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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] and let $I$ be a [[two-sided ideal|0DDA6CY7GJ]] of $R$.\nFor $a \\in R$, the coset of $a$ is $a + I = \\{a + x : x \\in I\\}$.\nThe quotient ring $R/I$ is the set of the cosets of $I$, with these operations:\n$$(a + I) + (b + I) = (a + b) + I, \\qquad (a + I)(b + I) = a b + I.$$\nThese operations do not depend on the choice of $a$ and $b$ in their cosets.\nWith them, $R/I$ is a ring with zero $0 + I = I$ and identity $1 + I$.\n\n## Remarks\n\nTwo elements $a, b \\in R$ have the same coset if and only if $a - b \\in I$.\nThe map $\\pi \\colon R \\to R/I$, $\\pi(a) = a + I$, is a surjective [[ring homomorphism|0GGE0Z0JQ1]] with [[kernel|0AAHE7GEW6]] $I$.\nFor $n \\ge 1$, the quotient $\\mathbb{Z}/n\\mathbb{Z}$ is the ring of integers modulo $n$.\nThe quotient $R/R$ is the [[zero ring|008TBCYDPX]].\n\n## Explanations\n\n### L1 human\n\nTo make the quotient ring, you declare all the elements of $I$ to be zero.\nTwo elements become equal when their difference is in $I$.\nFor $\\mathbb{Z}/5\\mathbb{Z}$, the numbers $2$ and $7$ become the same element.\n",
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  "name": "Quotient ring",
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