Generalized Derivation in Γ-Regular Ring

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1 International Mathematical Forum, Vol. 6, 2011, no. 10, Generalized Derivation in Γ-Regular Ring D. Krishnaswamy Associate Professor of Mathematics Department of Mathematics Annamalai University, India N. Kumaresan Assistant Professor of Mathematics Faculty of Marine Science CAS in Marine Biology Parangipettai, India Abstract In this papaer, we have defined Symmetric Martindale Γ Regular ring and generalized derivation in Γ Regular ring. Further, we have proved that the product of two generalized derivation in Γ Regular ring is again ageneralized derivation in Γ Regular ring. Mathematics Subject Classification: 15A33, 15A60 Keywords: Gamma ring, Generalized derivation, Regular ring, Γ Regular ring, Symmetric Martindale Γ Regular ring 1 Introduction The notion of a generalized derivation in rings was introduced by Bojan Hvala in 1998[2]. In 1936, the concept of a regular ring was introduced by Von-Neumann [6] and the characterization of a generalized derivation in regular ring was studied in [4]. Γ ring was introduced by Nobusawa [5] and further Barnes [1] studied various properties in Γ rings. In 2009, Krishnaswamy introduced Γ Regular ring and characterization of Γ Regular ring was studied in [3]. In this paper, we study the generalized derivation in Γ Regular ring.

2 494 D. Krishnaswamy and N. Kumaresan In view of the generalized derivation in Regular ring, we now define the generalized derivation in Γ Regular ring as follows: A function f : R Γ R R will be called a generalized derivation in Γ Regular ring, if thereexists a derivation D of R such that f(axa) =ea + ad for a R, x Γ and e lis a fixed element in R. 2 Preliminaries Definition 2.1 Let R and Γ be additive abelian groups. Then R is called a Γ ring if for any x, y, z R and α, β Γ, the following conditions are satisfied (i) xαy R (ii)(x + y)αz = xαz + yαz; x(α + β)z = xαz + xβz (iii)(xαy)βz = xα(yβz) Definition 2.2 (2) Let R be Γ ring and F : R R be an additive map. Tnen, F is called a generalized derivation if there exists a derivation D : R R such that F (xαy) =F (x)αy + xαd(y) for all x, y R and α Γ. Definition 2.3 (6) An element a of a ring R is said to be regular if and only if for x in R such that axa = a. The ring R is regular if and only if each element of R is regular. Definition 2.4 (4) Let R be an additive abelian group. A function f : R R will be called a generalized derivation in Regular ring, if there exist a derivation D of R such that f(axa) =ca + ad for all x, a R and c is a fixed element in R. Definition 2.5 (3) Let R and Γ be two additive abelian groups. An element a R is said to be Γ regular if and only if there exist an element x R such that axa = a. The ring R is said to be Γ regular ring if and only if each element of R is Γ regular ring. Definition 2.6 The Left Martindale Γ regular ring of quotients Q L R is characterized as the unique regular ring extension Q of R satisfying (i)if 0 q Q and 0 I,A R with Aq R, then Iq o. (ii)if 0 A R and f : A R, then there exist axq Q with axf = axq for all a A and x Γ. The Right Martindale Γ regular ring of quotients Q R R can be defined in the same manner. Definition 2.7 The Symmetric Martindale Γ regular ring of quotients Q S R is characterized as the unique regular ring extension Q of R satisfying

3 Generalized derivation in Γ-regular ring 495 (i)if 0 q Q and 0 I, A, B R with Aq, qb R, then Iq,qI o. and (ii)if 0 A, B R and f : A R, g : B R and (axf)b = ax(gb) for all a A, b B and x Γ, then there exist axq Q with axf = axq and gxb = gxq for all a A, b B and x Γ. Definition 2.8 The central closure R C is a Γ regular ring which contains the linear identities of R in the sense that if 0= a, b, c, d R with axb = cxd for all x Γ then there exist an element0 q C with a = cxq and b = qxd. Definition 2.9 The extend centroid C is a field and it is in the centre of both Q R (R) and Q S (R). The extend cetroid of R C is equal to C, where R C is equal to its central closure. Since R C is a Γ regular ring as well one can construct the Γ regular rings Q R (R) and Q S (R). Lemma 2.10 If c i,d i A satisfying c i xd i =0for all x Γ, then there exists c i s as well as d i s are C dependent unless all c i =0or d i =0. Lemma 2.11 Let c, d A, x Γ and let f :Γ A be defined by f(x) =cxd. Iff is a generalized derivation, then either c C or d C. Proof: Let a R and x Γ by the definition of 2.2, we have caxad = caxda + axδ(a), where δ is a derivation. We obtain cax(ad da) ax(δ(a)) = 0 for all a R and x Γ. Using Lemma 2.10, it follows that either c C or d C. Preposition : 2.12 Suppose that n i=1 f i (ax)αd i + k j=1 c j αh j (ax) = 0 for all a, α R, x Γ where f i,h j R and f i : R A and h j : R R C are any maps. If the sets n i=1 d i + k j=1 c j are C independent, then there exist q ij Q R (R C ) for i =1, 2,...n and j =1, 2,..k. Such that f i (ax) = k j=1 c j αaq ij and h j (ax) = n i=1 aq ij αd i for all a, α R, x Γ and i = 1, 2,...n and j =1, 2,..k. Lemma : 2.13 Let f : R R C be an additive map satisfying f(axa) =f(ax)a for all a R and x Γ. Then there exists q Q R (R) such that f(ax) =qax for all a R and x Γ. Proof: Let f : R R C be an extension of f according to f( a i x i λ i )= f(ai x i )λ i where a i R, x i Γ and λ i C. Let I be a non-zero ideal in R such that λ i I R for every i. Take a I and the factors in the sum a i x i (λ i α) lie in R. Therefore, we have 0 = f(ai x i )(λ i a)= f(ai x i λ i )(a) since this is true for all a I. We have f(ai x i )λ i = 0. Thus f is well defined. This fact that f(axa) = f(ax)a for all f(axa) = f(ax)a + axδ(a), can be verified by a direct computation. This proves that

4 496 D. Krishnaswamy and N. Kumaresan f : R C R C is a right R C module map, hence there exist q Q R (R C ) such that f(ax) =qax, for all x Γ. Since f is an extension of f. This proves the lemma. 3 Product of two Generalized derivation in Γ Regular ring The aim of this section is to prove that the product of two generalized derivation in Γ Regular ring is again a generalized derivation in Γ Regular ring. Theorem 3.1 Let R be a Γ Regular ring and let f 1,f 2 : R R be the generalized derivation. Then, the product f 3 = f 1 f 2 is again a generalized derivation if and only if the following conditions hold. (i)there exists α C such that either f 1 (axa) =αa and f 2 (axa) =αa. (ii)there exists p, q Q R (R C ) such that f 1 (axa) =pa and f 2 (axa) =qa. (iii)there exists p, q Q R (R C ) such that f 1 (axa) =ap and f 2 (axa) =aq. (iv)there exists p, q Q R (R C ) and β,γ C such that f 1 (axa) =pa + aq and f 2 (axa) =βa + γ(pa aq). Proof : From the definition it is clear that the product f 3 = f 1 f 2 is again a generalized derivation. To prove the converse, let f 3 = f 1 f 2 is a generalized derivation. We have f i (axa) =f i (ax)a + axd i (a), for i =1, 2, 3 and a R and x Γ for some derivations d i. Now f 3 (axa) =f 1 [f 2 (axa)], we obtain f 2 (ax)d 1 (a)+f 1 (ax)d 2 (a)+ax(d 1 d 2 d 3 )(a) = 0 for all a R and x Γ. Replacing ax by axα, we obtain f 2 (ax)αd 1 (a)+f 1 (ax)αd 2 (a)+ ax[d 1 (α)d 2 (a)+d 1 (a)d 2 (α)+α(d 1 d 2 d 3 )(a)] = 0 (1) for all a, α R and x Γ. Fix a and from proposition 2.12, we have the following two possibilities. (I) d 1 (a) and d 2 (a) are C dependent for all a R. (II) there exist p 1,p 2 Q R (R C ) such that f 1 (ax) = ap 1 and f 2 (ax) = ap 2. From case (II), write p = p 1 and q = p 2, it follows that the condition (iii) holds. From case (I), since d 1 (a) and d 2 (a) are C dependent, we have [d 1 (a), d 2 (a)] = 0 for all a R. It follows that

5 Generalized derivation in Γ-regular ring 497 α 1 d 1 (a)+α 2 d 2 (a) = 0 for some α 1,α 2 C (2) When d 1 = 0, implies that f 1 (axa) =f 1 (ax)a for all a R and x Γ. It follows from lemma 2.13, there exist p Q R (R C ) such that f 1 (axa) =pax. Equation (1) can be written in the form paxαd 2 (a) axαd 3 (a) = 0 for all a R, α C and x Γ. (3) In a similar manner, when d 2 = 0, yields f 2 (ax) =qax for some q Q R (R C ) for all a R and x Γ it follows that the condition (ii) holds. If d 2 0 and d 1 = 0, we choose a R and x Γ from (3) and axd 2 (a) 0 and hence equation (2) together with lemma 2.10 yields f 1 (axa) =αa, for some α C. Similarly, if d 2 = 0 and d 1 0, yields f 2 (axa) =αa for all a R, x Γ and hence (i) holds. Finally, let us consider d 1 0 and d 2 0 so that d 2 = αd 1, for some non-zero α C. Define F (ax) =f 2 (ax) αf 1 (ax) and note that F : R R C and F (axa) =F (ax)a for all a R and x Γ. Thus, Lemma 2.13 yields that there exists r Q R (R C ) such that F (axa) =rax and therefore f 2 (ax) =rax + αf 1 (ax) for all a R and x Γ. (4) Using equation (1), we obtain the result for the function [2αf 1 (ax)+pax]αd 1 (a)+ax[d 1 (a)g(a)+αh(a)] = 0 where g(a) =2αd 1 (α) and h(a) =(d 1 d 2 d 3 )(a). Pick a R such that d 1 (a) 0 and apply proposition 2.12 there exists ṕ Q R (R C ) such that 2αf 1 (ax) +pax = axṕ. This gives us f 1 (ax) = pax + axq where p = (2α) 1 p and q = (2α) 1 ṕ for all p, q Q R (R C ). Using equation (4), we obtain a similar result for the other function f 2 (ax) =(r + αp)ax + axqα. Now we compute the product f 1 f 2 and obtain f 3 (ax) =f 1 [f 2 (ax)] = (pr + αp 2 )a + aq 2 α +(r +2αp)aq. Since both the maps ax (pr + αp 2 )ax + axq 2 α and f 3 are generalized derivations. So, is their difference i.e., the map ax (r +2pα)axq. By using lemma 2.11, we have (r +2pα) C or q C. It is easy to notice that the condition (ii) leads to the conditin (iii) of the Theorem. So, we choose (r +2pα) =β C. Write γ = α and r = β +2pγ and we obtain f 2 (ax) =βax+ γ(pax axq). Hence, the condition (iv) holds. Corollary 3.2 Let f 1,f 2 : R R be the generalized derivations. Then, the product f 1.f 2 =0if and only if one of the possibilities holds. (i) either f 1 =0or f 2 =0. (ii)there exists p, q Q R (R C ) such that f 1 (axa) =pa, f 2 (axa) =qa and qp =0. (iii)there exists p, q Q R (R C ) such that f 1 (axa) =ap, f 2 (axa) =aq and pq =0. (iv)there exists p, q Q R (R C ) and β,γ C such that f 1 (axa) =pa + aq, f 2 (axa) =βa + γ(pa aq) and βp + γp 2 = γq 2 βq C.

6 498 D. Krishnaswamy and N. Kumaresan Proof : In all cases it is clear that f 1.f 2 =0. To prove the converse, if for f 1 and f 2 are non-zero generalized derivation, one of the conditions of Theorem 3.1 must hold. The condition (i) of the theorem 3.1 hold,f 1 (axa) =αa and f 2 (axa) =αa for all a R, x Γ and α C and therefore either f 1 =0or f 2 =0. If the condition (ii) of the Theorem 3.1 hold, there exist p, q Q R (R C ) such that f 1 (axa) =pa and f 2 (axa) =qa. We have f 3 = f 1 f 2 (axa) =qpa =0for all a R and x Γ and therefore qp =0. If the condition (iii) of the Theorem 3.1 hold, there exist p, q Q R (R C ) such that f 1 (axa) =ap and f 2 (axa) =aq. We have f 3 = f 1 f 2 (axa) =apq =0for all a R and x Γ and therefore pq =0. Finally from (iv) of the Theorem 3.1, we have f 1 [f 2 (axa)] = (βp + γp 2 )a + a(βq γq 2 )=0for all a R. where βp + γp 2 = γq 2 βq C follow from Lemma Corollary 3.3 Let f : R R be a generalized derivation and c, d C, if we have cf(axa)+f(axa)d =0for all a R, x Γ, then one of the following conditions hold. (i) c, d C and c + d =0. (ii)c C and there exist p Q R (R C ) such that f(axa) =pa and (c + d)p =0. (ii)d C and there exist p Q R (R C ) such that f(axa) =ap and p(c + d) =0. (iv)there exists p, q Q R (R C ) and β,γ C such that f(axa) =βa + γ(pa aq) and βp + γp 2 = γq 2 βq C. Proof: Define g(axa) =ca + ad for all a R and x Γ. So, we have gf =0according to corollary 3.2, this is possible in one of the four cases. The possibility g =0is equivalent to c, d C and c + d =0. The cases (ii) and (iii) of corollary 3.2 can be easily transformed in case (ii) and (iii) of corollary 3.3. Similarly, the case (iv) of corollary 3.2 gives us f(axa) =βp + γ(pa aq). Once we know this, write cf(axa)+f(axa)d =0 and use lemma 2.11 to derive βp + γp 2 = γq 2 βq C. Corollary 3.4 Let f : R R be a generalized derivation. Then f 2 is a generalized derivation if and only if there exist a Q R (R C ) such that either f(axa) =ax, x Γ or f(axa) =xa, x Γ. Proof : Suppose that f 2 is a generalized derivation by applying theorem 3.1 for f(axa) =f 1 (axa) =f 2 (axa). The conditions (i) - (iii) of Theorem 3.1 can be unified in saying that f is either left or right multiplication by an element of Q R (R C ).

7 Generalized derivation in Γ-regular ring 499 References [1] Barnes W.E, On the Γ rings of Nobusawa, Pacific Journal of Math.,18 (1966), [2] Bojan Hvala, Generalized derivation in rings, Communications in Algebra, 26(4) (1988), [3] Krishnaswamy D and Kumaresan N, On the fundamental theormes of Γ regular ring, Research Journal of Math. and Stat., 1(1) (2009), 1-3. [4] Krishnaswamy D and Kumaresan N, On the generalized derivation in regular ring, International Journal of Algebra 4(19) (2010), [5] Nobusawa. N, On a generalization of ring theory, Osaka Journal of Math., 1 (1964), [6] Von Neumann. J, On regular rings, Proc. National Academic Science U.S.A., 22 (1936), Received: September, 2010

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