Ideal (ring theory)

In abstract algebra, an ideal of a ring R is a subset of I which is closed with respect to R- linear combinations.

The term " ideal" is derived from the term " ideal number ": ideals can be viewed as a generalization of numbers.

• 5.1 constructions
• 5.2 Remarks

"Ideal Figures"

The origin of the ideals lies in the finding that does not apply in rings like the uniqueness of the decomposition into irreducible elements: thus

And the two factors each decomposition are irreducible. Ernst Eduard Kummer noted that uniqueness can sometimes restore them by one adds more, ideal numbers. In the example, is obtained by adding the number of the factorizations

( that the fractures on the right side are all you can see on their standards ) and

And the uniqueness is restored. From today's perspective, the introduction of ideal number corresponds to the transition to the ( whole ring ) between Hilbert class field in which all ideals ( the wholeness ring ) be an algebraic number field to principal ideals.

Richard Dedekind realized that you can avoid these ideal numbers by their place looking at the totality of all numbers divisible by it. Thus, the numbers and in the example the common ideal prime factor, and lying in multiples of this number are just the prime ideal

The ideal is a "real" common factor is present, consisting of just its multiples, ie it is a principal ideal. In wholeness rings of number fields (and more generally in the virtue of that fact named after him, the class of Dedekind rings ) are obtained in this way a unique decomposition of each ideal ( non-zero ) in prime ideals.

Definition

In order to have appropriate terms for noncommutative rings, one distinguishes between left -, right ideals and two-sided ideals:

It is a subset of a ring is then called a left ideal if:

Similarly, a right ideal, though valid for next 1 and 2:

Finally called two-sided ideal or only briefly Ideal if left and right ideal is ie 1, 2, 3L and 3R met.

Comments

• As an ideal includes, it is not empty. In fact suffices in place of condition 1 has a requirement that is not empty.
• The claims 1 and 2 are equivalent to the statement that a subgroup of the additive group.
• A left - as well as a right ideal in is nothing more than a sub-module of perceived as left- or - right module.
• When the ring is commutative, then fall together all three terms, but in a non-commutative ring may differ.

Examples

• The set of even integers is an ideal in the ring of all integers.
• The set of odd integers is not ideal in; not meet any of the three conditions.
• The set of all polynomials with real coefficients, which are divisible by that form an ideal in the polynomial ring. ( The body is isomorphic to the complex numbers and is even maximal ideal. )
• The ring of all continuous functions from to contains with the ideal of functions. Another ideal in the continuous functions with compact support, ie all functions that are equal to 0 for sufficiently large arguments.
• The non-commutative ring of Hurwitzquaternionen contains both left-and right - ideals as well as two -sided ideals. However, all of them are principal ideals.
• The quantities and are always ideals of a ring. This zero ideal and unit ideal is called. If and its only two -sided ideals are called simple. A commutative simple ring, the ring is not zero, is a field.

Generation of ideals

All links, all right ideals and all two -sided ideals each form a containment system, the associated ideal operators are also referred to as rare with.

Is a subset of the ring then it is called

Of the ideal generated, it is the smallest ( left, right or two-sided ) ideal contains the.

If a unit element is

And if in addition is still commutative, even applies:

The principal ideal generated by an element

Links of ideals

Constructions

For two ideals holds:

• The ( set-theoretic ) intersection is an ideal:

• The set-theoretic union is generally not ideal, but the sum is an ideal:

• Even the so-called complex product composed of the set of products from elements with elements of, in general, no ideal. As a product of the ideal and is therefore defined, which is generated by:

• The ratio of and is an ideal containing all for which the complex product is a subset of:

Comments

• The product of two ideals is always included in their section:
• The ideal ratio is often written in the literature in brackets:
• Use the links sum and average the set of all ideals of a ring forms a modular algebraic Association.
• Some important properties of these shortcuts are summarized in the Noether Isomorphiesätzen.

Special ideals

An ideal is called proper if it is not quite. This is when wrestling with exactly the case when not in.

A proper ideal is called maximal if there is no larger proper ideal, ie if for every ideal:

Using the lemma of anger can be shown that every real ideal of a ring is contained with in a maximal ideal. In particular, each ring has with (except the zero ring ) is a maximal ideal.

A proper ideal is called prime if for all ideals:

Every maximal ideal in a ring with is prime.

Factor rings and cores

Ideals are important because they act as nuclei of ring homomorphisms and allow the definition of factor rings.

A ring homomorphism from the ring in the ring is an illustration with

Of the core is defined as

The core is always a two-sided ideal of

If you start the other way around with a two-sided ideal of one can write the factor ring (pronounced " modulo ", not to be confused with a factorial ring ) define the elements of the form

For one have made ​​. The figure

Is a surjective ring homomorphism whose kernel is the ideal exactly. Thus the ideals of a ring are precisely the kernels of ring homomorphisms of

When the ring is commutative and is a prime ideal, then is an integral domain is a maximal ideal, then even a body.

The extreme examples of factor rings of a ring created by removing parts of the ideals or the factor ring is isomorphic to and is the trivial ring

Norm of an ideal

For wholeness rings of a number field can be a standard one ( whole ) ideals defined by ( and for the zero ideal). This standard is always a finite number, is related to the norm of the field extension, it is namely for principal ideals Moreover, this norm is multiplicative, ie. More generally, these standards are also considered for ideals in orders in number fields.