The Stacks project

Exercise 111.28.5. Regular local rings of dimension $1$. Suppose that $(A, {\mathfrak m}, k)$ is a regular Noetherian local ring of dimension $1$. Recall that this means that $A$ has dimension $1$ and embedding dimension $1$, i.e.,

\[ \dim _ k {\mathfrak m}/{\mathfrak m}^2 = 1. \]

Let $x\in {\mathfrak m}$ be any element whose class in ${\mathfrak m}/{\mathfrak m}^2$ is not zero.

  1. Show that for every element $y$ of ${\mathfrak m}$ there exists an integer $n$ such that $y$ can be written as $y = ux^ n$ with $u\in A^\ast $ a unit.

  2. Show that $x$ is a nonzerodivisor in $A$.

  3. Conclude that $A$ is a domain.

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