Noetherian $1$-dimensional domain with ideal generated by more than $2$ elements

Question

It is a classical theorem that in a Dedekind domain, every ideal can be generated by two or fewer elements. I was wondering what goes wrong if we omit the integrally closed property. I thought about, for example, $\mathbb{Z}[\sqrt{5}]$ which is not integrally closed since it is missing the integral element $\frac{1+\sqrt{5}}{2}$. However, all of the ideals in $\mathbb{Z}[\sqrt{5}]$ I can think of are still generated by at most two elements. For example, $$(2,1+\sqrt{5})^2=(4,2+2\sqrt{5},6+2\sqrt{5})=(4,2+2\sqrt{5}).$$ Anyone know of an example?

Answer 1

Let $A:=k[[t^3,t^5,t^7]], I:=(t^3,t^5,t^7), k$ a field. To show that $I$ is not generated by two elements, suppose $I=(f,g)$ where $f=f_3t^3+f_5t^5+f_6t^6+f_7t^7+\dots, g=g_3t^3+\dots$ . We have:$$\begin{cases} t^3=pf+qg, \\ t^5=rf+sg, \\ t^7=uf+vg. \end{cases} $$ By the first two equations, two vectors $(f_3,f_5),(g_3,g_5)$ are linearly independent. Then $u_0=v_0=0$ and $1=u_0f_7+v_0g_7$, contradiction.