A special type of subobject of an algebraic structure. The concept of
an ideal first arose in the theory of rings.
The name ideal derives from the concept of an
ideal number.
For an algebra, a ring or a semi-group
,
an
ideal
is a subalgebra, subring or sub-semi-group closed under multiplication by elements of
.
Here an ideal is said to be a
left
(or
right)
ideal
if it is closed under multiplication on the left (or right) by elements of
,
that is, if
where
An ideal that is simultaneously a left ideal and a right ideal
(that is, one that is preserved under multiplication by elements of
)
is said to be
two-sided.
These three concepts coincide in the commutative case. Every
assertion about left ideals has a corresponding dual assertion
about right ideals (subsequent statements will refer only to the
"left case" ).
Two-sided ideals in rings and algebras play exactly
the same role as do normal subgroups (cf.
Normal subgroup)
in groups. For every homomorphism
,
the kernel
(that is, the set of elements mapped to 0 by
)
is an ideal, and conversely every ideal
is the kernel of some homomorphism. Moreover, an ideal
determines a unique
congruence (in algebra)
on
of which it is the zero class, and thus determines the image
of the homomorphism
of which it is the kernel uniquely (up to an isomorphism):
is isomorphic to the quotient ring or quotient algebra
,
denoted also by
.
Ideals of multi-operator groups have similar properties
in relation to homomorphisms. In a multi-operator
-group
an ideal is defined to be a normal subgroup
of its additive group satisfying the following property: For every
-ary
operator
,
arbitrary elements
and
,
the relation
holds for
.
(This concept reduces to that of a two-sided ideal for rings and algebras.)
On the other hand, the two-sided ideals of a semi-group do not
give a description of all homomorphic images of the semi-group. If a homomorphism
of a semi-group
onto a semi-group
is given, then only in the case where
is a semi-group with zero it is possible to associate with
a two-sided ideal in a natural way, namely
;
however, this association need not determine
uniquely. Nevertheless, if
is an ideal of
,
then among the quotient semi-groups of
having the class of
as an element there exists a maximal one, written
(and called the
ideal quotient).
The elements of this semi-group are the elements of the set
and the ideal
itself, which is the zero in
.
For an arbitrary subset
one can define the
ideal
generated by
as the intersection of all ideals that contain
.
The set
is said to be a
basis of the ideal
.
Different bases can generate one and the same ideal. An ideal
generated by a single element is said to be a
principal ideal.
The intersection, and for semi-groups also the union, of left (two-sided)
ideals is again a left (two-sided) ideal. For rings and algebras,
the set-theoretical union of ideals need not be an ideal. Let
and
be left or two-sided ideals in a ring (or algebra)
.
The
sum of the ideals
and
is the ideal
;
it is the smallest ideal of
containing
and
.
The set of all (left or two-sided) ideals of a
ring (or algebra) forms a lattice under the operations of intersection
and taking sums. Many classes of rings and algebras are defined by
conditions on their ideals or on the lattice of ideals (see
Principal ideal ring;
Artinian ring;
Noetherian ring).
An ideal of the multiplicative semi-group of a ring may or
may not be an ideal of the ring. A semi-group
is a group if and only if
has no (left or two-sided) ideal other than
.
Thus, the abundance of ideals in a semi-group
characterizes the degree to which the semi-group differs from a group.
For a
-algebra
(an algebra over a field
),
an ideal of the ring
need not be an ideal of the algebra
.
For example, if
is a
-algebra
with zero multiplication, the set of ideals of the ring
is the set of subgroups of the additive group of
,
while the set of ideals of the algebra
is the set of all subspaces of the vector
-space
.
However, when
is an algebra with identity, these concepts of
an ideal coincide. Therefore many results have
identical statements for rings and algebras.
A ring not having any two-sided ideal is said to be a
simple ring.
A ring without proper one-sided ideals is a
skew-field.
Left ideals of a ring
may also be defined as submodules of the left
-module
.
Some properties of rings remain unchanged when right ideals
are substituted for left ideals. For example, the
Jacobson radical
defined in terms of left ideals is the same as the Jacobson radical
defined in terms of right ideals. On the other hand,
a left Noetherian ring can fail to be right Noetherian.
The study of ideals in commutative rings is an important part
of commutative algebra. With every commutative ring with
identity one can associate the topological space
whose elements are the proper prime ideals of
.
There is a one-to-one correspondence between the set of all radicals of ideals of
and the set of closed subspaces of
.
The concept of an ideal of a field occurs in commutative algebra, more
precisely, that of an ideal of a field relative to a ring. Here
is a commutative ring with identity and without zero divisors, and
is the field of fractions of
.
An
ideal of the field
is a non-zero subset
that is a subgroup of the additive group of
closed under multiplication by elements of
(that is,
whenever
and
)
and such that there exists an element
such that
.
An ideal is said to be an
integral ideal
if it is contained in
(and then it is an ordinary ideal of
);
otherwise it is a
fractional ideal.
An
ideal of a lattice
is a non-empty subset
of a lattice such that: 1) if
,
then
;
and 2) if
,
then
.
A
dual ideal
(or a
filter)
of a lattice is defined in the dual manner
(
implies
;
implies
).
The ideals of a lattice also form a lattice under inclusion. A maximal element
of the set of all proper ideals of a lattice is called a
maximal ideal.
If
is a homomorphism of a lattice onto a partially ordered set
with a zero, then the complete inverse image of the zero is an ideal. It is called the
kernel ideal
of
.
An ideal
of a lattice
is said to be a
standard ideal
if for arbitrary
,
,
the inequality
implies that
,
where
and
.
Every standard ideal is a kernel ideal. A
kernel ideal of a relatively complemented lattice (see
Lattice with complements)
is standard. An ideal
is called a
prime ideal
if
or
whenever
.
Each of the following conditions is equivalent to primality for an ideal
of a lattice
:
a) the complement
is a filter; or b)
is the complete inverse image of zero under some homomorphism of
onto a two-element lattice. Every maximal ideal of a distributive lattice is prime.
The concept of an
ideal in a partially ordered set
is not in full agreement with the preceding definition. In fact, instead of
1), a stronger condition is required to hold: For every subset of the
ideal, the supremum (join) of the set (if it exists) is also in
.
An
ideal object
of a category with null morphisms is a
subobject
of
such that
for some morphism
.
This ideal can be identified with the set of all
monomorphisms that are kernels of some morphism (see also
Normal monomorphism).
The concept of a
co-ideal object of a category
is defined in the dual way. The concept of an ideal for
-groups
is a special case of that of an ideal object in a category.
A
left ideal of a category
is a class of morphisms containing, with every morphism
of it, all products
with
,
if these are defined in
.
Right ideals of a category
are defined in the dual way. A
two-sided ideal
is a class of morphisms that is both a left ideal and a right ideal.