Ring and Field Axioms

The concept of a ring follows naturally from that of a group. A Ring satisfies the following axiomatic conditions.

A ring is a set R coupled with two binary operations, '+' and '*', in analogue to usual addition and multiplication so that;

1) (R , +) is an Abelian group, with identity element 0 (zero)

2) Multiplication under '*' satisfies the axioms of Associativity and Closure.

3) The Left and Right Distributive Laws hold for R, namely;
a * (b + c) = (a * b) + (a * c)   and   (a + b) * c = (a * c) + (b * c)

Simple huh?

There is no requirement for a ring to have an identity element for multiplication. When a ring does have a multiplicative identity, it is standard practice to write the identity element as '1', and such a ring is called a "Ring with Unity" in reference to the identity element.

There is also no need for multiplication to be commutative, and the set of multiplicative elements, (R without zero) need not be a group. Many rings have a subset of elements that multiplicatively form a group, and this group is called a "Group of Units" Units are the flashy term for elements that have a multiplicative inverse. The term 'Units' is not to be confused with 'Unity'.

The definition of a Field is as follows.

A Field is a Ring 'F', Where the set of all non-zero elements forms an abelian group under multiplication..

Note,
There are several easy to prove fundamental properties of rings, one given here is;
0 * x = x * 0 = 0 for all x in R

There is a special type of Ring, where all of the non-zero elements under '*' form a group, but one that is not abelian. Since division is possible, this structure is called a "Division Ring". Some texts call these "Skew Fields".

The group of units of a ring allow cancellation laws within the group, but other elements that are not units may not be cancelled in the usual fashion. solutions of x in the equation ax = b may not be unique if a is not a unit.

In usual terms, and in particular Fields as we most familiar with, We know that if    a*b = 0   Usual properties of integers and real numbers would imply that either a = 0 or b = 0 or both. This may not be the case for elements that are not units. Where this happens, where neither 'a' nor 'b' is zero, we term both elements 'a' and 'b' "Zero divisors". Neither can be units if this is so.

The set of integers under usual addition and multiplication is a Ring with Unity, but not a Field.

The set of all Fractions (Rational numbers) is a Field, as also is the set of Real numbers, (The complete set of all decimals).

All of these sets are infinite. Finite examples will be given in the section introducing modular arithmetic. One simple example of a finite field is the set of zero on its own. { 0 }. This is the 'Null Ring', or 'Zero-Ring'.

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