By Guo L.-T.

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If xy -I E R then v (x) 2:: v(y) by definition of the ordering. Also, 1 + xy -I E R, : so v(1 + xy -I ) 2:: v(I) , and we can multiply by v(y) to obtain v(x + y) v (y) = min{v(x) , v(y) } . Similarly if yx-I E R we obtain v (x + y) 2:: v (x) min{v(x) , v(y) } . Thus v is a valuation, and clearly R is the valuation ring v. 2:: = of 0 In the examples we gave above, the value group is infinite cyclic. Valuation rings arising from valuations where the value group is infinite cyclic are called discrete valuation rings, and are of particular interest.

The equivalence of ( 3) and (5) can be generalised as follows. 4. If (X, d) is a A-tree and Xo, . . , Xn E X , the following are equiv­ alent. ( 1 ) [XO , xn] = [XO, X l , . . Xn] ( 2 ) d(xo, xn) = L �= l d(Xi - l , Xi). Proof. ( 1 ) implies ( 2) by an easy induction using the remark preceding the lemma. We also use induction on n to show (2 ) implies ( 1 ) . This is trivial if n = 1 and follows from the preceding remark if n = 2. If n > 2 then by the triangle inequality, n- l d(xo, xn) ::; d(xo , Xn- l ) + d(Xn - l , xn) ::; L d(Xi - l , Xi) + d(Xn - l , Xn) i=l = d(xo , xn) so d(xo , xn - d = L �:ll d(Xi - l , Xi), whence [Xo , Xn - l ] = [Xo , X l , .

For X E F* , let x denote the coset xR* in F* I R*. Then it is easily checked that we can define a partial order on F* I R* by x 2: y if and only if xy - l E R, and that this makes F* I R* into a partially ordered abelian group (written multiplicatively) . 3. In the situation just described the following are equivalent. (1) R is a valuation ring. (2) a E F \ R implies a - I E R. (3) The set of ideals of R is linearly ordered by inclusion. (4) F* I R* is linearly ordered by the ordering defin ed above.

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