Regular local ring

Remarks

A set of $d$ generators of $\mathfrak m$ in a regular local ring is called a regular system of parameters.

Example: a field is regular, with $d = 0$ and $\mathfrak m = 0$. A Noetherian local domain of dimension $1$ is regular exactly when $\mathfrak m$ is principal, that is, when it is a discrete valuation ring, such as $\Z_{(p)}$. For a field $k$, the rings $k[x_1, \ldots, x_n]_{(x_1, \ldots, x_n)}$ and $k[\![x_1, \ldots, x_n]\!]$ have dimension $n$ and maximal ideal generated by the $n$ elements $x_1, \ldots, x_n$, so they are regular.

Non-example: $R = k[x]/(x^2)$ has dimension $0$, but its maximal ideal $(x) \ne 0$ needs one generator, so it is not regular. The cusp $k[x, y]_{(x, y)}/(y^2 - x^3)$ has dimension $1$ and embedding dimension $2$, so it is not regular.