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  "body": "\n## Definition\n\nLet $S$ be a set with two [[binary operations|0M418FE7BB]] $\\ast$ and $+$.\n\n1. The operation $\\ast$ is *left distributive* over $+$ if $a \\ast (b + c) = (a \\ast b) + (a \\ast c)$ for all $a, b, c \\in S$.\n2. The operation $\\ast$ is *right distributive* over $+$ if $(b + c) \\ast a = (b \\ast a) + (c \\ast a)$ for all $a, b, c \\in S$.\n3. The operation $\\ast$ is *distributive* over $+$ if it is both left and right distributive over $+$.\n\n## Remarks\n\nExample: matrix multiplication on the set of $n \\times n$ real matrices is distributive over matrix addition, since $A(B + C) = AB + AC$ and $(B + C)A = BA + CA$ by the entrywise formulas.\n\nNon-example: on the set $\\R_{\u003e0}$ of positive reals, take ordinary multiplication $\\cdot$ in place of $+$, and let $a \\ast b = a^b$.\nThen $\\ast$ is right distributive over $\\cdot$, since $(bc)^a = b^a c^a$.\nIt is not left distributive, since $2^{1 \\cdot 1} = 2$ while $2^1 \\cdot 2^1 = 4$.\n\nIf $\\ast$ is commutative, left and right distributivity are equivalent.\nIn a ring, multiplication is distributive over addition; in a Boolean algebra, each of $\\wedge$ and $\\vee$ is distributive over the other.\n",
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