Let $R = k[x, y] $ and $Q = k[x, y]_{(x,y)}$, the localization of $k[x, y]$ at $(0, 0)$. Let $I$ be an ideal of $Q$ generated by a regular sequence of length $2$. Assume additionally $I$ is $(x, y)$-primary, i.e., $(x, y)^n \subset I \subset (x, y)$ for some $n$.
Is it possible to find a regular sequence in $R$, say $a, b$, such that $I = Qa + Qb$, and $I \cap R = Ra + Rb$?
For example, let $I = (x^2 - y^3)Q + (x^3 - y^2)Q$. Easy to check $x^2, y^2\in I$ and $I \cap R = x^2 R + y^2 R$ (but $I \cap R \neq (x^2 - y^3)R + (x^3 - y^2)R$).
Thanks!