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

Let $A$ be the local ring at $(x, y)$ of $k[x, y]/(x^2, xy)$.

Show that $A$ has dimension 1.

Prove that every element of ${\mathfrak m}\subset A$ is a zerodivisor.

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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