Is \(\CC[x, y]\) a PID? Is \(\gens{ x, y }\) a prime ideals in it?
Solution
1. The ring \(\CC[x,y]\) is not a PID.
Consider the ideal \[ (x,y). \] If it were principal, say \((x,y)=(f)\), then \(f\) would divide both \(x\) and \(y\). Since \(\CC[x,y]\) is a UFD and \(x,y\) are nonassociate irreducibles, every common divisor of \(x\) and \(y\) is a unit. Thus \(f\) would be a unit, forcing \((x,y)=\CC[x,y]\), contradiction because \(1\notin(x,y)\).
2. The ideal \((x,y)\) is maximal, hence prime.
Evaluation at the origin gives a surjective homomorphism \[ \CC[x,y]\to\CC, \qquad f(x,y)\mapsto f(0,0), \] whose kernel is exactly \((x,y)\). Therefore \[ \CC[x,y]/(x,y)\cong\CC, \] a field. Hence \((x,y)\) is maximal and therefore prime.