$\mathbb{R}[x,y]/(x^5-y^2)$ is a principal ideal domain?

Question

$\mathbb{R}[x,y]/(x^5-y^2)$ is a PID? What is the general method to solve this kind of problem!?

Answer 1

I assume $R$ is a ring with unit.

Well, one way is to recall that every PID is a UFD and search for a violation of unique factorization. In the quotient, we have the equivalence $y^2 = x^5$, so maybe we can find a factorization of something close to this.

I note that $y^2 -1 = x^5 - 1$ and we have two distinct (by comparing $x$-degree and $y$-degree) factorizations: $$ x^5 - 1 = (x-1)(x^4+x^3+x^2+x+1) $$ and $$ y^2 - 1 = (y-1)(y+1) \text{.} $$