Consider the ideal $I=\{f \in \mathbb{R}: f(0,1)=f(1,0)=0\} \subset \mathbb{R}[x,y]$.
What is the factor ring $\mathbb{R}[x,y]/I$?
We have that $\mathbb{R}[x,y]/I = \{f(x,y) + I: f(x,y) \in \mathbb{R}[x,y]\}$ by definition. I want to be able to describe a bit more about the polynomials to answer the question but I am not sure where to go from here. Any hints?
In the factor ring two polynomials $g$ and $h$ are equivalent if their values at (0,1) and (1,0) are the same. Indeed, $(g-h)(0,1) = g(1,0) = 0 \Rightarrow g-h \in I$. Thus the elements of the factor ring are completely defined by their values at (1,0) and (0,1). This means that the factor ring is $\mathbb R^2$ with element-wise multiplication.