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:
“The boys think they can all be athletes, and the girls think they can all be singers. That’s the way to fame and success. ...as a group blacks must give up their illusions.”
—Kristin Hunter (b. 1931)
“With a group of bankers I always had the feeling that success was measured by the extent one gave nothing away.”
—Francis Aungier, Pakenham, 7th Earl Longford (b. 1905)