Let $p > 1$. Is $\mathbb{Z}[x,y]/(x^p -1)(y^p - 1)$ a UFD? Does this depend on the value of $p$?
It is about 10 years since I learnt about fields and rings - from memory I would say no, but I am struggling to recall why or find any theorems that would immediately tell me one way or the other.
I know that $\mathbb{Z}[x_1,\ldots,x_k]$ is a UFD - is there some result that can tell you when a quotient of a UFD is also a UFD?