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  "body": "\n## Definition\n\nA *rng* is a set $R$ with two binary operations $+$ and $\\cdot$ such that:\n\n1. $(R, +)$ is an [[abelian group|071DDFVG4J]];\n2. $(ab)c = a(bc)$ for all $a, b, c \\in R$;\n3. $a(b + c) = ab + ac$ and $(a + b)c = ac + bc$ for all $a, b, c \\in R$.\n\n## Remarks\n\nNo multiplicative identity is required; a ring is a rng that has one.\nOther names are *non-unital ring* and *pseudo-ring*.\nExample: the even integers $2\\Z$ with the usual operations form a rng; the clauses hold because they hold in $\\Z$ and $2\\Z$ is closed under both operations.\nIt has no identity, since $e \\cdot 2 = 2$ forces $e = 1 \\notin 2\\Z$.\nNon-example: $\\R^3$ with the cross product as multiplication fails clause 2.\nWith the standard basis $i, j, k$, one has $(i \\times i) \\times j = 0$ but $i \\times (i \\times j) = i \\times k = -j$.\nEvery rng $R$ embeds as an ideal in the ring $\\Z \\times R$ with the product $(m, a)(n, b) = (mn, mb + na + ab)$, whose identity is $(1, 0)$.\n",
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