Determining an $\mathbb{R}$-basis for $\mathbb{R}[x,y]/(x^2-x,y^2-y)$.

Question

Question:

Determine an $\mathbb{R}$-basis for $\mathbb{R}[x,y]/(x^2-x,y^2-y)$.

There is a similar question here Linear basis for a quotient ring, which involves determining a basis for $\mathbb{R}[x]/(x^2+k)$. But I am confused how to apply this method since the ideal we are quotienting by is generated by two polynomials.

Should we do something like: take an $f\in \mathbb{R}[x,y]$ and consider $(f \pmod{x^2-x})\pmod{y^2-y}$? But this seems very messy

Answer 1

Well, the monomial standard basis of $R={\Bbb R}[x,y]/\langle x^2-x,y^2-y\rangle$ is given by $\{1,x,y,xy\}$, since $x^2-x=0$ and $y^2-y=0$ in $R$.

Note for T. Stark: Consider all monomials $x^iy^j$ in the polynomial ring ${\Bbb R}[x,y]$ and apply the reduction $x^2=x$ and $y^2=y$ in $R$.

Answer 2

There is a surjective map from $\mathbb{R}[x,y]$ to the quotient, so an $\mathbb{R}$ basis of the polynomial ring will map to a generating set (not necessarily a basis) of the quotient. So take your favorite basis of $\mathbb{R}[x,y]$, and then figure out which of those elements become redundant after the map.