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
Formulation 0
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)
Formulation 1
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 $+$
Formulation 2
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)