ThmDex – An index of mathematical definitions, results, and conjectures.
 ▼ Set of symbols ▼ Alphabet ▼ Deduction system ▼ Theory ▼ Zermelo-Fraenkel set theory ▼ Set ▼ Binary cartesian set product ▼ Binary relation ▼ Map ▼ Operation ▼ N-operation ▼ Binary operation ▼ Enclosed binary operation ▼ Groupoid ▼ Ringoid ▼ Semiring
Definition D24
Ring

An D21: Algebraic structure $R = (X, f, g)$ is a ring if and only if
 (1) $\forall \, x, y \in X : f(x, y) \in X$ (D20: Enclosed binary operation) (2) $\forall \, x, y \in X : g(x, y) \in X$ (D20: Enclosed binary operation) (3) $\forall \, x, y, z \in X : g(x, f(y, z)) = f(g(x, y), g(x, z))$ (D555: Left-distributive binary operation) (4) (R3) $\forall \, x, y, z \in X : g(x, f(y, z)) = f(g(x, y), g(x, z))$ (D555: Left-distributive binary operation) (5) $\forall \, x, y, z \in X : f(f(x, y), z) = f(x, f(y, z))$ (D488: Associative binary operation) (6) $\forall \, x, y, z \in X : g(g(x, y), z) = g(x, g(y, z))$ (D488: Associative binary operation) (7) $\forall \, x, y \in X : f(x, y) = f(y, x)$ (D489: Commutative binary operation) (8) $\exists \, 0_R \in X : \forall \, x \in X : f(0_R, x) = f(x, 0_R) = x$ (D39: Identity element) (9) $\forall \, x \in X : \exists \, {-x} \in X: f(-x, x) = f(x, -x) = 0_R$ (D40: Inverse element)

An D21: Algebraic structure $R = (X, +, \times)$ is a ring if and only if
 (1) $G = (X, +)$ is an D23: Abelian group (2) $H = (X, \times)$ is a D264: Semigroup (3) The D20: Enclosed binary operation $\times$ is a D557: Distributive binary operation over $+$

An D21: Algebraic structure $R = (X, +, \times)$ is a ring if and only if
 (1) $\forall \, x, y \in X : x + y \in X$ (D20: Enclosed binary operation) (2) $\forall \, x, y \in X : x y \in X$ (D20: Enclosed binary operation) (3) $\forall \, x, y, z \in X : x (y + z) = (x y) + (x z)$ (D555: Left-distributive binary operation) (4) $\forall \, x, y, z \in X : (x + y) z = (x z) + (y z)$ (D556: Right-distributive binary operation) (5) $\forall \, x, y, z \in X : x + (y + z) = (x + y) + z$ (D488: Associative binary operation) (6) $\forall \, x, y, z \in X : x (y z) = (x y) z$ (D488: Associative binary operation) (7) $\forall \, x, y \in X : x + y = y + x$ (D489: Commutative binary operation) (8) $\exists \, 0_R \in X : \forall \, x \in X : 0_R + x = x + 0_R = x$ (D39: Identity element) (9) $\forall \, x \in X : \exists \, {-x} \in X : -x + x = x + (- x) = 0_R$ (D40: Inverse element)