Let $R = k[x, y]$ with $k$ algebraically closed, and $m = (x, y)$. Suppose $I$ is an $m$-primary ideal of $R$, i.e., $(x, y)^n \subset I \subset (x, y)$ for some $n$. If $I_m$ is generated by a regular sequence of length 2, i.e., $I_m = aR_m + bR_m$ where $a, b$ is a regular sequence of $R_m$. What can we say about the number of generators of $I$ in this case?

All my examples show that $I$ is generated by a regular sequence of length 2, yet not a proof is found.

Thanks,

Let $R = k[x, y]$ with $k$ algebraically closed, and $\mathfrak m = (x, y)$. Suppose $I$ is an $\mathfrak m$-primary ideal of $R$, i.e., $(x, y)^n \subset I \subset (x, y)$ for some $n$. If $I_{\mathfrak m}$ is generated by a regular sequence of length $2$, i.e., $I_{\mathfrak m} = aR_{\mathfrak m} + bR_{\mathfrak m}$ where $a, b$ is a regular sequence of $R_{\mathfrak m}$, what can we say about the number of generators of $I$ in this case?

All my examples show that $I$ is generated by a regular sequence of length $2$, yet not a proof is found.

Thanks,