“Ideal”的概念、定义、习题解题方法及翻译-数学辞典

单词

Ideal

释义

Ideal

A subset of elements in a Ring which forms an additive Group and has the property that, whenever belongs to and belongs to , then and belong to . For example, the set of EvenIntegers is an ideal in the Ring of Integers. Given an ideal , it is possible todefine a Factor Ring .

An ideal may be viewed as a lattice and specified as the finite list of algebraic integers that form a basis for the lattice. Anytwo bases for the same lattice are equivalent. Ideals have multiplication, and this is basically the Kronecker product of thetwo bases.

For any ideal , there is an ideal such that

where

is a Principal Ideal, (i.e., an ideal of rank 1). Moreover there is a finite list of ideals

such thatthis equation may be satisfied for every

. The size of this list is known as the Class Number. In effect, the aboverelation 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 classnumber is a measure of the failure of unique factorization in the original number ring.

Dedekind (1871) showed that every Nonzero ideal in the domain of Integers of a Fieldis a unique product of Prime Ideals.

See also Class Number, Divisor Theory, Ideal Number, Maximal Ideal, Prime Ideal, Principal Ideal

References

Malgrange, B. Ideals of Differentiable Functions. London: Oxford University Press, 1966.

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