In
mathematics, more specifically in
ring theory, a maximal ideal is an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
that is
maximal (with respect to
set inclusion
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
) amongst all ''proper'' ideals.
In other words, ''I'' is a maximal ideal of a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
''R'' if there are no other ideals contained between ''I'' and ''R''.
Maximal ideals are important because the
quotients of rings by maximal ideals are
simple rings, and in the special case of
unital commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
s they are also
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
s.
In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the
poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one sided maximal ideal ''A'' is not necessarily two-sided, the quotient ''R''/''A'' is not necessarily a ring, but it is a
simple module over ''R''. If ''R'' has a unique maximal right ideal, then ''R'' is known as a
local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the
Jacobson radical J(''R'').
It is possible for a ring to have a unique maximal two-sided ideal and yet lack unique maximal one sided ideals: for example, in the ring of 2 by 2
square matrices over a field, the
zero ideal is a maximal two-sided ideal, but there are many maximal right ideals.
Definition
There are other equivalent ways of expressing the definition of maximal one-sided and maximal two-sided ideals. Given a ring ''R'' and a proper ideal ''I'' of ''R'' (that is ''I'' ≠ ''R''), ''I'' is a maximal ideal of ''R'' if any of the following equivalent conditions hold:
* There exists no other proper ideal ''J'' of ''R'' so that ''I'' ⊊ ''J''.
* For any ideal ''J'' with ''I'' ⊆ ''J'', either ''J'' = ''I'' or ''J'' = ''R''.
* The quotient ring ''R''/''I'' is a simple ring.
There is an analogous list for one-sided ideals, for which only the right-hand versions will be given. For a right ideal ''A'' of a ring ''R'', the following conditions are equivalent to ''A'' being a maximal right ideal of ''R'':
* There exists no other proper right ideal ''B'' of ''R'' so that ''A'' ⊊ ''B''.
* For any right ideal ''B'' with ''A'' ⊆ ''B'', either ''B'' = ''A'' or ''B'' = ''R''.
* The quotient module ''R''/''A'' is a simple right ''R''-module.
Maximal right/left/two-sided ideals are the
dual notion to that of
minimal ideals.
Examples
* If F is a field, then the only maximal ideal is .
* In the ring Z of integers, the maximal ideals are the
principal ideals generated by a prime number.
* More generally, all nonzero
prime ideals are maximal in a
principal ideal domain
In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are princi ...
.
* The ideal
is a maximal ideal in ring
. Generally, the maximal ideals of
are of the form
where
is a prime number and
is a polynomial in
which is irreducible modulo
.
* Every prime ideal is a maximal ideal in a Boolean ring, i.e., a ring consisting of only idempotent elements. In fact, every prime ideal is maximal in a commutative ring
whenever there exists an integer
such that
for any
.
* The maximal ideals of the
polynomial ring