Valuation on Some Algebric Structures

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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(4): Pg.332-335

Valuation on Some Algebric Structurs M. H. Hosseini and M. H. Rezaeigol Academic Member of School of Mathematics, University of Birjand, IRAN. (Received on: July 4, 2014) ABSTRACT The purpose of this article is to introduce the notion of valuations on Some algebraic structures; valuation on ‍( )Ü´(ܯ‏the set of square matrices), on â€ŤÜŻâ€ŹáˆşŕŹˇáˆť (Ü´) (set of cubic matrices) over the field ܲ, valuation on simpleartinian rings, semisimple rings and finally valuation on a module overa commutative ring and their some basic results. Keywords: valuation ; matrix valuation;cubic matrix; semisimple ring; valuation on module.

1 INTRODUCTION In what follows, and ( ) will represent, an associative ring with unit and the set of all square matrices respectively. Definition 1.1.(see [1]) A valuation on ring is a function âˆś → âˆŞ {∞}, where is a totally ordered abelian additive group such that for all , ∈ , 1) ( ) = ( ) + ( ); 2) ( + ) ≼ { ( ), ( )}; 3) ( ) = ∞if and only if = 0. One can easily show that, − = ( )

(1)=0 and

The first generalization for valuation rings of division rings was obtained by Schilling in6. Remark 1.3.If = , then valuation onskewfield is called a discrete valuation. Example 1.2. Let be a prime number. Map âˆś → âˆŞ {∞}, by Ď‘( ௼ / ) = , where r ≼ 0 and p divides neither m nor n, is a discrete valuation. Also if = ( ), where is any field and be the set of all rational functions ௼ / , where ≼ 0, is a fixed polynomial that is irreducible over and and are arbitrary polynomials in [ ] not divisible by . Then map

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M. H. Hosseini, et al., J. Comp. & Math. Sci. Vol.5 (4), 332-335 (2014)

âˆś → âˆŞ ∞ , ( / ) = ( ) − ( ) is a discrete valuation. 2. MATRIX VALUATION Let ( )is the Set ofall square matrices over ring . For any two matrices and over we define the diagonal sum of and as:

A 0  A ⊕ B =   0 B Given two matrices = ( ଵ, ଶ, ଷ, ‌ , ௥, )and B= ( ଵ, ଶ, ଷ, ‌ , ௥, ) in ( ),The determinant sum of and , with respect to the first column, is defined by ∇ = ( ଵ + ଵ , ଶ, ଷ, ‌ , ௥, ) Definition 2.1.(see [3]). A valuation (or a classical valuation)on the set of all square matrices ( ) over a skew field is a mapping | |: → âˆŞ ∞ , satisfying the following conditions: (1) if ∈ ௥ and ( ) < , then | | = ∞; (2) if , ∈ ௥ , 1 ≤ ≤ , = − , and = ≠, 1 ≤ ≤ , then | | = | |; (3) if , ∈ ௥ ,which exists the determinantal sum , then | | ≼ {| |, | |}; (4) for any ∈ ௥ ∈ ௠, | ⊕ | = | | + | |; (5) |!| = 0, where ! is an identity matrix.

"#$%$&'('$) *. *. Let : → âˆŞ ∞ be a valuation on commutative ring R. mapping + âˆś → âˆŞ ∞ , by + ( ) = ( , ) is a matrix valuation. "#$%$&'('$) *. -. Let be a division ring with an abelian valuation ν. Then ν may be extended to a matrix valuation + on = ௥ ( ), for each ≼ 1 by the equation: + . = / 0 ,. , . 1 , whereâ€?Detâ€? denotes the Dieudonne determinant, together with the rule +(.) = ∞ when . is singular. Proof. (See [5]). Corollary 2.4. The correspondence / ↔ + in Proposition 2.2 andProposition 2.3 is a bijection between abelian valuations on and matrix valuations on = ௥ ( ). 3. VALUATIONS ON THE SET OF CUBIC MATRICES Let F be a field and áˆşŕŹˇáˆť (2) be the set of all cubic matrices over the field F. All necessary definitions and notations on cubic matrices can be found in (see [7, Chaps. I, II]). Definition 3.1.A valuation on the set of all cubic matrices áˆşŕŹˇáˆť (2) over a field 2 is a mapping 3: áˆşŕŹˇáˆť (2) → âˆŞ ∞ , satisfying the following conditions: (1)if ∈ áˆşŕŹˇáˆť (2) and ( ) < , 3( ) = ∞;

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

then

M. H. Hosseini, et al., J. Comp. & Math. Sci. Vol.5 (4), 258-161 (2014)

(2)if , ∈ áˆşŕŹˇáˆť (2), 1 ≤ ≤ , = − , and = ≠, 1 ≤ ≤ , then3( ) = 3( ); (3)if , ∈ áˆşŕŹˇáˆť (2),which exists the determinantal sum , then 3( ) ≼ {3( ), 3( )}; (4) for any ∈ ௥ 2 ∈ ௠2 , 3( ⊕ ) = 3( ) + 3( ); (5) |!| = 0, where ! is an identity matrix. Remark 3.2.Proposition 2.2, proposition 2.3 and corollary 2.4, similarly is established for the cubic matrices ring over a field. 4. VALUATIONS ON SEMISIMPLE RINGS In this section, we show that a valuation on simple artinian ring5 may be extended to a Valuation on semisimple ring by Wedderburn-artin theorem. Theorem 4.1.(Wedderburn-artin).Let R be any left semisimple ring. Then ≅ ௥భ (0ଵ ) Ă— ௥ఎ (0ଶ ) Ă— ‌ Ă— ௥ŕł? (0௼ ) for suitable division rings 0ଵ , 0ଶ , ‌ , 0௼ and positive integers ଵ , ଶ , ‌ , ௼ (up to a permutation).there are exactly mutually non isomorphic leftsimple module over R4. Now let 0ଵ , 0ଶ , ‌ , 0௼ are division rings with valuation4 ŕŻœ on0ŕŻœ (for i =1, 2,¡ ¡ ¡, r). Then there exists simple valuations3ŕŻœ on ŕŻœ = ௥ŕł” 0ŕŻœ (by proposition 2) such that for any.ŕŻœ 1 ŕŻœ , 3ŕŻœ .ŕŻœ = ŕŻœ 0 ,.ŕŻœ . Therefore we have : Theorem 4.2. Map 3 defined by ߤáˆşÜşŕŹľ , ܺଶ , ‌ , ܺ௼ áˆť = ൫ߤଵ áˆşÜşŕŹľ áˆť, ߤଶ áˆşÜşŕŹś áˆť, ‌ , ߤ௼ áˆşÜşŕŻĽ áˆťŕľŻis a valuation on semisimple ring

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≅ ௥భ (0ଵ ) Ă— ௥ఎ (0ଶ ) Ă— ‌ Ă— ௥ŕł? (0௼ ) 5. VALUATION ON MODULES In this section, the first we defined a valuations on a module, and show that if be a -module, then for any valuation on exist a Manis valuation on the ring . Moreover in case = , one can show that any classic valuation on , under a certain condition, is a valuation on -mod . Definition5.1.2 Let be an -module where is a ring and be an orderedset with maximum element ∞. A mapping 3of ontocalled a valuation on , if the following conditions are satisfied:

For any , ∈ , 3 + ≼ {3( ), 3( )};

If 3( ) ≤ 3( ), , ∈ , then 3( ) ≤ 3( ) for all ∈ ;

Put3 ିଵ ∞ = ∈ |3 = ∞ . If 3 5 ≤ 3 5 , where , ∈ , and 5 ∈ \3 ିଵ (∞), then 3( ) ≤ 3( ) for all ∈ ; ( ) For every ∈ \(3ିଵ (∞) âˆś ), there is an ᇹ ∈ such that 3(( ᇹ ) ) = 3( ). In this case is called the value set of 3, and3ିଵ (∞) is called the core of 3. Definition 5.23ିଵ (∞) is called the core of 3. 6789#: ;. -.For an ordered abelian group and an ordered set with maximum element ∞, a mapping 3 of onto is defined to be a valuation on , if the following conditions are satisfied:

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

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M. H. Hosseini, et al., J. Comp. & Math. Sci. Vol.5 (4), 332-335 (2014)

(1) For ∈ , 3( ) = 1 if and only if = 0; (2) For any , ∈ , 3( + ) ≥ {3( ), 3( )}; (3) For every nonzero integers , 3( ) = 3( ). Viewing as a −module, it is easy to see that such a mapping 3 is a valuationonM in the sense of valuations on a module. In this case, the coreof3 is {0}, andthe induced valuation pair is ( , {0}). Example 5.4.. Let V be a nonzero vector space over a field F (i.e.,a Fmodule) with base B and let w be an arbitrary valuation on F with value group Γ. For every nonzero α∈ V,α may be uniquely expressed as < = ଵ ଵ + ଶ ଶ + ⋯ + ௡ ௡ =ℎ ଵ , ଶ , … , ௡ ∈ ℬ ଵ , ଶ , … , ௡ ∈ 2\{0}. Then mapping ∶ + → = ℬ × ∪ {∞}

By < = > ଵ , ? ଵ @ and ∞ 4 AB , + 4 2 − BA = ,ℎ C 0 .

0 =

REFERENCES 1. P. M. Cohn, The construction of valuations on skew fields, J. of the India Math. Soc., V. 54, p. 1- 45 (1989). 2. Z. Guangxing, Valuations on a Module, Communications in Algebra, 35:8, 2341 – 2356. 3. M. H. Hosseini, H¨older rigidity for matrices, Journal of Mathematical Sciences, Vol. 140, No. 2, (2007). 4. T. Y. Lam, A first course in noncommutative Rings, Springer-Verlag, (1991). 5. M. Mahdavi-Hezavehi, Valuation on simple artinian rings, Scientia Iraniaca, 1(4): 347-351 (1995). 6. O. F. G. Schilling, Noncommutative valuation, Bull. Amer. Math. Soc. 51, 297- 304 (1945). 7. N. P. Sokolov, Space Matrices and Their Applications [in Russian], Fizmatlit, Moscow (1960).

Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)

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