International Journal of Pure and Applied Sciences and Technology

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1 Int. J. Pure Appl. Sci. Technol., 2(1) (2011), pp International Journal of Pure and Applied Sciences and Technology ISSN Available online at Research Paper A Note on α Derivations in Semirings M. Chandramouleeswaran 1,* and V. Thiruveni 2 1, 2 Saiva Bhanu Kshatriya College, Aruppukottai , India. * Corresponding author, moulee59@gmail.com (Received: ; Accepted: ) Abstract: Motivated by some results on derivations in rings, in [2] we have defined derivations and generalized derivations on semirings and investigated some fundamental results on the derivations in semirings. In this paper, we introduce the notion of α derivations on semirings and investigate some interesting results including the analogous theorem of Posner. Keywords: Semirings, derivations on rings, derivations. derivations on semirings, α 1. Introduction: The notion of semirings was first introduced in 1934 by H.S.Vandiver. A semiring is an algebraic structure, consisting of a nonempty set R on which we have defined two associative binary operations, addition (usually denoted by +) and multiplication (usually denoted by or by concatenation) such that the multipllication is distributive over addition. The notion of rings with derivations is quite old and plays a significant role in the integration of analysis, algebraic geometry and algebra. The study of derivations in rings though initiated long back, but got interested only after Posner who in 1957 established two very striking results on derivations in prime rings. [3] and [4] deals with the notion of derivations on commutative fields. The notion of derivation has also been generalized in various directions, such as Jordan derivation, generalized derivation [5], generalized Jordan derivation etc. Also there has been considerable interest in investigating commutativity of rings, more often that of prime and semiprime rings admitting those mappings which are centralizing or commuting on some appropriate subsets of the ring.

2 72 The notion of derivations in semirings is defined in [1] and nothing was said about it. In [2] we have defined derivations on semirings and investigated some fundamental results on the derivations in semirings. Also as in ring theory, we have defined the notion of generalized derivations on semirings and investigated some interesting simple results. Using generalized derivations some commutativity theorems were obtained. In this paper, we introduce the notion of α derivations on semirings and investigate some interesting results including the analogous theorem of Posner. 2. Preliminaries: In this section, basic definitions and results that are required for our work are recalled. Definition 2.1. A semiring (S, +, ) is a non-empty set S together with two associative binary operations, + and, such that the two distributive laws are satisfied. That is, a semiring (S, +, ) is a non-empty set S together with two binary operations, + and, such that (1) (S, +) is a semigroup. (2) (S, ) is a semigroup. (3) For all a, b, c S, a (b + c) = a b + a c and (b + c) a = b a + c a. Definition 2.2. A semiring (S, +, ) is said to be additively commutative if (S, +) is a commutative semigroup. A semiring (S, +, ) is said to be multiplicatively commutative if (S, ) is a commutative semigroup. It is said to be commutative if both (S, +) and (S, ) are commutative. Definition 2.3. The semiring (S, +, ) is said to be a semiring with zero, if it has an element 0 in S such that x + 0 = x = 0 + x x S. A semiring (S, +, ) is said to have an identity element 1, if 1 = 0 S such that 1 x = x = x 1 x S. Definition 2.4. Let (S, +, ) be a semiring. An element α of S is called additively left cancellative if for all γ, β S, α + β = α + γ β = γ. It is said to be additively right cancellative if for all γ, β S, β + α = γ + α β = γ. It is said to be additively cancellative if it is both left and right cancellative. If every element of the semiring S is additively left cancellative, it is called additively left cancellative. If every element of the semiring S is additively right cancellative, it is called additively right cancellative. Similarly we can define multiplicatively left cancellative element, right cancellative element, cancellative element and multiplicatively left(right and two-sided) cancellative semiring. Notation. Through out our work in this paper, we consider semirings with

3 73 absorbing zero and with identity 1 = 0. Definition 2.5. Let (S, +, ) be a semiring. Any nonempty subset A of S is said to be a subsemiring if it contains 0 and 1 and is closed under the operations of + and 0 0. Let (S, +, ) be a semiring. Any nonempty subset I of S is said to be a left ideal of S if the following conditions are satisfied. (1) 1 / I. (2) a, b I a + b I. (3) a I and s S sa I. Similarly, we can define right and two sided ideals in a semiring. Definition 2.6. Let R be an associative ring. An additive mapping d : R R is called a derivation if d(xy) = d(x)y + xd(y) x, y R. In particular, for a fixed a R, the mapping Ia : R R given by Ia (x) = [x, a] = xa ax is a derivation, called an inner derivation. An additive function F : R R is called a generalized derivation with associated derivation d if F (xy) = F (x)y + xd(y) x, y R. Definition 2.7. Let (S, +, ) be a semiring. A derivation on S is a map D : S S satisfying the following conditions (1) D(x + y) = D(x) + D(y), x, y S. (2) D(xy) = D(x)y + xd(y), x, y S. From condition (1) with b = 0, we see that D(0) = 0. From condition (2) with b = 1, we see that D(1) = 0. Any element whose derivative is zero is called a constant element. Example 2.8. Let Then under usual addition and multiplication of matrices, S is a semiring. Define a given by Then D is a derivation on S. Example 2.9. Let S be any semiring. Consider the set S[x] of polynomials in the ideterminate x over S. Under usual addition and multiplication of polynomials S[x] is a semiring. For any f (x) = a0 + a1x + a2x an x n S[x] define the map D : S S by D(f (x)) = a1 + 2a2 x + 3a3x nan x n 1. Then D is a derivation on S[x].

4 74 3. α -Derivations in Semirings: Definition 3.1. Let (S, +, ) be a semiring. Let α be a automorphism of S. An additive mapping D : S S is said to be an α -derivation if D(xy) = α(x)d(y) + D(x)y, x, y S. Example 3.2. Let Then under usual addition and multiplication of matrices, S is a semiring. Define a map given by Then D is a derivation on S. Define α : S S given by Then D is a α -derivatiion on S. Lemma 3.3. Let D be an additive endomorphism of a semiring S. Then D is an α - derivation iff D(xy) = D(x)y + α(x)d(y), x, y S. Proof. Let D be an α -derivation on S. Then x, y S D(xy) = α(x)d(y) + D(x)y Now, D(x(y + y)) = α(x)d(y + y) + D(x)(y + y) = α(x)(2d(y)) + D(x)2y = 2(α(x)D(y) + D(x)y) (1) Also D(xy + xy) = 2D(xy) = 2(α(x)D(y) + D(x)y) (2) From (1) and (2 ) we get α(x)d(y) + D(x)y = D(x)y + α(x)d(y). The converse follows easily. Remark 3.4. Let D be an α -derivation on a semiring S. Then x, y, x S (i) (α(x)d(y) + D(x)y)z = α(x)d(y)z + D(x)yz. (ii) (D(x)y + α(x)d(y))z = D(x)yz + α(x)d(y)z. Lemma 3.5. Let D be an α -derivation on a prime semiring S. Let a S is such that ad(x) = 0 (or D(x)a = 0), x S. Then a = 0 or D = 0. Proof. From hypothesis, x, y S, ad(xy) = 0 a(α(x)d(y) + D(x)y) = 0 aα(x)d(y) + ad(x)y = 0 aα(x)d(y) + 0 = 0. As ad(x) = 0, aα(x)d(y) = 0, x, y S asd(y) = 0, y S. Since S is prime, a = 0 or D = 0

5 75 Thus, if D(x)a = 0, we have D(yx)a = 0. Proceeding as above, we get the result. Lemma 3.6. Let S be a 2 torsion f ree prime semiring. Let D be an α -derivation on S such that Dα = αd. Then D 2 = 0 D = 0. Proof. Let D 2 = 0. Then for x, y S, D 2 (xy) = 0. D(D(xy)) = 0 Expanding weget α 2 (x)d 2 (y) + D((α(x)))D(y) + D((α(x)))D(y) + D 2 (x)y = 0 2D(α(x))D(y) = 0 (Since D 2 = 0 ) D(α(x))D(y) = 0 (Since S is 2 torsion f ree ) D(x)D(y) = 0 ( Since α is automorphism) D = 0. (Using 3.5) Lemma 3.7. S Let D be an α -derivation on a semiring S. Let β be an automorphism of which commutes with D. Then αβ(x)d(β(y) = βα(x)βd(y), x, y S. Proof. Let x, y S. Then βd(xy) = β(α(x)d(y) + D(x)y) = βα(x)β(d(y)) + βd(x)β(y) Also Dβ(xy) = D(β(x)β(y)) = α(β(x))d(β(y)) + D(β(x))β(y) Since β commutes with D, from the above equations we get βα(x)βd(y) = αβ(x)βd(y). From the above lemmas, one can establish further. the following result which we are exploring Theorem 3.8. Let D 1 be a α -derivation and D 2 be a β - derivation on a 2-torsion free prime semiring S such that α, β commute with D 1 and with D 2. Then D 1 D 2 is an αβ - derivation iff D 1 = 0 or D 2 = 0. Proof. Let D 1 D 2 be an αβ -derivation. For x, y S (D 1 D 2 )(xy) = (αβ(x)d 1 D 2 (y) + D 1 D 2 (x)y (1) Now, (D 1 D 2 )(xy) = D 1 (D 2 (xy)) = D 1 (β(x)d 2 (y) + D 2 (x)y) = D 1 (β(x)d 2 (y) + D 1 (D 2 (x)y)

6 76 = α(β(x))d 1 (D 2 (y)) + D 1 (β(x))d 2 (y) + α(d 2 (x))d 1 )(y) + D 1 (D 2 (x))y (2) From (1) and (2),we get D 1 β(x)d 2 (y) + αd 2 (x)d 1 (y) = 0 (3) Replacing x by xd 2 (z) in (3), we get (D 1 β)[xd 2 (z)d 2 (y) + (αd 2 )(xd 2 (z)]d 1 (y) = 0 (βd 1 )[xd 2 (z)d 2 (y) + (αd 2 )(xd 2 (z)]d 1 (y) = 0 By Lemma 3.3 β[d 1 (x)d 2 (z) + α(x)d 1 D 2 (z)]d 2 (y) + α[β(x)d 2 (z) 2 + D 2 (x)d 2 (z)]d 1 (y) = 0 [βd 1 (x)βd 2 (z) + βα(x)βd 1 D 2 (z)]d 2 (y) + [αβ(x)d 2 (z) + αd 2 (x)αd 2 (z)]d 1 (y) = 0 By Lemma 3.7 [D 1 β(x) + D 2 β(x) + αβd 1 βd 2 (z)]d 2 (y) + [αβ(x)d 2 α(z) 2 + D 2 α(x)d 2 α(z)]d 1 (y) = 0 By Lemma 3.5 D 1 β(x)d 2 β(z)d 2 (y) + αβ(x)[d 1 βd 2 (z)d 2 (y) + D 2 α(z)d 2 1 (y)] + D 2 α(x)d 2 α(z)d 1 (y) = 0 (4) Replacing x by D 2 (z) in (3), we get (D 1 β)(d 2 (z))d 2 (y) + (αd 2 )(D 2 (z))d 1 (y) = 0 D 1 (βd 2 (z))d 2 (y) + D 2 (α(z))d 1 (y) = 0 (5) 2 From equations (4) and (5), we get D 1 β(x)d 2 β(z)d 2 (y) + D 2 α(x)d 2 α(z)d 1 (y) = 0 (6) Replacing x by z in (3), we get (D 1 β)(z)d 2 (y) + αd 2 (z)d 1 (y) = 0 (7) Replacing y by β(z) in (3), we get (D 1 β)(x)d 2 (β(z)) + (αd 2 )(x)d 1 (β(z)) = 0 (8) Premultiply equation (7) by D 2 α(x), we get D 2 α(x)d 1 β(z)d 2 (y) + D 2 α(x)d 2 α(z)d 1 (y) = 0 (9) 2 Post multiply equation (8) by D 2 (y), we get D 1 β(x)d 2 β(z)d 2 (y) + αd 2 (x)d 1 β(z)d 2 (y) = 0 (10) Adding equations (6), (9) and (10), and simplifying 2[D 1 β(x)d 2 β(z)d 2 (y) + D 2 α(x)d 2 α(z)d 1 (y) + D 2 α(x)d 1 β(z)d 2 (y)] = 0 Since S is 2-torsion free, we have [D 1 β(x)d 2 β(z)d 2 (y) + D 2 α(x)d 2 α(z)d 1 (y) + D 2 α(x)d 1 β(z)d 2 (y)] = 0 D 2 α(x)d 2 α(z)d 1 (y) = 0 If D 2 = 0, by Lemma 3.5, we get D 1 = 0, thus completing the proof.

7 77 4. Conclusions: In this paper we discussed the notion of α derivations on semirings and has proved the analogues of Posner s theorem. The notion of derivations on semirings can be used to study the commutativity conditions of the semirings. References [1] E. Howard, Bell and Nadeem-Ur Rehman, Generalized Derivations with Commutativity and Anti- commutativity Conditions, Math. J. Okayama University, 49(2007), [2] I. Kaplansky, An Introduction to Differential Algebra, Publications De L Institute De Mathematique, De L University, De Nancago, [4] Joseph Fels Ritt, Differential Algebra, AMS Colloquium Publications, Vol. 33, [3] Jonathan S. Golan, Semirings and their Applications, Kluwer Academic Press(1969). [4] M. Chandramouleeswaran and V.Thiruveni, On Derivations of Semirings, Advances in Algebra, Vol. 3(1) (2010),