Definition. A ring is a setwith the operations of addition and multiplication.
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.
This implies that a ring is an Abelian group under addition.
For anythere is an element
Multiplication is associative, i.e. for all
Multiplication is distributive over addition, i.e.
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.