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text: '[summary: A ring is a kind of [-3gx] which we obtain by considering [3gd groups] as being "things with addition" and then endowing them with a multiplication operation which must interact appropriately with the pre-existing addition. Terminology varies across sources; we will take "ring" to refer to "commutative ring with $1$".]\n\n[summary(Technical): A ring $R$ is a triple $(X, \\oplus, \\otimes)$ where $X$ is a [3jz set] and $\\oplus$ and $\\otimes$ are binary [set_theory_operation operations] subject to the ring axioms. We write $x \\oplus y$ for the application of $\\oplus$ to $x, y \\in X$, which must be defined, and similarly for $\\otimes$. Terminology varies across sources; our rings will have both operations [3jb commutative] and will have an [-54p] under multiplication, denoted $1$.]\n\nA ring $R$ is a triple $(X, \\oplus, \\otimes)$ where $X$ is a [3jz set] and $\\oplus$ and $\\otimes$ are binary [set_theory_operation operations] subject to the ring axioms. We write $x \\oplus y$ for the application of $\\oplus$ to $x, y \\in X$, which must be defined, and similarly for $\\otimes$. It is standard to abbreviate $x \\otimes y$ as $xy$ when $\\otimes$ can be inferred from context. The ten ring axioms (which govern the behavior of $\\oplus$ and $\\otimes$) are as follows:\n\n1. $X$ must be a [3h2 commutative group] under $\\oplus$. That means:\n * $X$ must be [3gy closed] under $\\oplus$.\n * $\\oplus$ must be [associative_function associative].\n * $\\oplus$ must be [commutative_function commutative].\n * $\\oplus$ must have an identity, which is usually named $0$.\n * Every $x \\in X$ must have an inverse $(-x) \\in X$ such that $x \\oplus (-x) = 0$.\n2. $X$ must be a [3h3 monoid] under $\\otimes$. That means:\n * $X$ must be [3gy closed] under $\\otimes$.\n * $\\otimes$ must be [associative_function associative].\n * $\\otimes$ must have an identity, which is usually named $1$.\n3. $\\otimes$ must [distributive_property distribute] over $\\oplus$. That means:\n * $a \\otimes (x \\oplus y) = (a\\otimes x) \\oplus (a\\otimes y)$ for all $a, x, y \\in X$.\n * $(x \\oplus y)\\otimes a = (x\\otimes a) \\oplus (y\\otimes a)$ for all $a, x, y \\in X$.\n \nThough the axioms are many, the idea is simple: A ring is a [3h2 commutative group] equipped with an additional operation, under which the ring is a [3h3 monoid], and the two operations play nice together (the monoid operation [distributive_property distributes] over the group operation).\n\nA ring is an [3gx algebraic structure]. To see how it relates to other algebraic structures, refer to the [5dz tree of algebraic structures].\n\n# Examples\n\nThe integers $\\mathbb{Z}$ form a ring under addition and multiplication.\n\n[fixme: Add more example rings.]\n\\[work in progress.\\]\n\n# Notation\n\nGiven a ring $R = (X, \\oplus, \\otimes)$, we say "$R$ forms a ring under $\\oplus$ and $\\otimes$." $X$ is called the [3gz underlying set] of $R$. $\\oplus$ is called the "additive operation," $0$ is called the "additive identity", $-x$ is called the "additive inverse" of $x$. $\\otimes$ is called the "multiplicative operation," $1$ is called the "multiplicative identity", and a ring does not necessarily have multiplicative inverses.\n\n# Basic properties\n\n[fixme: Add the basic properties of rings.]\n\\[work in progress.\\]\n\n# Interpretations, Visualizations, and Applications\n\n[fixme: Add (links to) interpretations, visualizations, and applications.]\n\\[work in progress.\\]',
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