Local ring and isomorphism problem

Question

I have a local ring $R$ with maximal ideal $\mathfrak{m}$. Fixing some $x\in\mathfrak{m}$, I want to show that $\mathfrak{m}^{k-1} \subset (\mathfrak{m}^k : x)$ and conclude that $R/(\mathfrak{m}^k : x)\cong (x)/(x)\cap\mathfrak{m}^k$ for all $k>0$ integer.

For the first part, let $a_1\cdot\ldots\cdot a_{k-1} \in\mathfrak{m}^{k-1}$, where each $a_i\in\mathfrak{m}$. Then it's clear that $a_1\cdot\ldots\cdot a_{k-1}\cdot x\in\mathfrak{m}^k$, so $a_1\cdot\ldots\cdot a_{k-1} \in (\mathfrak{m}^k : x)$. Is this correct?

The second part I don't know how to proceed, I need most help there. Hope someone can help me on this. Thank you very much.

Answer 1

1. Your argument for $\mathfrak{m}^{k-1} \subset (\mathfrak{m}^k : x)$ is correct, but this follows immediately from $\mathfrak m^{k-1}x\subseteq\mathfrak m^k$.

2. Define $\varphi:R\to (x)/(x)\cap\mathfrak{m}^k$ by $\varphi(a)=\overline{ax}$ and notice that $\ker\varphi=(\mathfrak{m}^k : x)$.