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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0B1261B5ED]] with identity and let $\\sigma \\colon R \\to R$ be a [[ring endomorphism|0VTH8KPCG2]] with $\\sigma(1) = 1$.\nA *$\\sigma$-derivation* of $R$ is an additive map $\\delta \\colon R \\to R$ with $\\delta(ab) = \\sigma(a)\\delta(b) + \\delta(a) b$ for all $a, b \\in R$.\nGiven such $\\sigma$ and $\\delta$, the *Ore extension* $R[x; \\sigma, \\delta]$ is the ring $S$ with these properties:\n\n1. $S$ contains $R$ as a subring and contains an element $x$.\n2. As a left $R$-[[module|0NZ8JA25WK]], $S$ is [[free|0DZ9DKAKZR]] with basis $1, x, x^2, \\dots$.\n3. $x a = \\sigma(a) x + \\delta(a)$ for all $a \\in R$.\n\n## Remarks\n\nFor every endomorphism $\\sigma$ and $\\sigma$-derivation $\\delta$, such a ring exists and is unique up to an isomorphism fixing $R$ and $x$.\nIts elements are written $a_0 + a_1 x + \\cdots + a_n x^n$, and rule 3 together with associativity determines all products.\nExample: let $k$ be a field, $R = k[y]$, $\\sigma = \\id$ and $\\delta = d/dy$.\nThe map $\\delta$ satisfies $\\delta(ab) = a \\delta(b) + \\delta(a) b$, the Leibniz rule, so it is an $\\id$-derivation, and rule 3 reads $x y = y x + 1$; this ring is the first [[Weyl algebra|98QE7KY55W]].\nNon-example: take $R = k[y]$, $\\sigma = \\id$ and $\\delta = \\id$.\nThen $\\delta(1 \\cdot 1) = 1$ but $1 \\cdot \\delta(1) + \\delta(1) \\cdot 1 = 2$, and $1 \\ne 2$ in every field, so $\\delta$ is not an $\\id$-derivation.\nNo ring as above exists, since rule 3 would give $x = x \\cdot 1 = 1 \\cdot x + \\delta(1) = x + 1$.\nWhen $\\delta = 0$ one writes $R[x; \\sigma]$, the skew polynomial ring; when $\\sigma = \\id$ one writes $R[x; \\delta]$, a differential polynomial ring.\nIf $R$ is a [[domain|0Q4CBD3HX0]] and $\\sigma$ is injective, then $R[x; \\sigma, \\delta]$ is a domain.\nIf $R$ is left Noetherian and $\\sigma$ is an automorphism, then $R[x; \\sigma, \\delta]$ is left Noetherian, a form of the Hilbert basis theorem.\n",
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