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Ideals (in ring theory)

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 typeset structure be a ring with identity. A left ideal typeset structure in typeset structure is a subset of typeset structure which is a subgroup of the additive group of typeset structure, and such that typeset structure is a subset of typeset structure (since typeset structure contains identity, typeset structure). To define a right ideal we require that typeset structure. A two-sided ideal  or shortly an ideal of typeset structure is a subset which is both a left and a right ideal of typeset structure.

If typeset structure are two ideals of a ring typeset structure and typeset structure, then we say that typeset structure is a subideal of typeset structure, or that typeset structure is an overideal of typeset structure. At the beginning of the development of the ring theory, when the notion of the ideal was extracted from the number theoretical background, if typeset structure was subideal of typeset structure then typeset structure was called divisible by typeset structure (Dedekind’s definition) and the notation typeset structure was used.

We call typeset structure a proper ideal if it is a proper subset of R, that is, typeset structure but typeset structure It is called a genuine ideal if typeset structure proper and typeset structure, where typeset structure is the called zero ideal of typeset structure. The ideal typeset structure is often called the unit ideal, since if typeset structure has identity typeset structure then typeset structure

If typeset structure is a ring with identity and typeset structure, then the set typeset structure is an left ideal of typeset structure termed left principal ideal generated by typeset structure. Similarly we define the right principal ideal generated by typeset structure as typeset structure or a principal ideal generated by typeset structure as the set typeset structure.  In all these cases we denote the this left (resp. right, resp. two sided) principal ideal of typeset structure generated  by element typeset structure by typeset structure.

Note that If the ring typeset structure does not contain the identity element then

(a) = {r a + n a : r ∈ R, n ∈ Z} .(1)

Hier typeset structure with typeset structure is an abbreviation for the sum the sum typeset structure of typeset structure elements typeset structure. The right ideal generated by typeset structure has obviously the form

(a) = { a r + n a : r ∈ R, n ∈ Z} .(2)

Similarly for the left ideal generated by typeset structure.

If typeset structure is an arbitrary system of left (resp. right, resp. two-sided) ideals of ring typeset structure then also typeset structure is again a left (resp. right, resp. two-sided) ideal of typeset structure. This statement not true for a union of ideals. If typeset structure and typeset structure are two left (resp. right, resp. two-sided) ideals of a ring typeset structure then typeset structure is necessarily a left (resp. right, resp. two-sided)  ideal of typeset structure. However of typeset structure and typeset structure are two left (resp. right, resp. two sided) ideals of a ring typeset structure then the set

a + b = {a + b : a ∈ a, b ∈ b}(3)

is a left (resp. right, resp. two-sided)  ideal of typeset structure. This is the smallest left (resp. right, resp. two-sided)  ideal of typeset structure that contains both ideals typeset structure and typeset structure. The smallest means that if typeset structure is a left (resp. right, resp. two sided) ideal of ring typeset structure then typeset structure.

The smallest left (resp. right, resp. two-sided) ideal of a ring typeset structure containing a given subset typeset structure is said to be generated by typeset structure. The left (resp. right, resp. two-sided) ideal generated by typeset structure is the intersection of all left (resp. right, resp. two-sided) ideals of typeset structure containing typeset structure. The  left (resp. right, resp. two-sided) ideal typeset structure of  typeset structure generated by typeset structure has the form

{r _ 1 a _ 1 + ... + r _ k a _ k : r _ 1, ...r _ k ∈ R, a _ 1, ...a _ k ∈ X}, (res ... ... + r _ k a _ k t _ k : r _ 1, ...r _ k, t _ 1, ...t _ k ∈ R, a _ 1, ...a _ k ∈ X},)(4)

If typeset structure does not contain the identity then the left ideal generated by typeset structure has the form

{r _ 1 a _ 1 + ... + r _ k a _ k + n _ 1 a _ 1 + ... + n _ k a _ k : r _ 1, ...r _ k ∈ R, a _ 1, ...a _ k ∈ X, n _ 1, ..., n _ k ∈ Z},

and similarly for the right or two-sided ideal. If the set typeset structure is finite then we write typeset structure instead of typeset structure.

If typeset structure is a  left (resp. right, resp. two-sided) ideal of a ring typeset structure and there exists a finite set  typeset structure  such that typeset structure, then the ideal typeset structure is said to be finitely generated, and the elements typeset structure are called generators.

Ideals are important because they appear as kernels of ring homomorphisms and allow one to define factor rings   ↑.

Operations on ideals

If typeset structure are two ideals of a ring typeset structure, then their sum typeset structuretypeset structure. As noticed above, this is the smallest ideal containing both typeset structure and typeset structure. The definition can be extended to an arbitrary system if ideals, say typeset structure. Then typeset structure is the set of all element of the form typeset structure, where typeset structure for every typeset structure, and where all but a finite number of summads typeset structure are zero.

If typeset structure we say that ideals typeset structure are coprime.

The intersection typeset structure of two ideals is again an ideal of typeset structure. The same is true for the intersection of an arbitrary family of ideals typeset structure.

The product typeset structure of two ideals typeset structure is defined as the ideal generated by the set typeset structure. Actually

FormBox[RowBox[{ab, =, RowBox[{{Underscript[∑, i] a _ i b _ i : a _ i ∈ a, b _ i ∈ b, and     i < ℵ _ 0}, , ., Cell[]}]}], TraditionalForm](5)

The definition of a product can be extended to any finite set of ideals typeset structure. The product typeset structure is an ideal generated by products of the form typeset structure, where typeset structure, for typeset structure.

For example, Let typeset structure, the ring of integers, this a principal ideal ring. ↑  Then typeset structure, where typeset structure; typeset structure, where typeset structure is the least common multiple of typeset structure and typeset structure. In particular, if typeset structure are coprime (typeset structure), then typeset structure. In general case we have only the relation typeset structure.

If the ring typeset structure is commutative, then these operations are commutative and associative, and the distribution law is true typeset structure.

If typeset structure, then also typeset structure and typeset structure are distributive with each other. In general only the called modular law is valid

a ∩ (b + c) = a ∩ b + a ∩ c, provided     a ⊃ b    or a ⊃ c .(6)

Let typeset structure be a commutative ring having elements which are not divisors of zero. Then the quotient of two ideals typeset structure is defined by

(a : b) = {x ∈ R : x b ⊂ a} .(7)

This is an ideal of typeset structure. If typeset structure is a principal ideal, then we write typeset structure instead of typeset structure. The quotient typeset structure is called the annihilator of ideal typeset structure, and it is denoted by typeset structure.

For instance, the set of all zero divisors of a ring typeset structure can be written in the form typeset structure.

If typeset structure, then typeset structure where typeset structure.

Types of ideals

A proper left (resp. right, resp. two-sided)  ideal typeset structure is called a maximal left (resp. right, resp. two-sided)  ideal of a ring typeset structure if there exists no proper ideal typeset structure such that  typeset structure. The factor ring of a maximal ideal is a field ↑.

A proper ideal typeset structure is called a prime ideal of typeset structure if for any typeset structure such that typeset structure at least one of  the factors typeset structure and typeset structure belongs to typeset structure. ↑

An ideal typeset structure is called primary ideal of typeset structure if for any typeset structure such that typeset structure at least one of typeset structure and typeset structure belongs to typeset structure for some positive integer typeset structure. Every prime ideal is primary, but not conversely.  A principal ideal of typeset structure generated by a power of a prime number is primary.

Special rings defined by ideals

The subset typeset structure and typeset structure are ideals in every ring typeset structure. If typeset structure is a division ring or a field, then these are its only ideals. A ring typeset structure with no two-sided ideals different from typeset structure and typeset structure is called simple.

A commutative ring with identity typeset structure such that its every ideal is principal is called a principal ideal ring (PIR).

Cite this web-page as:

Štefan Porubský: Ideal.

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