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Ideal


An ideal is a subset I of elements in a ring R that forms an additive group and has the property that, whenever x belongs to R and y belongs to I, then xy and yx belong to I. For example, the set of even integers is an ideal in the ring of integers Z. Given an ideal I, it is possible to define a quotient ring R/I. Ideals are commonly denoted using a Gothic typeface.

A finitely generated ideal is generated by a finite list a_1, a_2, ..., a_n and contains all elements of the form sumc_ia_i, where the coefficients c_i are arbitrary elements of the ring. The list of generators is not unique, for instance <4,10>=<2> in the integers.

IdealLattice

In a number ring, ideals can be represented as lattices, and can be given a finite basis of algebraic integers which generates the ideal additively. Any two bases for the same lattice are equivalent. Ideals have multiplication, and this is basically the Kronecker product of the two bases. The illustration above shows an ideal in the Gaussian integers generated by 2 and 1+i, where elements of the ideal are indicated in red.

From the perspective of algebraic geometry, ideals correspond to varieties.

For any ideal I, there is an ideal I_i such that

 II_i=z,
(1)

where z is a principal ideal, (i.e., an ideal of rank 1). Moreover there is a finite list of ideals I_i such that this equation may be satisfied for every I. The size of this list is known as the class number. In effect, the above relation imposes an equivalence relation on ideals, and the number of ideals modulo this relation is the class number. When the class number is 1, the corresponding number ring has unique factorization and, in a sense, the class number is a measure of the failure of unique factorization in the original number ring.

In 1871, Dedekind showed that every nonzero ideal in the domain of integers of a field is a unique product of prime ideals, and in fact all ideals of Z are of this form and therefore principal ideals.

Ideals can be added, multiplied and intersected. The union of ideals usually is not an ideal since it may not be closed under addition. From the perspective of algebraic geometry, the addition of ideals corresponds to the intersection of varieties and the intersection of ideals corresponds to the union of varieties. Also, the multiplication of ideals corresponds to the union of varieties.

Intersection and multiplication are different, for instance consider the ideal a=<x> in Z[x,y]. Then

 a^2=a·a=<x^2>.
(2)

Sometimes they are the same. If b=<y>, then

 ab=a intersection b=<xy>.
(3)

There is also an analog of division, the ideal quotient (a:b), and there is an analog of the radical, also called the ideal radical r(a). Given a ring homomorphism f:A->B, ideals in A extend to ideals in B, while ideals in B contract to ideals in A.

The following formulas summarize operations on ideals, where x^c denotes contract, x^e denotes ideal extension, and (a:b) denotes an ideal quotient:

a(b+c)=ab+ac
(4)
(a:b)b subset a
(5)
( intersection a_i:b)= intersection (a_i:b)
(6)
(a:sumb_i)= intersection (a:b_i)
(7)
a subset r(a)
(8)
r(r(a))=r(a)
(9)
r(ab)=r(a intersection b)=r(a) intersection r(b)
(10)
r(a+b)=r(r(a)+r(b))
(11)
a subset a^(ec)
(12)
b^(ce) subset b
(13)
b^c=b^(cec)
(14)
a^e=a^(ece)
(15)
(a_1+a_2)^e=a_1^e+a_2^e
(16)
b_1^c+b_2^c subset (b_1+b_2)^c
(17)
(a_1 intersection a_2)^e subset a_1^e intersection a_2^e
(18)
b_1^c intersection b_2^c=(b_1 intersection b_2)^c
(19)
a_1^ea_2^e=(a_1a_2)^e
(20)
b_1^cb_2^c subset (b_1b_2)^c
(21)
(a_1:a_2)^e subset (a_1^e:a_2^e)
(22)
(b_1:b_2)^c subset (b_1^c:b_2^c)
(23)
r(a)^e subset r(a^e)
(24)
r(b)^c=r(b^c)
(25)

If A is an algebra, a left (right) ideal of A is a subspace I of A such that ax in I(xa in I) whenever a in A and x in I. A two-sided ideal is a subset of A that is both a left and right ideal. For each algebra A and an element a in A, the sets Aa={ba:b in A} and aA={ab:b in A} are examples of left and right ideals respectively, and AaA={bac:b,c in A} is an example of a two-sided ideal.


See also

Dedekind Ring, Hilbert's Nullstellensatz, Left Ideal, Maximal Ideal, Minkowski's Lemma, Prime Ideal, Right Ideal Explore this topic in the MathWorld classroom

Portions of this entry contributed by Todd Rowland

Portions of this entry contributed by Mohammad Sal Moslehian

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References

Atiyah, M. F. and Macdonald, I. G. Introduction to Commutative Algebra. Menlo Park, CA: Addison-Wesley, 1969. Bonsall, F. F. and Duncan, J. Complete Normed Algebras. New York: Springer-Verlag, 1973.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002.

Referenced on Wolfram|Alpha

Ideal

Cite this as:

Moslehian, Mohammad Sal; Rowland, Todd; and Weisstein, Eric W. "Ideal." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Ideal.html

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