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Encyclopedia > Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept generalizes in an appropriate way some important properties of integers like "even number" or "multiple of 3". In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... The integers are commonly denoted by the above symbol. ...


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 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. 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 that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... Coprime - Wikipedia /**/ @import /skins-1. ... Several related results in number theory and abstract algebra are known under the name Chinese remainder theorem. ... 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 of problems that arise from their study. ... In abstract algebra, a Dedekind domain is a Noetherian integral domain which is integrally closed in its fraction field and which has Krull dimension 1. ... In number theory, the fundamental theorem of arithmetic (or unique factorization theorem) states that every natural number either is itself a prime number, or can be written as a unique product of prime numbers. ...


An ideal can be used to construct a factor ring in a similar way as a normal subgroup in group theory can be used to construct a factor group. The concept of an order ideal in order theory is derived from the notion of ideal in ring theory. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ... In mathematical order theory, an ideal is a special subset of a partially ordered set. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ...

Contents

History

Ideals were first proposed by Dedekind in 1876 in the third edition of his book Vorlesungen über Zahlentheorie (English: Lectures on Number Theory). They were a generalization of the concept of ideal numbers developed by Ernst Kummer. Later the concept was expanded by David Hilbert and especially Emmy Noether. Julius Wilhelm Richard Dedekind (October 6, 1831 - February 12, 1916) was a German mathematician and Ernst Eduard Kummers closest follower in arithmetic. ... Year 1876 Pick up Sticks(MDCCCLXXVI) was a leap year starting on Saturday (link will display the full calendar) of the Gregorian Calendar (or a leap year starting on Thursday of the 12-day slower Julian calendar). ... Vorlesungen über Zahlentheorie (Lectures on Number Theory) is a textbook of number theory written by German mathematicians P.G.L. Dirichlet and Richard Dedekind, and published in 1863. ... 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 by Kummer, and lead to Dedekinds definition of ideals for rings. ... Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ... David Hilbert (January 23, 1862, Königsberg, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ... Amalie Emmy Noether [1] (March 23, 1882 – April 14, 1935) was a German-born mathematician, said by Einstein in eulogy to be [i]n the judgment of the most competent living mathematicians, [...] the most significant creative mathematical genius thus far produced since the higher education of women began. ...


Definitions

Let R be a ring, with (R, +) the underlying additive group of the ring. A subset I of R is called right ideal of R if In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... An additive group is a group, and any group can be written as an additive group, so the adjective additive does not describe a class of groups, but rather the notation used to write the group operation. ...

  • (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 G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... Look up module in Wiktionary, the free dictionary. ...


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 G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... Look up module in Wiktionary, the free dictionary. ...


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 the principal right and left ideals generated by p. To remember which is which, note that right ideals are stable under right-multiplication (IRI) and left ideals are stable under left-multiplication (RII). 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 element a of R. More specifically: a left principal ideal of R is a subset of R of the form Ra := {ra : r in R...


The left ideals in R are exactly the right ideals in the opposite ring Ro and vice versa. 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. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...


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


Examples

  • The even integers form an ideal in the ring Z of all integers; it is usually denoted by 2Z. 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.
  • 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. 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. 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.

The integers are commonly denoted by the above symbol. ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

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. I is the smallest left (resp. right, resp. two-sided) ideal of R containing X. In mathematics, a subset S of a algebraic structure G is a generating set of G (or G is generated by S) if the smallest subset of G that includes S and is closed under the algebraic operations on G is G itself. ...


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 rR and aX (resp. ar, where rR and aX, 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 + ··· + rnan (resp. a1r1 + ··· + anrn, resp. r1a1r′1 + ··· + rnanr′n)

with each ri,r′i in R and each ai 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 aR, 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 rR. 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. 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 abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ... In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of...

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. Different types of ideals are studied because they can be used to construct different types of factor rings. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ...

  • 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. The factor ring of a maximal ideal is a field.
  • 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. The factor ring of a prime ideal is an integral domain.
  • 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. Every prime ideal is primary, but not conversely.
  • Principal ideal: An ideal generated by one element.
  • Primitive ideal: is the annihilator of a simple left module. 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, more specifically in ring theory a maximal ideal is an ideal which is maximal (with respect to set inclusion) amongst all proper ideals, i. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... 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 abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... In mathematics, an ideal in a commutative ring is a primary ideal if for all elements , we have that if , then either or for some This is clearly a generalization of the notion of a prime ideal, and (very) loosely mirrors the relationship in between prime numbers and prime powers. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... 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 element a of R. More specifically: a left principal ideal of R is a subset of R of the form Ra := {ra : r in R... A left primitive ideal is the annihilator of a simple left module. ... Annihilators are a concept that occurs in ring theory, a branch of mathematics. ... In abstract algebra, a (left or right) module S over a ring R is called simple if it is not the zero module and if its only submodules are 0 and S. Understanding the simple modules over a ring is usually helpful because they form the building blocks of all... 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 to lie in a field, the scalars may lie in an arbitrary ring. ... In abstract algebra, a left primitive ring R is a ring with a faithful simple left module R-module. ...

Properties

  • Because zero belongs to it, any ideal is nonempty.
  • 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. 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. If R is commutative, then all three sorts of module are the same, just as all three sorts of ideal are the same.

This article does not cite any references or sources. ... In mathematics, especially order theory, a partially ordered set (or poset) is a set equipped with a partial order relation. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... Zorns lemma, also known as the Kuratowski-Zorn lemma, is a proposition of set theory that states: Every non-empty partially ordered set in which every chain (i. ... In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set. ... In abstract algebra, a module is a generalization of a vector space. ... 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. ...

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. A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ...


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. 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. However, the union of two ideals is not necessarily an ideal. 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, all elements of B that also belong to A), but no other elements. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...


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 isomorphisms. ...


See also


  Results from FactBites:
 
Ideal (ring theory) - Wikipedia, the free encyclopedia (1382 words)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.
The map p from R to R/I defined by p(a) = a + I is a surjective ring homomorphism (or ring epimorphism) whose kernel is the original ideal I.
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 ring forms a lattice.
Ring theory: Definition and Links by Encyclopedian.com - All about Ring theory (562 words)
In mathematics, Ring theory is that branch of mathematics concerned with the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers.
The theory of commutative rings resembles the theory of numbers in several respects, and various definitions for commutative rings are designed to recover properties known from the integers.
In commutative ring theory, numbers are often replaced by ideals, and the definition of prime ideal tries to capture the essence of prime numbers.
  More results at FactBites »

 
 

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