Regular sequence

Remarks

Example: in $R = K[x_1, \ldots, x_n]$ with $K$ a field, the variables form a regular sequence, since $R/(x_1, \ldots, x_{i-1}) \cong K[x_i, \ldots, x_n]$ is a domain in which $x_i \ne 0$, and $R/(x_1, \ldots, x_n) \cong K \ne 0$. Non-example: in $K[x, y]$ the sequence $x, xy$ fails clause 1, because $xy$ is zero in $K[x,y]/(x)$ and so kills the nonzero element $1$. A sequence that satisfies only clause 1 is called weakly regular. The order matters in general: in $K[x, y, z]$, the sequence $x, y(1-x), z(1-x)$ is regular but $y(1-x), z(1-x), x$ is not. If $R$ is a Noetherian local ring, $M$ is finitely generated and all $x_i$ lie in the maximal ideal, then every permutation of an $M$-regular sequence is $M$-regular.