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  "body": "\n## Summary\n\nAn Ore extension $R[x; \\sigma, \\delta]$ of a [[ring|0B1261B5ED]] $R$ is a ring of polynomials $\\sum a_i x^i$ in which the variable does not commute with the coefficients.\nInstead, $x a = \\sigma(a) x + \\delta(a)$ for an endomorphism $\\sigma$ of $R$ and a twisted [[derivation|3J89REV9WQ]] $\\delta$.\nOre extensions give many of the basic examples of [[noncommutative rings|D1SA8CHJV0]], such as the [[Weyl algebras|98QE7KY55W]] and quantum planes.\nThey keep good properties of $R$, such as being a domain or being Noetherian, under mild conditions.\n\n## Explanations\n\n### L1 human\n\nIn an ordinary polynomial ring the variable $x$ commutes with every coefficient.\nIn an Ore extension, moving $x$ past a coefficient $a$ changes $a$ to $\\sigma(a)$ and adds a correction term $\\delta(a)$.\nEvery element can still be written uniquely as $a_0 + a_1 x + \\cdots + a_n x^n$, so the ring looks like a polynomial ring, only with a new rule of multiplication.\nFor example, with $R = k[y]$, $\\sigma$ the identity and $\\delta = d/dy$, the rule $xy = yx + 1$ gives the first Weyl algebra.\n",
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