Cmjv05i03p0258

JOURNAL OF COMPUTER AND MATHEMATICAL SCIENCES An International Open Free Access, Peer Reviewed Research Journal www.compmath-journal.org

ISSN 0976-5727 (Print) ISSN 2319-8133 (Online) Abbr:J.Comp.&Math.Sci. 2014, Vol.5(3): Pg.258-261

The Hölder Rigidity Theorem for Valuations M. H. Hosseini Department of Mathematics, Birjand University, P. O. Box 11111, Birjand, IRAN. (Received on: March 16, 2014) ABSTRACT In 1999 Garsia proved Hölder rigidity theorem for absolute4. In this papers we study generalized valuation on a field or a commutative ring with unit element satisfying an approximate maximize inequality an approximate multiplicative property. Also we prove Hölder rigidity theorem for valuations on ring, that they are always Hölder equivalent to a classical valuations.

Keywords: Valuation, (C1 , C2 ) -Hölder valuation, (C0 , α ) -Hölder equivalent, Hölder rigidity.

1. INTRODUCTION AND PRELIMINARIES The theory of valuations may be viewed as a branch of topological algebra.

The development of valuation theory has spanned over more than a hundred years.

First introduced the notion of valuations on fields. Details on valuations on fields can be found in many monographs, e.g Endeler2, Ribenboim8, and Schilling9. Then Manis introduced the notion of valuations in the category of commutative rings can be found in Manis7, Huckab5, and Knebusch and Zhang6. A similar generalization for square marices ring by M. H. Hosseini is proved in3.

Definition 1.1. A valuation of K is any (onto) map v : K Γ∪{∞}, which satisfies the following properties for all a, b in K: 1) v(ab) = v(a) + v(b); 2) v(a) = ∞ if, and only if, a = 0; 3) v(a + b) ≥ min(v(a), v(b)); • in which is a field with multiplicative subgroup ൈ \ 0 , and • (Γ, +, ≥) is an abelian totally ordered group (which could also be iven in multiplicative notation as (Γ, ·, ≥)).

The ordering and group law on Γ are extended to the set Γ∪{∞} by the rules • ∞ ≥ α for all α in Γ; ∞ + α = α + ∞ = ∞ for all α in Γ.

Journal of Computer and Mathematical Sciences Vol. 5, Issue 3, 30 June, 2014 Pages (258-331)

259

M. H. Hosseini, J. Comp. & Math. Sci. Vol.5 (3), 258-161 (2014)

Remark 1.2. It is possible to give a dual definition of the same concept using the multiplicative notation for Γ: if, instead of ∞, an element 0 is given and the ordering and group law on Γ are extended by the rules: • 0 ≤ Îą for all Îą in Γ; • 0 ¡ Îą = Îą ¡ 0 = 0 for all Îą in Γ, then Definition 1.3. A valuation of K is any (onto) map v : K → Î“âˆŞ{0}, satisfying the following properties for all a, b in K:

Definition 2.1. We say that two valuations and on are (C 0 , α ) -Hölder equivalence, if there exists 0 and ଴

1 such that for any x ∈ R :

C0âˆ’Îą ( Âľ ( x ))Îą ≤ ν ( x ) ≤ C1Îą ( Âľ ( x ))Îą .

We would show that any (C1 , C2 ) HĂślder valuation on a division ring or a commutative ring with unit element is HĂślder equivalent to a classical one.

1) v(a) = 0 if, and only if, a = 0; 2) v(ab) = v(a) ¡ v(b); 3) v(a + b) ≤ max(v(a), v(b)).

Theorem 2.2. Let

In this case ொ ={ ; 1 and ொ ; 1 are valuation ring and maximal ideal of ொ , respectively.

ring or a commutative ring R with unit element.There exists a classical valuation, Âľ : R → R+ âˆŞ{0} , which is (C1Îą , Îą ) -

ν : R → R+ âˆŞ{0} be a

(C1 , C2 ) -HĂślder valuation on a division

HÜlder valuation to ν with ι = log 2 2 , i.e.

2C

Definition

1.4.

Let

C1 ≼ 1, C2 ≼ 1. A

(C1 , C2 ) -HĂślder valuation on ring R is mapping ν : R → R+ U {0} that satisfying the following conditions : (H1) For x ∈ R , ν (x) = 0 if and only if x = 0; (H2) For x, y ∈ R ,

ν ( x + y) ≤ C2 Max{ν ( x),ν ( y)};

(H3) For x, y ∈ R ,

C1−1ν ( x )ν ( y ) ≤ ν ( xy ) ≤ C1ν ( x )ν ( y ) .

Remark 1.5. Clearly any (1,1 ) -HĂślder valuation on ring R is a classical valuation on R. 2. HĂślder RIGIDITY for VALUATIONS

for x ∈ R , C1âˆ’Îą (Âľ ( x))Îą ≤ ν ( x) ≤ C1Îą (Âľ ( x))Îą .

Moreover Ďˆ can be defined by

Âľ ( x ) = limnâ†’âˆžÎ˝ ( x n )Îą/n .

N. Bourbaki1 has studied (1, C2 ) HÜlder absolute values. E.Munoz Garsia[3] generalized Bourbaki result as (C1 , C2 ) HÜlder absolute values. Our theorem prove Garsia’s idea for valuations. For the proof of the theorem we need several lemmas and propositions. The proof of the theorem consists of two parts. First we prove that the limit defining ¾ always exists. Second we show that ¾ is a classical valuation on R. Proof. Let R be a division ring or a commutative ring R with unit element endowed with a (C1 , C2 ) -HÜlder valuation, denoted by ν , with C1 ≼ 1, C2 ≼ 1.

Journal of Computer and Mathematical Sciences Vol. 5, Issue 3, 30 June, 2014 Pages (258-331)

M. H. Hosseini, J. Comp. & Math. Sci. Vol.5 (3), 258-161 (2014)

Proposition 2.3. The map

260

ω : R → R+ ∪{0} Lemma 2.7 (Dyadic trick : [10]) Let

by ω ( x ) = ν ( x )α . is a (C1α , C 2α ) -Hölder valuation on R. Proposition 2.4. Let x ∈ R, then (C1−α ) ( n−1)/n ω( x) ≤ (ω( x n ))1/n ≤ (C1α ) ( n−1)/n ω ( x).

Proof. By proposition 2.3, (C1−α ) n−1 (ω( x)) n ≤ ω ( x n ) ≤ (C1α ) n−1 (ω ( x)) n , therefore (C1−α ) ( n−1)/n ω( x) ≤ (ω( x n ))1/n ≤ (C1α ) ( n−1)/n ω ( x).

Consider x ∈ R , x ≠ 0 , and define the sequence of the positive numbers ( an ) n ≥ 0 by a n = ω ( x n )1/n .Then : Proposition 2.5. The sequence ( a n ) n ≥ 0 has

a limit. For any x ∈ R, x ≠ 0 , define

| . |′: R → R+ ∪{0} is a mapping, such that for x, y ∈ R , we have | x + y |′≤ Mmax{| x |′, | y |′},

for some positive constant M. Then for x1 , x2 ,⋅ ⋅ ⋅, xn ∈ R , we have

|

∑ xi |' ≤ M

[ log 2n ]+1

1≤i ≤ n

max1≤i ≤ n (| xi |' ),

where [a ] denotes the integer part of a.

Proposition 2.8.

(1) For x ∈ R, µ ( x ) = 0 ⇔ x = 0 (2) For x, y ∈ R, µ ( xy ) = µ ( x ) µ ( y ).

Proposition 2.9. Let Z be the image of Z in R.For n ∈ N,ψ ( n ) ≤ n .(see [3])

µ ( x ) = limn→∞ an = limn→∞ (ω ( x n ))1/n , and also define µ (0) = 0 . Using

Proposition 2.10. For

Proposition 2.5 we prove :

Proof. By using proposition 2.6 and proposition 2.3 we have: µ ( x + y ) ≤ C1α ω ( x + y ) ≤ C1α C 2α max{ω ( x), ω ( y )} ≤

Proposition 2.6. For x ∈ R ,

C1−α ω ( x ) ≤ µ ( x ) ≤ C1α ω ( x ).

Proof. By Proposition 2.4, we have : (C1−α ) ( n−1)/n ω( x) ≤ (ω( x n ))1/n ≤ (C1α ) ( n−1)/n ω ( x).

Therefore for x ∈ R , we have C1−α ω ( x) ≤ limn→∞ an = limn→∞ (ω ( x n ))1/n ≤ C1α ω ( x).

So we only need to prove that µ is indeed a classical valuation on R. The only delicate part is proving the inequality (H2).

x, y ∈ R, µ ( x + y ) ≤ max{µ ( x), µ ( y )} .

≤ (C1C2 )α max{C1α µ ( x), C1α µ ( y )} = (C12C 2 )α max{µ ( x), µ ( y)}.

We consider first the case when R is a commutative ring with unit element. Let x, y ∈ R and n ≥ 1. Let m = [log 2n ] + 1 .Using lemma 2.7 we have

µ (( x + y ) n ) = µ ( ∑ C (n, i) x i y n−i ) 0≤i ≤ n

α m

≤ ((C C 2 ) ) max 0≤i≤n {µ (C ( n, i ) x i y n −i )}.

2 1

Journal of Computer and Mathematical Sciences Vol. 5, Issue 3, 30 June, 2014 Pages (258-331)

261

M. H. Hosseini, J. Comp. & Math. Sci. Vol.5 (3), 258-161 (2014)

Now using proposition 2.8 and proposition 2.9,

and µ (0) = 0 ), is a classical valuation on R.

µ (( x + y) n ) ≤ ((C12C2 )α ) m max0≤i ≤ n

≤ ((C12C 2 )α ) m max0≤i ≤ n {C (n, i)( µ ( x)) i

Proof. By proposition 2.8 and 2.10 µ , is a classical valuation on R. Now using proposition 2.5, for x ∈ R , we have

( µ ( y)) n −i }.

C1−α (ν ( x))α = C1−α (ω ( x))α ≤ µ ( x) ≤ C1α ω ( x)

{µ (C (n, i))(µ ( x))i ( µ ( y)) n −i } ≤

k

Now let n = 2 , we have 2

max0≤i≤ n C (2k , i ) = C (2k , k ) = (2k )! /( k!) .

Moreover, let µ ( x) ≥ µ ( y ) . Then we have : µ (( x + y) 2k ) ≤ ((C12C2 )α ) m (2k )!/(k!) 2 (µ ( x)) 2k .

Finally exploiting the multiplicative property for µ , µ ( x + y ) = ( µ (( x + y ) 2 k ))1/2 k ≤ (((C12 C 2 )α )

[ log 22 k ]+1

((2 k )! /( k!) 2 ))1/2 k µ ( x ),

and passing to the limit n → ∞ we get inequality µ ( x + y ) ≤ µ ( x) .

Similarly, if µ ( y ) ≥ µ ( x) then µ ( x + y ) ≤ µ ( y ). Hence for x, y ∈ R ,

µ ( x + y ) ≤ max{µ ( x), µ ( y )}.

In the case where R is a division ring we can assume that x ≠ 0 and we obtain in the same way : µ ((1 + x −1 y ) 2 k ) ≤ ((C12 C2 )α )

[ log 22 k ]+1

((2k )! /(k!) 2 )) µ ((1 + ( µ ( x))) −1 ( µ ( y )) 2 k .

So multiplying by ( µ ( x)) n and using the multiplicative property we prove the same inequality as before.

Proposition 2.11. The map

µ : R → R+ ∪{0}

by µ = limn→∞ a n = limn→∞ (ω ( x n )1/n (for

x≠0

= C1α (ω ( x))α .

i.e µ is (C1α , α ) -Hölder equivalent to ν .

Moreover proposition 2.11 implies that the map ψ = limn →∞ a n = limn →∞ (ϕ ( x n )1/n , is a classical valuation on R. By this is proved Theorem 2.2.

REFERENCES 1. N. Borbaki, commutative algebra Elements of Mathematics, capter VI, section 6.1, Addison-Wosloy, (1972).

2. O. Endler, valuation Theory. New York: Springer-Verlag (1972).

3. M. H. Hosseini, H¨older rigidity for matrices, Journal of Mathematical Sciences, Vol. 140, No. 2, (2007).

4. E. Munoz Garcia, Holder absolute values are equiavalent to classical, ones Proc. Amer. Math. Soc, 127, No. 7, 1967-1971 (1999).

5. J.A. Huckaba, Commutative Rings with zero divisors, New York: Marcel Dekker, Inc (1988).

6. M. Knebusch, D. Zhang, Manis valuations and Prufer Extensions I. A new chapter in Commutative algebra.

Lecture Notes in Math. 1791. Berlin: Springer-Verlage (2002).

Journal of Computer and Mathematical Sciences Vol. 5, Issue 3, 30 June, 2014 Pages (258-331)

M. H. Hosseini, J. Comp. & Math. Sci. Vol.5 (3), 258-161 (2014)

7. M. Manis, valuations on a commutative ring. Proc. Amer. Marh Soc. 20:193-198 (1969).

8. P. Ribenboim. The Theorie des valuations. Montreal: Less presses de I’ Universite de Montreal (1964).

9. O.F.G. Schilling. The Theory of valuations New York: Amer. Math. SOC Thomas, J. J (1978). set Theory. New York: Academic press, Inc (1952).

10. D. Welsh, codes and cryptography.

Oxford Science publications, MR 89i:94001 (1988).

Journal of Computer and Mathematical Sciences Vol. 5, Issue 3, 30 June, 2014 Pages (258-331)

262