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  "body": "\n## Definition\n\nLet $Q$ be a set with two binary operations $+$ and $\\cdot$, and let $Q^* = Q \\setminus \\set{0}$.\nThe triple $(Q, +, \\cdot)$ is a *(left) quasifield* if the following hold.\n\n1. $(Q, +)$ is a [[group|G5S1M9]] with identity element $0$.\n2. $a \\cdot 0 = 0 \\cdot a = 0$ for all $a \\in Q$.\n3. $(Q^*, \\cdot)$ is a loop: there is $1 \\in Q^*$ with $1 \\cdot a = a \\cdot 1 = a$ for all $a \\in Q^*$, and for $a, b \\in Q^*$ the equations $a \\cdot x = b$ and $y \\cdot a = b$ have unique solutions in $Q^*$.\n4. $a \\cdot (b + c) = a \\cdot b + a \\cdot c$ for all $a, b, c \\in Q$.\n5. For all $a, b, c \\in Q$ with $a \\ne b$, the equation $a \\cdot x = b \\cdot x + c$ has exactly one solution $x \\in Q$.\n\nA *right quasifield* is defined in the same way with $(b + c) \\cdot a = b \\cdot a + c \\cdot a$ in clause 4 and $x \\cdot a = x \\cdot b + c$ in clause 5.\n\n## Remarks\n\nExample: every [[division ring|0EA5BTA0CX]] $D$ is a quasifield.\nClauses 1 to 4 are axioms of a division ring.\nFor clause 5, by distributivity $a x = b x + c$ is equivalent to $(a - b) x = c$, which has the unique solution $x = (a - b)^{-1} c$ since $a \\ne b$.\n\nNon-example: the integers $\\Z$ with the usual operations.\nClause 3 fails, since $2 \\cdot x = 1$ has no solution in $\\Z \\setminus \\set{0}$.\n\nOne can prove that the additive group $(Q, +)$ of a quasifield is always abelian.\nA quasifield in which both distributive laws hold is a [[semifield|AHH97EJ6VH]], and a quasifield whose multiplication is associative is a [[near-field|DSE72WQ5A2]] (on the matching side).\n",
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