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In the most general form ideals are subobjects of special type of some algebraic structures. The prototype of this notions are ideals of a ring. They were introduced by R.Dedekind in the third edition of his Vorlesungen übew Zahlentheorie (Lectures on number theory) in 1876. The name has its origin in the concept of ideal numbers introduced by E.E.Kummer. The theory of ideal was later developed by D.HIlbert and especially by Emmy Noether.
Let be a ring with identity. A left ideal
in
is a subset of
which is a subgroup of the additive group of
, and such that
is a subset of
(since
contains identity,
). To define a right ideal we require that
. A two-sided ideal or shortly an ideal of
is a subset which is both a left and a right ideal of
.
If are two ideals of a ring
and
, then we say that
is a subideal of
, or that
is an overideal of
. At the beginning of the development of the ring theory, when the notion of the ideal was extracted from the number theoretical background, if
was subideal of
then
was called divisible by
(Dedekind’s definition) and the notation
was used.
We call a proper ideal if it is a proper subset of R, that is,
but
It is called a genuine ideal if
proper and
, where
is the called zero ideal of
. The ideal
is often called the unit ideal, since if
has identity
then
If is a ring with identity and
, then the set
is an left ideal of
termed left principal ideal generated by
. Similarly we define the right principal ideal generated by
as
or a principal ideal generated by
as the set
. In all these cases we denote the this left (resp. right, resp. two sided) principal ideal of
generated by element
by
.
Note that If the ring does not contain the identity element then
![]() | (1) |
Hier with
is an abbreviation for the sum the sum
of
elements
. The right ideal generated by
has obviously the form
![]() | (2) |
Similarly for the left ideal generated by .
If is an arbitrary system of left (resp. right, resp. two-sided) ideals of ring
then also
is again a left (resp. right, resp. two-sided) ideal of
. This statement not true for a union of ideals. If
and
are two left (resp. right, resp. two-sided) ideals of a ring
then
is necessarily a left (resp. right, resp. two-sided) ideal of
. However of
and
are two left (resp. right, resp. two sided) ideals of a ring
then the set
![]() | (3) |
is a left (resp. right, resp. two-sided) ideal of . This is the smallest left (resp. right, resp. two-sided) ideal of
that contains both ideals
and
. The smallest means that if
is a left (resp. right, resp. two sided) ideal of ring
then
.
The smallest left (resp. right, resp. two-sided) ideal of a ring containing a given subset
is said to be generated by
. The left (resp. right, resp. two-sided) ideal generated by
is the intersection of all left (resp. right, resp. two-sided) ideals of
containing
. The left (resp. right, resp. two-sided) ideal
of
generated by
has the form
![]() | (4) |
If does not contain the identity then the left ideal generated by
has the form
and similarly for the right or two-sided ideal. If the set is finite then we write
instead of
.
If is a left (resp. right, resp. two-sided) ideal of a ring
and there exists a finite set
such that
, then the ideal
is said to be finitely generated, and the elements
are called generators.
Ideals are important because they appear as kernels of ring homomorphisms and allow one to define factor rings ↑.
If are two ideals of a ring
, then their sum
. As noticed above, this is the smallest ideal containing both
and
. The definition can be extended to an arbitrary system if ideals, say
. Then
is the set of all element of the form
, where
for every
, and where all but a finite number of summads
are zero.
If we say that ideals
are coprime.
The intersection of two ideals is again an ideal of
. The same is true for the intersection of an arbitrary family of ideals
.
The product of two ideals
is defined as the ideal generated by the set
. Actually
![]() | (5) |
The definition of a product can be extended to any finite set of ideals . The product
is an ideal generated by products of the form
, where
, for
.
For example, Let , the ring of integers, this a principal ideal ring. ↑ Then
, where
;
, where
is the least common multiple of
and
. In particular, if
are coprime (
), then
. In general case we have only the relation
.
If the ring is commutative, then these operations are commutative and associative, and the distribution law is true
.
If , then also
and
are distributive with each other. In general only the called modular law is valid
![]() | (6) |
Let be a commutative ring having elements which are not divisors of zero. Then the quotient of two ideals
is defined by
![]() | (7) |
This is an ideal of . If
is a principal ideal, then we write
instead of
. The quotient
is called the annihilator of ideal
, and it is denoted by
.
For instance, the set of all zero divisors of a ring can be written in the form
.
If , then
where
.
A proper left (resp. right, resp. two-sided) ideal is called a maximal left (resp. right, resp. two-sided) ideal of a ring
if there exists no proper ideal
such that
. The factor ring of a maximal ideal is a field ↑.
A proper ideal is called a prime ideal of
if for any
such that
at least one of the factors
and
belongs to
. ↑
An ideal is called primary ideal of
if for any
such that
at least one of
and
belongs to
for some positive integer
. Every prime ideal is primary, but not conversely. A principal ideal of
generated by a power of a prime number is primary.
The subset and
are ideals in every ring
. If
is a division ring or a field, then these are its only ideals. A ring
with no two-sided ideals different from
and
is called simple.
A commutative ring with identity such that its every ideal is principal is called a principal ideal ring (PIR).
Cite this web-page as:
Štefan Porubský: Ideal.