Value Group
The units D* of D comprise a group under multiplication, which is a subgroup of the units F* of F, the nonzero elements of F. These are both abelian groups, and we can define the quotient group V = F*/D*, which is the value group of D. Hence, we have a group homomorphism ν from F* to the value group V. It is customary to write the group operation in V as +.
We can turn V into a totally ordered group by declaring the residue classes of elements of D as "positive". More precisely, V is totally ordered by defining if and only if where and are equivalence classes in V.
Read more about this topic: Valuation Ring
Famous quotes containing the word group:
“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–?)
“The government of the United States at present is a foster-child of the special interests. It is not allowed to have a voice of its own. It is told at every move, “Don’t do that, You will interfere with our prosperity.” And when we ask: “where is our prosperity lodged?” a certain group of gentlemen say, “With us.””
—Woodrow Wilson (1856–1924)