A maximal ideal of a ring 
is an ideal 
, not equal to 
, such that there are no ideals "in
between" 
and 
. In other words, if 
is an ideal which contains 
as a subset,
then either 
or 
. For example, for a positive
integer 
,
is a maximal ideal of 
iff 
is prime, where 
is the ring of integers.
A commutative ring with multiplicative identity has exactly one maximal ideal iff it is a local ring. In the ring
of integers, 
is a maximal ideal whenever 
is a prime number.
In a commutative ring 
with multiplicative
identity, a maximal ideal 
is always a prime ideal, and
the quotient ring 
is a field. In general, not all
prime ideals are maximal. For a noncommutative
ring, the quotient by a maximal two-sided ideal
is a simple ring and need not be a field.
