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Exercise 111.28.2. Cohen-Macaulay rings of dimension 1. Part I: Theory.

  1. Let $(A, {\mathfrak m})$ be a local Noetherian with $\dim A = 1$. Show that if $x\in {\mathfrak m}$ is not a zerodivisor then

    1. $\dim A/xA = 0$, in other words $A/xA$ is Artinian, in other words $\{ x\} $ is a system of parameters for $A$.

    2. $A$ is has no embedded prime.

  2. Conversely, let $(A, {\mathfrak m})$ be a local Noetherian ring of dimension $1$. Show that if $A$ has no embedded prime then there exists a nonzerodivisor in ${\mathfrak m}$.

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