Example of invertible maximal ideal that is not generated by one element Could anyone give me an example of an invertible maximal ideal of some integral domain which is not generated by one element?
 A: In the coordinate ring    $A=\mathbb R[x,y]\stackrel {def}{=}\mathbb R[X,Y]/(X^2+Y^2-1)$ of the circle  the maximal ideal $\mathfrak m=(x-1,y)\subset A$ cannot be generated by a single element, as Möbius so successfully showed us with his band (no, no, he didn't conduct an orchestra).  
Complements
1) The square $$\mathfrak m^2=((x-1)^2,(x-1)y, y^2)=((x-1)^2,(x-1)y,1-x^2)=(x-1)(x-1, y,-x-1))=(1-x)(-2)=(1-x)$$ of the above ideal $\mathfrak m$ is generated by a single element, namely $1-x$. 
2) Every  maximal ideal in  $A\otimes_\mathbb R \mathbb C=\mathbb C[X,Y]/(X^2+Y^2-1)$ can be generated by a single element since  that complexified ring is a PID.   
Edit
At Rodrigo's request in the comment, I'll explain why $A_\mathbb C=\mathbb C[X,Y]/(X^2+Y^2-1)$ is a PID.
It is just the  change of variables $U=X+iY, V=X-iY$ which shows that $A_\mathbb C\cong \mathbb C[U,V]/(UV-1)=\mathbb C[U,\frac 1U]$, which is a PID like any ring of fractions $\\$ of the PID $\mathbb C[U]$ .
A: $$(2, 1+\sqrt{-5}) \subseteq \mathbf Z[\sqrt{-5}]$$
A: Going by standard examples in alg. num. theory, $(2,1+\sqrt{-5})$ in the number ring $\Bbb Z[\sqrt{-5}]$.
