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is a commutative ring, and 
is a non empty subset of 
then, 
is said to have a proper ideal if both the following are true.
and 
is a commutative ring, 
is a prime ideal if 
is true and 
is a commutative ring, and 
is a non empty subset of 
then, 
has a maximal ideal if all ideal 
are 
is a commutative ring, and 
is a non empty subset of 
then, 
is said to have a proper ideal if both the following are true.
and 
is a commutative ring, 
is a prime ideal if 
is true and 
is a commutative ring, and 
is a non empty subset of 
then, 
has a maximal ideal if all ideal 
are 
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