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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]] and let $M$ be an [[$R$-module|0NZ8JA25WK]].\nA sequence $x_1, \\ldots, x_n$ of elements of $R$ is an *$M$-regular sequence* if:\n\n1. for each $i = 1, \\ldots, n$, the element $x_i$ is a non-zero-divisor on $M / (x_1, \\ldots, x_{i-1}) M$, that is, $x_i m = 0$ in this quotient implies $m = 0$ in it;\n2. $M / (x_1, \\ldots, x_n) M \\ne 0$.\n\nA *regular sequence* of $R$ is an $R$-regular sequence.\n\n## Remarks\n\nExample: 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 $R/(x_1, \\ldots, x_n) \\cong K \\ne 0$.\nNon-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 $1$.\nA sequence that satisfies only clause 1 is called *weakly regular*.\nThe 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.\nIf $R$ 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.\n",
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