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)