I want to find the elements in the quotient ring ${F_2}\left[ {x,y} \right]/\left( {xy + 1} \right)$. I think that the elements of this ring are of the form ${a_0} + {a_1}{x^n} +{a_2} {y^m}$, ${a_i} \in {F_2}$ since the leading term $xy$ doesn't divide both ${x^n}$ and ${y^m}$. Here, $n$ and $m$ are nonnegative integers.
I don't know whether I'm right or not. Also, can we determine the form of elements for general case ${F_2}\left[ {x,y} \right]/ \left( {f\left( {x,y} \right)} \right)$, where ${f\left( {x,y} \right)}$ is a polynomial with the leading term ${x^n}{y^m}$.
I'll be pleasure if one gives an answer.
I think the remainder must be in the form $${a_0} + {a_1}f\left( x \right) + {a_2}g\left( y \right).$$