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  "body": "\n## Definition\n\nA *(right) near-ring* is a set $N$ with two binary operations $+$ and $\\cdot$ such that:\n\n1. $(N, +)$ is a [[group|G5S1M9]], not necessarily abelian, with identity element $0$.\n2. $(N, \\cdot)$ is a [[semigroup|0V3CFG1KK9]]: $(a \\cdot b) \\cdot c = a \\cdot (b \\cdot c)$ for all $a, b, c \\in N$.\n3. $(a + b) \\cdot c = a \\cdot c + b \\cdot c$ for all $a, b, c \\in N$.\n\n## Remarks\n\nA *left near-ring* asks instead $c \\cdot (a + b) = c \\cdot a + c \\cdot b$; the two notions are mirror images.\nEvery ring is a near-ring.\nExample: for a group $(G, +)$, let $M(G)$ be the set of all maps $G \\to G$, with $(f + g)(x) = f(x) + g(x)$ and $f \\cdot g = f \\circ g$.\nClause 1 holds pointwise, clause 2 because composition is associative, and clause 3 because $((f + g) \\circ h)(x) = f(h(x)) + g(h(x))$.\nThe other distributive law fails in $M(\\Z)$: for the constant map $f = 1$, $f \\circ (g + h) = 1$ but $f \\circ g + f \\circ h = 2$.\nNon-example: $\\N = \\set{0, 1, 2, \\ldots}$ with the usual operations satisfies clauses 2 and 3, but fails clause 1, since $1$ has no additive inverse.\n",
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