A structure which can operate addition, subtraction, multiplication and division.
All instance of Ring must satisfy following conditions:
Associative on addition: for all x, y and z; additive.combine(additive.combine(x, y), z) equals to additive.combine(x, additive.combine(y, z))
Identity on addition: for all x; additive.combine(additive.identity, x) equals to x.
Inverse on addition: for all x; exists y; additive.combine(x, y) equals to additive.identity.
Commutative on addition: for all x and y; additive.combine(x, y) equals to additive.combine(y, x).
Associative on multiplication: for all x, y and z except zero; multiplication.combine(multiplication.combine(x, y), z) equals to multiplication.combine(x, multiplication.combine(y, z))
Identity on multiplication: for all x except zero; multiplication.combine(multiplication.identity, x) equals to multiplication.combine(x, multiplication.identity) and x.
Inverse on multiplication: for all x except zero; exists y; multiplication.combine(x, y) equals to multiplication.combine(y, x) and multiplication.identity.
Distributive: for all x, y and z; multiplication.combine(x, additive.combine(y, z)) equals to additive.combine(multiplication.combine(x, y), multiplication.combine(x, z))
A structure which can operate addition, subtraction, multiplication and division.
All instance of
Ringmust satisfy following conditions:x,yandz;additive.combine(additive.combine(x, y), z)equals toadditive.combine(x, additive.combine(y, z))x;additive.combine(additive.identity, x)equals tox.x; existsy;additive.combine(x, y)equals toadditive.identity.xandy;additive.combine(x, y)equals toadditive.combine(y, x).x,yandzexcept zero;multiplication.combine(multiplication.combine(x, y), z)equals tomultiplication.combine(x, multiplication.combine(y, z))xexcept zero;multiplication.combine(multiplication.identity, x)equals tomultiplication.combine(x, multiplication.identity)andx.xexcept zero; existsy;multiplication.combine(x, y)equals tomultiplication.combine(y, x)andmultiplication.identity.x,yandz;multiplication.combine(x, additive.combine(y, z))equals toadditive.combine(multiplication.combine(x, y), multiplication.combine(x, z))