Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $x,y\in \mathfrak m$ such that $y$ is not a zero-divisor on $R$. Then, is it true that
$\cap_{n=1}^\infty (x,y^n) \subseteq (x)$ ?
By Krull intersection theorem, we would be done if we can show $\cap_{n=1}^\infty (x,y^n) \subseteq (x) \cap (\cap_{n=1}^\infty y^n )$ , but unfortunately, I am not sure if this later inclusion is true.
Please help.
Let $a\in(x,y^n)$. Then $\bar a\in(\bar y)^n$ in $\bar R=R/(x)$. But $\bar R$ is a Noetherian local ring and we can use the Krull Intersection Theorem for the ideal $(\bar y)$.