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  "body": "\n## Summary\n\nA nonassociative ring is an abelian group with a multiplication that distributes over addition on both sides, without the requirement that it be associative.\n\"Nonassociative\" means \"not necessarily associative\", so associative rings are a special case.\nLie algebras, [[Jordan algebras|6C2MCVB8C7]] and the [[octonions|1HRMT3DSX2]] are the main examples where associativity fails.\nA nonassociative algebra over a field is the same notion with a compatible vector space structure.\n\n## Explanations\n\n### L1 human\n\nIn most familiar number systems, $(ab)c = a(bc)$, so brackets in products do not matter.\nA nonassociative ring drops this rule and keeps the others: addition works as usual, and multiplication distributes over addition.\nThe cross product of vectors in space is an example: $(e_1 \\times e_1) \\times e_2 = 0$, but $e_1 \\times (e_1 \\times e_2) = e_1 \\times e_3 = -e_2$.\nSuch products are typical in geometry and physics, where they measure rotations or symmetries rather than repeated scaling.\n",
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