System of parameters

Remarks

The ideal $(x_1, \dots, x_d)$ is $\mathfrak m$-primary if and only if $\mathfrak m^n \subseteq (x_1, \dots, x_d)$ for some $n \ge 1$, and if and only if $R/(x_1, \dots, x_d)$ has finite length. Example: in $R = k[x, y]_{(x, y)}$, with $k$ a field, the elements $x^2, y^3$ form a system of parameters, because $\dim R = 2$ and $\mathfrak m^4 \subseteq (x^2, y^3)$: each monomial $x^a y^b$ with $a + b = 4$ has $a \ge 2$ or $b \ge 3$. Non-example: in the same ring, $x, x + x^2$ is not a system of parameters, because it generates the ideal $(x)$, and no power of $y$ lies in $(x)$. If $x_1, \dots, x_d$ generate $\mathfrak m$ itself, then $R$ is a regular local ring and the list is called a regular system of parameters.