Let $(R, \mathfrak m,k)$ be a Noetherian local ring such that the residue field $k$ is infinite. Let $n=\mu(\mathfrak m)$. Then $n=\dim\_k(\mathfrak m/\mathfrak m^2)$ . By fixing $x\_1,...,x\_n \in \mathfrak m$ such that $\bar x\_1,...,\bar x\_n\in \mathfrak m/\mathfrak m^2$ gives a $k$-vector space basis, we can identify $\mathfrak m/\mathfrak m^2$ with $\mathbb A^n(k)$ by sending $\bar x\_i \to e\_i$. So we can transfer the classical Zariski topology of $\mathbb A^n(k)$ to $\mathfrak m/\mathfrak m^2$. 


I have two similar kind of questions : 


(1) If $I$ is $\mathfrak m$-primary i.e. $\sqrt I=\mathfrak m$ and the set $\{\bar x\in \mathfrak m/\mathfrak m^2: x\in \mathfrak m \setminus \mathfrak m^2 \space \text{and} \space (\mathfrak mI:x)=I \}$ is non-empty, then how to show that the set is Zariski Open ?


(2) If $I=\overline I$ is $\mathfrak m$-primary i.e. $\sqrt I=\mathfrak m$ and the set $\{\bar x\in \mathfrak m/\mathfrak m^2: x\in \mathfrak m \setminus \mathfrak m^2 \space \text{and} \space (\overline{\mathfrak mI}:x)=I \}$ is non-empty, then how to show that the set is Zariski Open ?


Here for an ideal $I$, by $\overline I$ we denote the integral closure of $I$ <https://en.m.wikipedia.org/wiki/Integral_closure_of_an_ideal>



On a condition on ideals viwed as a Zariski open condition on co-tangent space
Case 1 was treated carefully in Appendix A of [J. Watanabe's paper](https://projecteuclid.org/download/pdf_1/euclid.nmj/1118780704) "$m$-full ideals". Case 2 can be treated the same way, as sketched below. 


Let $J$ be either $mI$ or $\overline{mI}$ and $A=R/J$. Since $J:x \supset I$, we have that $l(A/xA)= l(0:\_Ax) \geq l(I/J)$. Thus we need to show that the set of $x$ such that the length $A/xA$ is minimal possible is Zariski open. Then the idea is to consider the "universal generic element". Let $S=A[Y\_1,\dots, Y\_n]$ and $A'=S\_{mS}$. Consider the element $X= \sum Y\_ix\_i$ in $A'$. Prove that $l(A/xA)\geq l(A'/XA')$ with equality happens if and only if the ideal $(Y\_1-a\_1,\dots Y\_n-a\_n)$ does not contain certain radical ideal in $k[Y\_1,\dots, Y\_n]$ where $x = (a\_1,\dots, a\_n) \in m/m^2=\mathbb A^n\_k$ 


