Let $(R,m)$ be a Noetherian local domain whose integral closure $S$ is too. Also assume that $S$ is module-finite over $R$.


Let $x\in m^k\setminus m^{k+1}$ and $u\in S^\times$ such that $ux \in R$. Is $ux$ necessarily in $m^k\setminus m^{k+1}$?


The case I care about is when $R$ is an affine semigroup ring over $\Bbb C$ with maximal (multigraded) homogeneous ideal $m$ (or rather the localization of this stuff at $m$).



Adic filtration and integral closure
In general, this can be wrong. Consider $R=\mathbb{R}[[x,y]]/(x^2+y^4)$ (or $\mathbb{R}[x,y]\_{(x,y)}/(x^2+y^4)$). Then $u=x/y^2$ is in the integral closure $S$ and it is a unit. $y^2\in\mathfrak{m}^2$, $uy^2=x\in\mathfrak{m}$, but not in its square.


