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  "body": "\n## Definition\n\nA ring is a set $R$ with two binary operations and two elements $0, 1 \\in R$ that satisfy the axioms below.\nThe operations are the addition $+$ and the multiplication $\\cdot$.\n\n1. $(R, +)$ is an abelian [[group|G5S1M9]] with identity element $0$.\n2. The multiplication is associative: $(a \\cdot b) \\cdot c = a \\cdot (b \\cdot c)$ for all $a, b, c \\in R$.\n3. The element $1$ is a multiplicative identity: $1 \\cdot a = a \\cdot 1 = a$ for all $a \\in R$.\n4. The multiplication is distributive over the addition: $a \\cdot (b + c) = a \\cdot b + a \\cdot c$ and $(b + c) \\cdot a = b \\cdot a + c \\cdot a$ for all $a, b, c \\in R$.\n\n## Remarks\n\nSome authors do not require the element $1$. They call the structure of this definition a ring with identity, or a unital ring.\nThe axioms give $0 \\cdot a = a \\cdot 0 = 0$ and $(-a) \\cdot b = -(a \\cdot b)$ for all $a, b \\in R$.\nThe integers $\\mathbb{Z}$, the integers modulo $n$ and the $n \\times n$ real matrices are rings.\nA ring in which $a \\cdot b = b \\cdot a$ for all $a, b$ is a [[commutative ring|0GYGHKX834]].\n\n## Explanations\n\n### L1 human\n\nA ring is a place where you can add, subtract and multiply, as with the integers.\nYou cannot always divide.\nThe order of a product can matter: for two matrices $A$ and $B$, $AB$ and $BA$ are often different.\n",
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