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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 $\\sigma(1) = 1$.\nThe *skew polynomial ring* $R[x; \\sigma]$ is the [[Ore extension|VF3PV13BA1]] $R[x; \\sigma, 0]$.\nSo it is a free left $R$-module with basis $1, x, x^2, \\dots$, it contains $R$ as a subring, and $x a = \\sigma(a) x$ for all $a \\in R$.\n\n## Remarks\n\nThe product of monomials is $(a x^i)(b x^j) = a \\sigma^i(b) x^{i+j}$.\nExample: let $R = \\C$ and let $\\sigma$ be complex conjugation.\nThen $x z = \\bar z x$ for $z \\in \\C$, and $x^2 z = x \\bar z x = z x^2$, so $x^2$ commutes with all of $\\C$ and with $x$; it lies in the center.\nExample: for a field $k$, $q \\in k$ nonzero, $R = k[y]$ and $\\sigma(y) = q y$, the ring $R[x; \\sigma]$ has $x y = q y x$; it is the quantum plane.\nNon-example: the first Weyl algebra, with $x y = y x + 1$, is an Ore extension $k[y][x; \\id, d/dy]$ with a nonzero derivation, so it is not of the form $R[x; \\sigma]$ with this $R$ and this $x$.\nWhen $\\sigma = \\id$ the ring $R[x; \\sigma]$ is the ordinary polynomial ring $R[x]$.\nSome books call every Ore extension $R[x; \\sigma, \\delta]$ a skew polynomial ring.\n",
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