Exercise 111.13.1. In the following cases, please answer yes or no. No explanation or proof necessary.

Let $p$ be a prime number. Is the local ring $\mathbf{Z}_{(p)}$ a Cohen-Macaulay local ring?

Let $p$ be a prime number. Is the local ring $\mathbf{Z}_{(p)}$ a regular local ring?

Let $k$ be a field. Is the local ring $k[x]_{(x)}$ a Cohen-Macaulay local ring?

Let $k$ be a field. Is the local ring $k[x]_{(x)}$ a regular local ring?

Let $k$ be a field. Is the local ring $(k[x, y]/(y^2 - x^3))_{(x, y)} = k[x, y]_{(x, y)}/(y^2 - x^3)$ a Cohen-Macaulay local ring?

Let $k$ be a field. Is the local ring $(k[x, y]/(y^2, xy))_{(x, y)} = k[x, y]_{(x, y)}/(y^2, xy)$ a Cohen-Macaulay local ring?

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