Relation With The Group of Units
It was remarked above that the ideal class group provides part of the answer to the question of how much ideals in a Dedekind domain behave like elements. The other part of the answer is provided by the multiplicative group of units of the Dedekind domain, since passage from principal ideals to their generators requires the use of units (and this is the rest of the reason for introducing the concept of fractional ideal, as well):
Define a map from K× to the set of all nonzero fractional ideals of R by sending every element to the principal (fractional) ideal it generates. This is a group homomorphism; its kernel is the group of units of R, and its cokernel is the ideal class group of R. The failure of these groups to be trivial is a measure of the failure of the map to be an isomorphism: that is the failure of ideals to act like ring elements, that is to say, like numbers.
Read more about this topic: Ideal Class Group
Famous quotes containing the words relation, group and/or units:
“We must get back into relation, vivid and nourishing relation to the cosmos and the universe. The way is through daily ritual, and is an affair of the individual and the household, a ritual of dawn and noon and sunset, the ritual of the kindling fire and pouring water, the ritual of the first breath, and the last.”
—D.H. (David Herbert)
“Unless a group of workers know their work is under surveillance, that they are being rated as fairly as human beings, with the fallibility that goes with human judgment, can rate them, and that at least an attempt is made to measure their worth to an organization in relative terms, they are likely to sink back on length of service as the sole reason for retention and promotion.”
—Mary Barnett Gilson (1877–?)
“Even in harmonious families there is this double life: the group life, which is the one we can observe in our neighbour’s household, and, underneath, another—secret and passionate and intense—which is the real life that stamps the faces and gives character to the voices of our friends. Always in his mind each member of these social units is escaping, running away, trying to break the net which circumstances and his own affections have woven about him.”
—Willa Cather (1873–1947)