I need to answer the following question:
Let $R$ be the ring of real polynomial functions on the circle:
$R=\mathbb R[X,Y]/I$, with $I=(X^2+Y^2-1)$
Let $x:=X\pmod I$ and $y:= Y\pmod I$
a) Prove that every residue class mod $I$ has a unique representative of the form $A(X)+B(X)Y$.
Now I don't really know how to get started. I'm not really understanding what is happening here and how I can see or simplify this ring of twovalued polynomials modulo this circle-ideal. Could anybody help me?