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:
“He hung out of the window a long while looking up and down the street. The worlds second metropolis. In the brick houses and the dingy lamplight and the voices of a group of boys kidding and quarreling on the steps of a house opposite, in the regular firm tread of a policeman, he felt a marching like soldiers, like a sidewheeler going up the Hudson under the Palisades, like an election parade, through long streets towards something tall white full of colonnades and stately. Metropolis.”
—John Dos Passos (18961970)
“JuryA group of twelve men who, having lied to the judge about their hearing, health, and business engagements, have failed to fool him.”
—H.L. (Henry Lewis)