Ideal (ring theory) - Citizendia

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real The ideal concept generalizes in an appropriate way some important properties of integers like "even number" or "multiple of 3". The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

For instance, in rings one studies prime ideals instead of prime numbers, one defines coprime ideals as a generalization of coprime numbers, and one can prove a generalized Chinese remainder theorem about ideals. In Mathematics, a prime ideal is a Subset of a ring which shares many important properties of a Prime number in the Ring of integers In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than The Chinese remainder theorem is a result about congruences in Number theory and its generalizations in Abstract algebra. In a certain class of rings important in number theory, the Dedekind domains, one can even recover a version of the fundamental theorem of arithmetic: in these rings, every nonzero ideal can be uniquely written as a product of prime ideals. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes In Abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an Integral domain in which every nonzero Proper In Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written

An ideal can be used to construct a quotient ring in a similar way as a normal subgroup in group theory can be used to construct a quotient group. In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G The concept of an order ideal in order theory is derived from the notion of ideal in ring theory. In mathematical Order theory, an ideal is a special subset of a Partially ordered set (poset Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying

1 History
2 Definitions
3 Motivation
4 Examples
5 Ideal generated by a set
- 5.1 Example
6 Types of ideals
7 Properties
8 Ideal operations
9 Ideals and congruence relations
10 See also

History

Ideals were first proposed by Dedekind in 1876 in the third edition of his book Vorlesungen über Zahlentheorie (English: Lectures on Number Theory). Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important Year 1876 ( MDCCCLXXVI) was a Leap year starting on Saturday (link will display the full calendar of the Gregorian Calendar (or a Leap year de ''[[Vorlesungen über Zahlentheorie]]'' ( German for Lectures on Number Theory) is a textbook of Number theory written by German mathematicians They were a generalization of the concept of ideal numbers developed by Ernst Kummer. In Mathematics an ideal number is an Algebraic integer which represents an ideal in the ring of integers of a Number field; the idea was developed Ernst Eduard Kummer ( 29 January 1810 - 14 May 1893) was a German Mathematician. Later the concept was expanded by David Hilbert and especially Emmy Noether. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Amalie Emmy Noether, ˈnøːtɐ (23 March 1882 – 14 April 1935 was a German Mathematician known for her groundbreaking contributions to Abstract algebra and

Definitions

Let R be a ring, with (R, +) the underlying additive group of the ring. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, an additive group may be an Abelian group, when it is written using the symbol + for its Binary operation A subset I of R is called right ideal of R if

(I, +) is a subgroup of (R, +)
xr is in I for all x in I and all r in R

Equivalently, a right ideal of R is a right R-submodule of R. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars

A subset I of R is called left ideal of R if

(I, +) is a subgroup of (R, +)
rx is in I for all x in I and all r in R

Equivalently, a left ideal of R is a left R-submodule of R. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars

For example, if p is in R, then pR is a right ideal and Rp is a left ideal of R. These are called, respectively, the principal right and left ideals generated by p. In Ring theory, a branch of Abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single To remember which is which, note that right ideals are stable under right-multiplication (IR ⊆ I) and left ideals are stable under left-multiplication (RI ⊆ I).

The left ideals in R are exactly the right ideals in the opposite ring R^o and vice versa. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real A two-sided ideal is a left ideal that is also a right ideal, and is often called an ideal except to emphasize that there might exist single-sided ideals. When R is a commutative ring, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone.

We call I a proper ideal if it is a proper subset of R, that is, I does not equal R.

Motivation

Intuitively, the definition can be justified as follows: Suppose we have a subset of elements Z of a ring R and that we would like to obtain a ring with the same structure as R, except that the elements of Z should be zero (they are in some sense "negligible").

But if $z 1 = 0$ and $z 2 = 0$ in our new ring, then surely $z 1 + z 2$ should be zero too, and $r z 1$ as well as $z 1 r$ should be zero for any element $r$ (zero or not).

The definition of an ideal is such that the ideal I generated by Z is exactly the set of elements that are forced to become zero if Z becomes zero, and the quotient ring R/I is the desired ring where Z is zero, and only elements that are forced by Z to be zero are zero. In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the The requirement that R and R/I should have the same structure (except that I becomes zero) is formalized by the condition that the projection from R to R/I is a (surjective) ring homomorphism. In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication

Examples

The even integers form an ideal in the ring Z of all integers; it is usually denoted by 2Z. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French This is because the sum of any even integers is even, and the product of any integer with an even integer is also even. Similarly, the set of all integers divisible by a fixed integer n is an ideal denoted nZ.
The set of all polynomials with real coefficients which are divisible by the polynomial x² + 1 is an ideal in the ring of all polynomials. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations
The set of all n-by-n matrices whose last column is zero forms a left ideal in the ring of all n-by-n matrices. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally It is not a right ideal. The set of all n-by-n matrices whose last row is zero forms a right ideal but not a left ideal.
The ring C(R) of all continuous functions f from R to R contains the ideal of all continuous functions f such that f(1) = 0. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output Another ideal in C(R) is given by those functions which vanish for large enough arguments, i. e. those continuous functions f for which there exists a number L > 0 such that f(x) = 0 whenever |x| > L.
{0} and R are ideals in every ring R. If R is a division ring or a field, then these are its only ideals. In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division

Ideal generated by a set

Let R be a ring.

Any intersection of left (resp. right, resp. two-sided) ideals of R is again a left (resp. right, resp. two-sided) ideal of R. Therefore, if X is any subset of R, the intersection of all left (resp. right, resp. two-sided) ideals of R containing X is a left (resp. right, resp. two-sided) ideal I of R, said to be generated by X. In Mathematics, the expressions generator, generate, generated by and generating set can have several closely related technical meanings I is the smallest left (resp. right, resp. two-sided) ideal of R containing X.

If R is commutative, the left, right and two-sided ideals generated by a subset X of R are the same, since the left, right and two-sided ideals of R are the same. We then speak of the ideal of R generated by X, without further specification. However, if R is not commutative they may not be the same.

The left (resp. right, resp. two-sided) ideal of R generated by a subset X of R is the set of all finite sums of elements of R of the form ra, where r ∈ R and a ∈ X (resp. ar, where r ∈ R and a ∈ X, resp. rar′, where r,r′ ∈ R and a ∈ X). That is, the left (resp. right, resp. two-sided) ideal generated by X is the set of all elements of the form

r₁a₁ + ··· + r_na_n (resp. a₁r₁ + ··· + a_nr_n, resp. r₁a₁r′₁ + ··· + r_na_nr′_n)

with each r_i,r′_i in R and each a_i in X.

By convention, 0 is viewed as the sum of zero such terms, agreeing with the fact that the ideal of R generated by ∅ is {0} by the previous definition.

If a ∈ R, then the left (resp. right, resp. two-sided) ideal of R generated by {a} is denoted by Ra (resp. aR, resp. RaR). Ra is the set of elements of R of the form ra for r ∈ R. An analogous statement holds for aR, but not for RaR.

If an ideal I of R is such that there exists a finite subset X of R (necessarily a subset of I) generating it, then the ideal I is said to be finitely generated.

Example

In the ring Z of integers, every ideal can be generated by a single number (so Z is a principal ideal domain), and the ideal determines the number up to its sign. In Abstract algebra, a principal ideal domain, or PID is an Integral domain in which every ideal is principal i The concepts of "ideal" and "number" are therefore almost identical in Z. In an arbitrary principal ideal domain this is also true, except that instead of differing only by sign, the various generators of a given ideal may differ multiplicatively by any invertible element of the ring. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to

Types of ideals

To simplify the description all rings are assumed to be commutative. The non-commutative case is discussed in detail in the respective articles

Ideals are important because they appear as kernels of ring homomorphisms and allow one to define factor rings. In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the Different types of ideals are studied because they can be used to construct different types of factor rings.

Maximal ideal: A proper ideal I is called a maximal ideal if there exists no other proper ideal J with I a subset of J. In Mathematics, more specifically in Ring theory, a maximal ideal is an ideal which is maximal (with respect to set inclusion amongst all proper ideals The factor ring of a maximal ideal is a field. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division
Prime ideal: A proper ideal I is called a prime ideal if for any a and b in R, if ab is in I, then at least one of a and b is in I. In Mathematics, a prime ideal is a Subset of a ring which shares many important properties of a Prime number in the Ring of integers The factor ring of a prime ideal is an integral domain. In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such
Primary ideal: An ideal I is called primary ideal if for all a and b in R, if ab is in I, then at least one of a and bⁿ is in I for some natural number n. In Mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be written as an intersection In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an Every prime ideal is primary, but not conversely.
Principal ideal: An ideal generated by one element. In Ring theory, a branch of Abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single
Primitive ideal: is the annihilator of a simple left module. In Mathematics, a left primitive ideal in Ring theory is the annihilator of a simple left module. In Mathematics, specifically Module theory, annihilators are a concept that formalizes torsion and generalizes torsion and Orthogonal complement In Abstract algebra, a (left or right module S over a ring R is called simple or irreducible if it is not the Zero In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars A right primitive ideal is defined similarly. Note that (despite the name) left and right primitive ideals are always two-sided ideals. Factor rings constructed with primitive ideals are primitive rings. In Mathematics, especially in the area of Abstract algebra known as Ring theory, the concept of left primitive ring generalizes that of Matrix algebra

Properties

An ideal is proper if and only if it does not contain 1.

The proper ideals can be partially ordered via subset inclusion and therefore as a consequence of Zorn's lemma every ideal is contained in a maximal ideal. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement Zorn's lemma, also known as the Kuratowski-Zorn lemma, is a proposition of Set theory that states Every Partially ordered set in which

Because zero belongs to it, any ideal is nonempty. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members

The ring R can be considered as a left module over itself, and the left ideals of R are then seen as the submodules of this module. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars Similarly, the right ideals are submodules of R as a right module over itself, and the two-sided ideals are submodules of R as a bimodule over itself. In Abstract algebra a bimodule is an Abelian group that is both a left and a right module, such that the left and right multiplications are compatible If R is commutative, then all three sorts of module are the same, just as all three sorts of ideal are the same.

Ideal operations

The sum and product of ideals are defined as follows. For I and J ideals of R,

$I+J:=\{a+b \,|\, a \in I \mbox{ and } b \in J\}$

and

$IJ:=\{a_1b_1+ \dots + a_nb_n \,|\, a_i \in I \mbox{ and } b_i \in J, i=1, 2, \dots, n; \mbox{ for } n=1, 2, \dots\},$

i. e. the product of two ideals I and J is defined to be the ideal IJ generated by all products of the form ab with a in I and b in J. The product IJ is contained in the intersection of I and J.

The sum and the intersection of ideals is again an ideal; with these two operations as join and meet, the set of all ideals of a given commutative ring forms a lattice. In Mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' Also, the union of two ideals is a subset of the sum of those two ideals, because for any element a inside an ideal, we can write it as a+0, or 0+a, therefore, it is contained in the sum as well. In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets However, the union of two ideals is not necessarily an ideal.

Important properties of these ideal operations are recorded in the Noether isomorphism theorems. In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural

Ideals and congruence relations

There is a bijective correspondence between ideals and congruence relations (equivalence relations that respect the ring structure) on the ring:

Given an ideal I, let x ~ y iff x-y ∈ I. See Congruence (geometry for the term as used in elementary geometry

Conversely, given a congruence relation ~, let $I=\lbrace x: x \sim 0 \rbrace\$ .

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