The Ring of Integers, Euclidean Rings and Modulo Integers

The binary operation multint on Z is defined as follows: (Def. 1) For all elements a, b of Z holds (multint)(a, b) = ·R(a, b). The unary operation compint on Z is defined as follows: (Def. 2) For every element a of Z holds (compint)(a) = −R(a). The double loop structure INT.Ring is defined by: (Def. 3) INT.Ring = 〈Z, +Z,multint, 1(∈ Z), 0(∈ Z)〉. Let us mention that INT.Ring is strict and non empty. Let us mention that INT.Ring is Abelian add-associative right zeroed right complementable well unital distributive commutative associative integral domain-like and non degenerated. Let a, b be elements of the carrier of INT.Ring. The predicate a ¬ b is defined by: