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Ideal

In mathematics , an ideal is a structure algebraic set in a ring. The ideals fruitfully generalize the study of divisibility for integers. It is thus possible to formulate very general versions of theorems of arithmetic such as the Chinese remainder theorem or the fundamental theorem of arithmetic , valid for ideals. We can also compare this concept with that of subgroup for the algebraic structure of group in that it defines the notion of quotient ring.

Summary

Definitions

Several notions of ideals coexist, which coincide in the case of a commutative ring , but play very different roles without assuming commutativity of multiplication.

Ideals as sub-modules: left ideals and right

Part I of a ring A is a left ideal of A if:

  • I is a subgroup of additive A.
  • \ Forall (a, x) \ in A \ times I: a \ times x \ in I
    The product, left, an element of I by an element of A belongs to I.

and is a right ideal of A if:

  • I is a subgroup of additive A.
  • \ Forall (x, a) \ in I \ times A: x \ times a \ in I
    The product, right, an element of I by an element of A belongs to I.

Ideals as nuclei: two-sided ideals and quotient rings,

Main article: Ring quotient.

A two-sided ideal is a perfect left and right. In a commutative ring, the notions of ideal to right, left ideal and two-sided ideal blend and it is called simply ideal.

A two-sided ideal play for the rings, the same role as normal subgroups for groups.

  • Let A and B be two rings and a morphism of A into B, then the kernel of is a two-sided ideal.
  • Let A be a ring and I a two-sided ideal of A, then the quotient group A / I can be equipped with a single ring structure such that the canonical surjection from A to A / I is a morphism of rings. See section below.
  • Let A and B be two rings, a morphism from A to B ring. Let s be the canonical mapping of A in the quotient ring A / I and i the morphism (A) in B b to b associates. Then i is an injection, a surjection s and there exists a bijection b such that:
\ Phi = i \ circ b \ circ s

Examples of ideals

  • For any integer k, k \ mathbb {Z} is an ideal of \ Mathbb {Z} .
  • If A is a ring, {0} and A are ideals of A. Called ideal own a different ideal of A.
  • If A is a ring and if I is an ideal containing 1 then I = A. More generally, if I contains an element invertible then I = A
  • The only ideals in a field K are the trivial ideals.

Calculating the ideals

Action morphisms

  • Let A and B be two rings and \ Varphi morphism ring of A in B. Then:
    • If J is a two-sided ideal of B then \ Varphi ^ {-1} (J) is a two-sided ideal of A. If, moreover, J is a prime ideal of B, then \ Varphi ^ {-1} (J) is a prime ideal of A. There is however no similar result for the maximal ideals.
    • If is a surjective morphism of rings from A to B, then for any two-sided ideal I of A, (I) is a two-sided ideal of B.
    • The above property is not generally true if is not surjective. We can take eg A = , B = and the canonical inclusion. Then (I) is an ideal that if I is the zero ideal.

Operations on ideals

In what follows, we assume that the ideals are considered the same type (eg. All two-sided)

Sum: if I and J are two ideals of a ring then the set I + J = \ {x + y | x \ in I \ and \ y \ in J \} is an ideal. More generally, for any family of ideals {I} , their sum, denoted

I

is the set of sums \ Sum_ {\ alpha} x_ {\ alpha} \ x_ {\ alpha} \ in I_ {\ alpha} that do intervene whenever a finite number of elements x . This set is an ideal.

Intersection: an intersection of any ideals remains an ideal.

The set of ideals of A equipped with two operations then forms a lattice.

Ideal generated: the second law can implement the concept. If P is part of a ring, called ideal generated by P the intersection of all ideals of A containing P

Examples:
  • For a commutative ring A and one element of this ring, the ideal generated by {a} is aA (eg ideal \ Z generated by {n} n \ Z ).
  • For I and J two ideals of A, the subset I \ cup J of A is not an ideal (trivial exceptions). The ideal of A generated by this part is I + J. This remains valid for any family of ideals.

Product: if I and J are two-sided ideals of a ring, called the product of I and J the ideal IJ equal to all finite sums

x k y k
k

where x_k \ in I and y_k \ in J . Was IJ \ subset I \ cap J

Example: in the ring \ Mathbb Z , The product of ideals n \ mathbb Z and p \ mathbb Z is the ideal np \ mathbb Z and the latter is included in n \ mathbb Z \ cap p \ mathbb Z .

Ideals of finite type, chain conditions

Here we enumerate the properties of ideals clearly linked to their exposure as submodules.

Principal ideal : the ideal generated by an of the ring is by definition the smallest ideal containing We note (a). An ideal I of a ring A is principal if there is an element of A such that I = (a).

An integral domain in which all ideals are principal is said key ring. For example, \ Mathbb {Z} or ring K Ideals in commutative algebra

Ideals individuals

Maximal ideal : an ideal in a commutative ring is called maximal if there are exactly two ideals containing at & itself, ie if is a commutative field.

Prime ideal : In a commutative ring, an ideal is prime if is different from and for all and of such that ab \ in I , It has the property that if a \ notin I then b \ in I . Thus, I is prime if and only if A / I is integrated.

Irreducible ideal: In a commutative ring, an ideal is irreducible if it can not be written as an intersection of two different ideals

(In a commutative ring, every maximal ideal is prime and every prime ideal is irreducible.)

Ideal Primary: In a commutative ring, an ideal is primary if for all a and b of A such that ab \ in I If a \ notin I then there exists a natural number n such that b ^ n \ in I .

Decomposable ideal: In a commutative ring, an ideal is decomposable if it is the finite intersection of primary ideals.

Radical of an ideal in a commutative ring

If I is an ideal in a commutative ring, called radical of noted \ Sqrt {I} , All elements x of A such that there exists an integer n such that x ^ n \ in I . It is an ideal of A.

Example: 30 \ mathbb Z is the radical 360 \ mathbb Z

If A is a commutative ring, we have the following properties

  • \ Sqrt {I} \ supset I
  • \ Sqrt {\ sqrt {I}} = \ sqrt {I}
  • \ Sqrt {IJ} = \ sqrt {I \ cap J} = \ sqrt {I} \ cap \ sqrt {J}
  • \ Sqrt {I + J} = \ sqrt {\ sqrt {I} + \ sqrt {J}}
  • If moreover A is unitary \ Sqrt {I} = A \ I = A Leftrightarrow

Purely inseparable ideal: an ideal is purely inseparable if it is equal to its radical. This applies, for example, any prime ideal in a commutative ring.

Historical aspects

The ideal theory is relatively recent since it was created by Richard Dedekind in the late nineteenth century. At that time, some of the mathematical community was interested in algebraic numbers and especially to algebraic integers.

The question was whether the algebraic integers behave like integers , especially the prime factorization unique way. It seemed, from the early nineteenth century, it was not always the case: 6 for example can be broken down in the ring \ Mathbb Z <span style = Some concepts homonyms

Mathematics in other contexts, different objects are also called ideals, all related or similar in a sense to the ideals of rings discussed in this article:


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