So by the division algorithm, for any ring R and R[x], and let f $\in$ R[x], g $\in$ R[x] and deg(g)=n, then f=hg+r for some r $\in$ R[x] and deg(r) < deg(g).
However, I don't know how the similar thing can work on multivariate polynomial rings. For example, in A = k[x, y], let f be an arbitrary polynomial in A, what properties should its remainder of $y-x^2$ have; what properties should its remainder of $x^2y-y^5$ have, etc.
In general, if we know a polynomial in a multivariate polynomial ring, then what can we conclude about the remainder of any polynomial in the same ring?
Thanks and I appreciate your insights.
