Consider the ring $R=\mathbb R[x,y]$. Let $\mathfrak m=(x,y)$. Let $n\ge 3$ be an odd integer and let $I_n=(x^n,y^n)$. What is the smallest integer $s\ge 1 $ (obviously depending on $n$) such that $(I_n\mathfrak m^s:\mathfrak m)\subseteq I_n$?
(For ideals $I,J$ in a ring $R$, $(I:J):=\{r\in R \mid rJ \subseteq I\}$; https://en.wikipedia.org/wiki/Ideal_quotient.)
NOTE:
$(I_n\mathfrak m^s : \mathfrak m)$
$=(y^{n+s}R +\sum_{j=0}^s x^{n+j-1}y^{s-j}R + \sum_{j=1}^s x^{j-1}y^{n+s-j}R ) \cap (x^{n+s}R +\sum_{j=0}^{s-1} x^{n+j}y^{s-j-1}R+ \sum_{j=0}^s x^{j}y^{n+s-j-1}R) $