Ring Axioms

Definition. A ring is a setwith the operations of addition and multiplication.

Addition

For anythere is an element, so that R is closed under addition.

Addition is associative, i.e. for all

There is an element ofcalled the zero element and written 0,  which has the property that

for allso that 0 is the identity element of the addition operation.

Every elementhas a negative, an element ofwrittenwhich satisfies

so that the additive inverse ofis

These together imply that a ring is a group under addition with identity 0.

Addition is commutative, i.e.

for all

This implies that a ring is an Abelian group under addition.

Multiplication

For anythere is an element

Multiplication is associative, i.e. for all

Multiplication is distributive over addition, i.e.

andfor all

A ring may or may not have a multiplicative identity. This means that the elements of the ring with the operation of multiplication may not necessarily be a group. Neither must the multiplication operation be commutative.

Examples:are all rings but the set of all ordered pairswith first component equal to 1 is not sinceand the first component is not 1 so the operation of addition is not closed.