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  "body": "\n## Summary\n\nA near-field is an algebraic structure like a [[division ring|0EA5BTA0CX]], except that only one of the two distributive laws is required.\nIt is a near-ring with identity in which every nonzero element has a multiplicative inverse.\nNear-fields were first constructed by Dickson, and they coordinatize certain projective planes that are not Desarguesian.\n\n## Explanations\n\n### L1 human\n\nIn a field one can add, subtract, multiply and divide, and multiplication distributes over addition from both sides.\nA near-field keeps everything except one side of distributivity: $(a + b)c = ac + bc$ always holds, but $a(b + c) = ab + ac$ may fail.\nEvery field and every division ring is a near-field.\nThe smallest near-field that is not a division ring has $9$ elements.\nIt is built from the [[finite field|0ZXZ82FTX2]] with $9$ elements by twisting its multiplication.\n",
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