,y,z\right]\right)^{6}}{\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 & 0 & 0\newline 0 & 1+x& 1+y & 0 & 0 & 0 \end{array}\right)} \end{align}

where $\mathbb{Z}_{2}\left[x,y,z\right]^{6}$ is a polynomial ring in variables $x,y,z$ over field $\mathbb{F}_2$. $\mathbb{Z}_2$. I am interested in calculating the Groebner basis of the submodule in the denominator using Sage and I can do the rest. I am finally interested in finding the vector space basis of the quotient module or its dimension. If that is also possible directly using Sage, it will be great.

### vector space basis for a quotient module

For my question, let's say I have the following quotient module as an example,

\begin{align} & \frac{\left(\mathbb{Z}_{2}\left[x,y,z\right]\right)^{6}}{\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 & 0 & 0\newline 0 & 1+x& 1+y & 0 & 0 & 0 \end{array}\right)} \end{align}

where $\mathbb{Z}_{2}\left[x,y,z\right]^{6}$ is a polynomial ring in variables $x,y,z$ over field $\mathbb{Z}_2$. I am interested in calculating the Groebner basis of the submodule in the denominator using Sage and I can do the rest. I am finally interested in finding the vector space basis of the quotient module or its dimension. If that is also possible directly using Sage, it will be great.

### vector space basis for a quotient module

For my question, let's say I have the following quotient module as an example,

\begin{align} & \frac{\left(\mathbb{Z}_{2}\left[x,y,z\right]\right)^{6}}{\left(\begin{array}{cccccc} \frac{\left(\mathbb{Z}_{2}\left[x,y,z\right]\right)^{3}}{\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 & 0 & 0\newline 0 & 1+x& 1+y & 0 & 0 & 0 \end{array}\right)} \end{align}

where $\mathbb{Z}_{2}\left[x,y,z\right]^{6}$ $\mathbb{Z}_{2}\left[x,y,z\right]^{3}$ is a polynomial ring in variables $x,y,z$ over field $\mathbb{Z}_2$. I am interested in calculating the Groebner basis of the submodule in the denominator using Sage and I can do the rest. I am finally interested in finding the vector space basis of the quotient module or its dimension. If that is also possible directly using Sage, it will be great.

### vector space basis for a quotient module

For my question, let's say I have the following quotient module as an example,

\begin{align} & \frac{\left(\mathbb{Z}_{2}\left[x,y,z\right]\right)^{3}}{\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 & 0 & 0\newline 0 & 1+x& 1+y & 0 & 0 & 0 \end{array}\right)} \end{align}

where $\mathbb{Z}_{2}\left[x,y,z\right]^{3}$ is a polynomial ring in variables $x,y,z$ over field $\mathbb{Z}_2$. I am interested in calculating the Groebner basis of the submodule in the denominator using Sage and I can do the rest. I am finally interested in finding the vector space basis of the quotient module or its dimension. If that is also possible directly using Sage, it will be great. 9 retagged FrédéricC 2409 ●3 ●28 ●48

### vector space basis for a quotient module

For my question, let's say I have the following quotient module as an example,

\begin{align} & \frac{\left(\mathbb{Z}_{2}\left[x,y,z\right]\right)^{3}}{\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 & 0 & 0\newline 0 & 1+x& 1+y & 0 & 0 & 0 \end{array}\right)} \end{align}

where $\mathbb{Z}_{2}\left[x,y,z\right]^{3}$ is a polynomial ring in variables $x,y,z$ over field $\mathbb{Z}_2$. I am interested in calculating the Groebner basis of the submodule in the denominator using Sage and I can do the rest. I am finally interested in finding the vector space basis of the quotient module or its dimension. If that is also possible directly using Sage, it will be great.