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, 
is a maximal ideal of 
iff 
is prime, where 
is the ring of integers.
Only in a local ring is there just one maximal ideal. For instance, in the integers, 
is a maximal ideal whenever 
is prime.
A maximal ideal 
is always a prime ideal, and the quotient
ring 
is always a field. In general, not all prime ideals are
maximal.
