What is the difference between the ring the field and the group in abstract algebra?

Answer 1

A Group is the simplest of these algebraic structures. It is a set, G, of elements (numbers) with a binary operation (addition) that combines any two elements such that the following four axioms are satisfied:

1. Closure: if x and y belong to G then x + y belongs to G.
2. Associativity: if x, y and z belong to G then (x + y) + z = x + (y + z) and so either can be written as x + y + z without ambiguity.
3. Identity: there is an element, 0, in G such that x + 0 = 0 + x = x for all x in G.
4. Invertibility: for any element x in G, there is an element -x such that x + -x = -x + x = 0.

A Ring, R, is an Abelian group which has a second binary operation (multiplication) that is defined on its elements. This second operation is distributive over the first.

1. Abelian: for all x and y in R, x + y = y + x (also known as commutativity).
2. Distributive: for all x, y and z in R, x*(y + z) = x*y + x*z.

A Field is a Ring over which division - by non-zero numbers - is defined.