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 Even Integers is an ideal in the Ring of Integers. Given an ideal , it is possible to define 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. Any two bases for the same lattice are equivalent. Ideals have multiplication, and this is basically the Kronecker product of the two bases.
For any ideal , there is an ideal such that
Dedekind (1871) showed that every Nonzero ideal in the domain of Integers of a Field is a unique product of Prime Ideals.
See also Class Number, Divisor Theory, Ideal Number, Maximal Ideal, Prime Ideal, Principal Ideal
Malgrange, B. Ideals of Differentiable Functions. London: Oxford University Press, 1966.
© 1996-9 Eric W. Weisstein