Let A be a set on which there is a binary operation 
.
An element e of this set is called a left identity if for all
, 
. Similarly, an element f is a
right identity if 
for each .
An element which is both a right and left identity is called an
identity. (Some authors use the term two sided identity.)
For example 0 is an identity for the usual addition on the
real numbers.
Given a binary operation on a set there might be no identity element. There
might be many. There might be left identities which are not right identities
and vice-versa. We tend to be familiar with the situation in which there is
a unique identity. Also note that an identity (left or right or both) for one
operation does not have to be an identity for another operation. Think of
addition and multiplication on the reals where the identities are 0 and 1
respectively. This can be quite dangerous for unusual binary operations on
familiar sets. For example, suppose we have a "new" binary operation on the
real numbers and you are told it has an identity. Do NOT
assume this identity
is 0 or 1 (it might turn out to be 2.7182818284590 
).
One final warning: Being an identity is a global property in the sense that
it must work for ALL elements of the set. It might happen that
for some b, 
and 
and yet e is not an identity
(i.e. for some c, 
).