Exercise 111.28.3 (02EH)—The Stacks project

Table of contents
Part 9: Miscellany
Chapter 111: Exercises
Section 111.28: Cohen-Macaulay rings of dimension 1
Exercise 111.28.3 (cite)

Exercise 111.28.3. Cohen-Macaulay rings of dimension 1. Part II: Examples.

Let $A$ be the local ring at $(x, y)$ of $k[x, y]/(x^2, xy)$.
1. Show that $A$ has dimension 1.
2. Prove that every element of ${\mathfrak m}\subset A$ is a zerodivisor.
3. Find $z\in {\mathfrak m}$ such that $\dim A/zA = 0$ (no proof required).
Let $A$ be the local ring at $(x, y)$ of $k[x, y]/(x^2)$. Find a nonzerodivisor in ${\mathfrak m}$ (no proof required).

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