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  "body": "\n## Definition\n\nA *nonassociative ring* is an [[abelian group|071DDFVG4J]] $(A, +)$ with a binary operation $A \\times A \\to A$, $(a, b) \\mapsto ab$, such that\n$$a(b + c) = ab + ac \\quad \\text{and} \\quad (a + b)c = ac + bc$$\nfor all $a, b, c \\in A$.\nNo associativity and no unit element are required.\n\n## Remarks\n\nThe two distributive laws say that the multiplication is biadditive.\nAn associative ring is a nonassociative ring whose product is associative; the word \"nonassociative\" means \"not necessarily associative\".\nSome authors, and the Lean library Mathlib (`NonAssocRing`), also require a two-sided unit element.\nExample: $\\R^3$ with the cross product $\\times$.\nThe product is bilinear, hence biadditive, but $(e_1 \\times e_1) \\times e_2 = 0 \\ne -e_2 = e_1 \\times (e_1 \\times e_2)$, so it is not associative.\nNon-example: $\\Z$ with the product $a * b = a - b$.\nDistributivity fails: $1 * (1 + 1) = -1$, while $1 * 1 + 1 * 1 = 0$.\n",
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