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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $S = k[x_1, \\ldots, x_n]$ be the [[polynomial ring|0SDC934R52]] with the variables ordered $x_1 \u003c x_2 \u003c \\cdots \u003c x_n$.\nLet $p \\in S \\setminus k$.\nIts *main variable* $\\operatorname{mvar}(p)$ is the largest variable that occurs in $p$.\nIts *initial* $\\operatorname{init}(p)$ is the [[leading coefficient|0V5GYWR8R8]] of $p$ as a polynomial in $\\operatorname{mvar}(p)$.\nA *triangular set* is a finite set $T$ of elements of $S \\setminus k$ with pairwise distinct main variables.\nFor a triangular set $T$ with product of initials $h_T$, the *saturated ideal* is the [[ideal quotient|16J9DM8V57]] $\\operatorname{sat}(T) = (T) : h_T^\\infty = \\set{ f \\in S : h_T^m f \\in (T) \\text{ for some } m \\ge 0 }$.\nA triangular set $T$ is a *regular chain* if either $T = \\emptyset$, or $T = T' \\cup \\set{t}$ where:\n\n1. $\\operatorname{mvar}(t)$ is larger than the main variable of every element of $T'$.\n2. $T'$ is a regular chain.\n3. $\\operatorname{init}(t)$ is not a [[zero divisor|0NNRDW10RZ]] in the [[quotient ring|0076HWHGHQ]] $S/\\operatorname{sat}(T')$.\n\n## Remarks\n\nFor $T = \\emptyset$, the saturated ideal is $(0)$, so a set $\\set{t}$ with one element is always a regular chain.\n\nExample: in $k[x, y]$ with $x \u003c y$, let $T = \\set{x^2 - 1, \\; xy - 1}$.\nHere $T' = \\set{x^2 - 1}$ has initial $1$, so $\\operatorname{sat}(T') = (x^2 - 1)$.\nThe initial of $t = xy - 1$ is $x$, and $x$ is a unit in $k[x, y]/(x^2 - 1)$, because $x \\cdot x = 1$ there; so it is not a zero divisor, and $T$ is a regular chain.\n\nNon-example: $T = \\set{x^2 - x, \\; xy - 1}$ fails clause 3.\nHere $\\operatorname{sat}(\\set{x^2 - x}) = (x^2 - x)$, and $x \\cdot (x - 1) = 0$ in $k[x, y]/(x^2 - x)$ with $x - 1 \\ne 0$, so the initial $x$ is a zero divisor.\n",
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