Journal of Mathematical Extension Vol. 10, No. 3, (2016), 119-135 ISSN: 1735-8299 Journal of Mathematical Extension of Mathematical Extension URL: Journal http://www.ijmex.com Vol.Vol. 10, 10, No.No. 3, (2016), 119-135 3, (2016), 119-135 ISSN: 1735-8299 ISSN: 1735-8299 URL: http://www.ijmex.com URL: http://www.ijmex.com A General Characterization of Additive Maps on Semiprime Rings AA General Characterization of of Additive General Characterization Additive A. Hosseini Maps onon Semiprime Rings Maps Semiprime Rings Kashmar Higher Education Institute A. A. Hosseini Hosseini Kashmar Higher Education Institute RKashmar be a 2-torsion freeEducation semiprime ring and T : R → R be Abstract. Let Higher Institute 1. a Jordan left centralizer associated with a l฀semi Hochschild 2฀cocycle ฀ : R฀R → R. Then, T is a left centralizer associated with ฀. Applying Let Let R be 2-torsion free free semiprime ringring andand T :R Abstract. Raprove be a 2-torsion semiprime T :→ RR →be R Abstract. this main result, we that every Jordan generalized derivation onbe a Jordan left left centralizer associated withwith a l฀semi Hochschild 2฀cocycle a Jordan centralizer associated a l฀semi Hochschild 2฀cocycle a 2-torsion free semiprime ring is a generalized derivation. ฀ : R฀R → R. T isTa is left centralizer associated withwith ฀. Applying ฀ : R฀R → Then, R. Then, a left centralizer associated ฀. Applying this main result, we prove thatthat every Jordan generalized on on this main result, we prove every Jordan generalized derivation AMS Subject Classi฀cation: 47B47; 17B40; 16N60 derivation a Keywords 2-torsion free semiprime ring is a isgeneralized derivation. a 2-torsion free semiprime ring a generalized derivation. and Phrases: Jordan derivation, jordan centralizer, l-semi hochschild 2-cocycle, 2-torsion free semiprime ring AMS Subject Classi฀cation: 47B47; 17B40; 16N60 AMS Subject Classi฀cation: 47B47; 17B40; 16N60 Keywords and Phrases: Jordan derivation, jordan centralizer, l-semi Keywords and Phrases: Jordan derivation, jordan centralizer, l-semi hochschild 2-cocycle, 2-torsion freefree semiprime ringring hochschild 2-cocycle, 2-torsion semiprime Introduction this paper, R denotes an associative ring with the center 1.Throughout 1. Introduction Introduction Z(R), T is considered as an additive map on R and ฀ is a biadditive map from Rthis × this Rpaper, into R. anan integer n  2, ring a ring Rwith isthe said tocenter be Throughout R Given denotes associative with center Throughout paper, R denotes an associative ring the n-torsion free, if for x ∈ R, nx = 0 implies that x = 0. We denote by Z(R), T is considered as an additive map on R and ฀ is a biadditive Z(R), T is considered as an additive map on R and ฀ is a biadditive map from Rcommutator ×RR×into [x,map y], the ฀Given yx,an foran allinteger x, yn∈nR. that aisring R.xy Given integer 2,Recall a2,ring R isRsaid to R be from R into R. a ring said toisbe prime if for x, y ∈ R, n-torsion free, if for xRy = {0} implies that either x = 0 or y = 0, and x ∈ R, nx = 0 implies that x = 0. We denote by n-torsion free, if for x ∈ R, nx = 0 implies that 0. We denote by is y], semiprime in the case that xthat = 0. [x, thethe commutator xythat ฀ for = all{0} x, yimplies Recall aAn ring R isR is [x, y], commutator xyyx, ฀xRx yx, for all x,∈yR. ∈ R. Recall that aadditive ring prime if for x, y ∈ R, map D : R → R is called a derivation if D(xy) = D(x)y + xD(y) xRy = {0} implies that either x = 0 or y = 0, and prime if for x, y ∈ R, xRy = {0} implies that either x = 0 or y =holds 0, and 2) = isfor semiprime case thatthat x, y ∈inRthe and is case called a Jordan in the D(x xRx = {0} implies that x case =x0.=that An additive isall semiprime in the xRx =derivation {0} implies that 0. An additive map D(x)x is called fulfilled for all x ∈ifR. every derivation is D :+ a derivation D(xy) = D(x)y + xD(y) holds map DRxD(x) :→ RR → isR is called a derivation ifObviously, D(xy) = D(x)y + xD(y) holds 2 2 a for Jordan the is not in true, in general. AD(x wellfor all x, is called a Jordan derivation the case thatthat D(x ) =) = allyx,∈derivation, yR∈and R and isbut called aconverse Jordan derivation in the case known ofisHerstein [8] for states in Obviously, the caseevery that D(x)x +result xD(x) fulfilled for all all x ∈xthat R. Obviously, derivation is is R derivation is a prime D(x)x + xD(x) is fulfilled ∈ R. every a Jordan derivation, but the converse is not true, in general. A wella Received: Jordan December derivation, but the converse is not true, in general. A well2015; Accepted: March 2016 2015 known result of Herstein [8] [8] states thatthat in the casecase thatthat R is known result of Herstein states in the R aisprime a prime 119 Received: December 2015; Accepted: March 2015 Received: December 2015; Accepted: March 2015 119119 120 A. HOSEINI ring of characteristic not 2, then every Jordan derivation D : R → R is a derivation. A brief proof of this result has been presented in [5]. Cusack [6] has extended Herstein’s result to 2-torsion free semiprime rings (see also [4] for an alternative proof). An additive map T : R → R is called a left (resp. right) centralizer if T (xy) = T (x)y (resp. T (xy) = xT (y)) holds for all x, y ∈ R. We call T a centralizer whenever T is both a left and a right centralizer. An additive map T : R → R is called a Jordan left (right) centralizer when T (x2 ) = T (x)x (resp. T (x2 ) = xT (x)) holds for all x ∈ R. Following some ideas from [4], Zalar [11] proved that any left (resp. right) Jordan centralizer on a 2-torsion free semiprime ring is a left (resp. right) centralizer. By using the main results of [4] and [11], Vukman [10] proved that every Jordan generalized derivation on a 2-torsion free semiprime ring is a generalized derivation. By using the main result of this paper, we offer an alternative proof for this result of Vukman. A bi-additive map ฀ : R × R → R is said to be a l-semi Hochschild 2-cocycle if ฀(xy, z) ฀ ฀(x, yz) + ฀(x, y)z = 0, for all x, y, z ∈ R. A l-semi Hochschild 2-cocycle ฀ is said to be symmetric (resp. anti symmetric) if ฀(x, y) = ฀(y, x) (resp. ฀(x, y) = ฀฀(y, x)) for all x, y ∈ R. We say that an additive map T : R → R is a left centralizer associated with ฀, if there exists a bi-additive map ฀ : R × R → R such that T (xy) = T (x)y + ฀(x, y) holds for all x, y ∈ R. Clearly, in this case we have ฀(xy, z) ฀ ฀(x, yz) + ฀(x, y)z = T (xyz) ฀ T (xy)z ฀ T (xyz)+ T (x)yz + T (xy)z ฀ T (x)yz = 0, for all x, y, z ∈ R. It means that ฀ is a l-semi Hochschild 2-cocycle. Let ฀ : R × R → R be a l-semi Hochschild 2-cocycle. An additive map T : R → R is said to be a Jordan left centralizer associated with ฀, whenever T (x2 ) = T (x)x + ฀(x, x) for all x ∈ R. A bi-additive map λ : R × R → R is called a r-semi Hochschild 2-cocycle if λ(z, xy) ฀ λ(zx, y) + zλ(x, y) = 0 for all x, y, z ∈ R. An additive map T : R → R A A GENERAL GENERAL CHARACTERIZATION CHARACTERIZATIONOF OFADDITIVE ADDITIVE...... 121 121 is is said said to to be be aa right right centralizer centralizer associated associated with withλ,λ,ififthere thereexists existsaabibiadditive additive map map λλ :: R R× ×R R→ →R R such such that thatTT(xy) (xy)==xT xT(y) (y)++λ(x, λ(x,y)y)for forall all x, x, yy ∈ ∈ R. R. Obviously, Obviously, λλ isis aa r-semi r-semi Hochschild Hochschild2-cocycle. 2-cocycle. Let Let TT(x (x22)) = = TT(x)x (x)x + +฀(x, ฀(x,x) x) for for all all xx ∈∈RR (1). (1).Replacing Replacingxxby byxx++yy in in (1), (1), we we get get TT(xy (xy + +yx) yx) = = TT(x)y (x)y++TT(y)x (y)x++฀(x, ฀(x,y) y)++฀(y, ฀(y,x)x)for forall all x, x, yy ∈ ∈R R (2). (2). Note Note that that ifif R R isis aa 2-torsion 2-torsionfree freering, ring,then then(1) (1)and and(2) (2) are are equivalent. equivalent. Some Some authors authors define defineaaJordan Jordanleft leftcentralizer centralizerasasfollows: follows: An An additive additive map map TT :: R R→ →R R isis called called aa Jordan Jordan left left centralizer centralizerifif TT(xy (xy + + yx) yx) = = TT(x)y (x)y + + TT(y)x (y)x holds holds for for all all x,x,yy ∈∈ R. R. With Withthis thishyhypothesis, pothesis, we we have have TT(xy) (xy)฀ ฀TT(x)y (x)y == ฀(T ฀(T(yx) (yx)฀฀TT(y)x) (y)x) (3). (3).IfIfwe wededefine fine ฀(x, ฀(x,y) y) = = TT(xy) (xy)฀ ฀TT(x)y (x)y (x, (x,yy ∈∈ R), R), then thenititfollows followsfrom from(3) (3)that that ฀(x, ฀(x,y) y) = = ฀฀(y, ฀฀(y,x) x) and and itit means means that that ฀฀ isis anti anti symmetric. symmetric. Suppose Suppose that that TT :: R R→ →R R isis aa left left centralizer centralizer associated associated with with฀,฀,i.e. i.e.TT(xy) (xy)== TT(x)y (x)y + + ฀(x, ฀(x,y) y) for for all all x, x,yy ∈∈ R. R. IfIf ฀฀ isis anti anti symmetric, symmetric,then thenwe wehave have TT(xy) (xy) ฀ ฀ TT(x)y (x)y = = ฀T ฀T(yx) (yx) + + TT(y)x (y)x and and consequently, consequently, TT(xy (xy++yx) yx) == TT(x)y (x)y + +TT(y)x (y)x for for all all x, x,yy ∈∈R. R.ItItmeans meansthat thatTT isisaaJordan Jordanleft leftcentralcentralizer. izer. Therefore, Therefore, IfIf ฀฀ :: R R× ×R R→ →R R defined definedby by฀(x, ฀(x,y) y)==TT(xy) (xy)฀฀TT(x)y (x)y is is anti anti symmetric, symmetric, then then TT isis aa Jordan Jordanleft leftcentralizer centralizerififand andonly onlyififTTisis aa left left centralizer centralizer associated associated with with the thel-semi l-semiHochschild Hochschild2-cocycle 2-cocycle฀.฀. Similar Similar to to the the approach approach presented presented in in [11], [11],we weprove provethe thefollowing followingmain main result: result: Let Let R R be be aa 2-torsion 2-torsion free free semiprime semiprime ring ring and andTT : :RR→ →RRbe beaaJordan Jordan left left centralizer centralizer associated associated with with ฀, ฀, where where ฀฀ isis aa l-semi l-semi Hochschild Hochschild2-2cocycle. cocycle. Then, Then, TT isis aa left left centralizer centralizerassociated associatedwith with฀. ฀. In In this this paper, paper, we we show show that that derivations, derivations, generalized generalized derivations, derivations, σ-σderivations, derivations, generalized generalizedσ-derivations, σ-derivations,(σ, (σ,ττ)-derivations )-derivationsand andθ-centralizers θ-centralizers are are left left centralizer centralizer associated associatedwith withaasuitable suitablel-semi l-semiHochschild Hochschild2-cocycle. 2-cocycle. This This means means that that the the aforementioned aforementioned concepts concepts can can be be unified unifiedand andinintegrated tegrated together. together. By By reviewing reviewing some some papers papers (( [1], [1], [2], [2], [3], [3],and andreferreferences ences therein) therein) about about Jordan Jordan left left(θ, (θ,φ)-derivations, φ)-derivations,(θ, (θ,φ)-derivations, φ)-derivations,τ τ- centralizers centralizers and and ฀-centralizers ฀-centralizers on on 2-torsion 2-torsion free free semiprime semiprimerings, rings,ititisis observed observed that that the the maps maps like like θ,θ, φ, φ, and and ττ are are supposed supposed asas homomorhomomorphism phism and and automorphism. automorphism. We We believe believe this this assumption assumptioncan canreduce reducethe the generality generality of of the the topic. topic. In In this this paper, paper,therefore, therefore,we weare aregoing goingtotopresent present 122 A. HOSEINI some results about Jordan σ-derivations, Jordan (σ, τ )-derivations, Jordan generalized derivations and Jordan generalized σ-derivations based on the new type of left centralizers, while τ , σ and θ are not supposed homomorphism, necessarily. It is one of the reasons that obviously proves performance and application of this type of centralizers. 2. Centralizer Associated with Semi Hochschild 2-Cocycle Definition 2.1. A biadditive map ฀ : R × R → R is said to be a lsemi Hochschild 2-cocycle if ฀(xy, z) ฀ ฀(x, yz) + ฀(x, y)z = 0 for all x, y, z ∈ R. Definition 2.2. For a biadditive map ฀ : R × R → R, an additive map T : R → R is said to be a left centralizer associated with ฀ if T (xy) = T (x)y + ฀(x, y) for all x, y ∈ R. As we mentioned in the introduction, such ฀ is a l-semi Hochschild 2cocycle. Example 2.3. Every derivation D : R → R is a left centralizer associated with ฀ : R×R → R defined by ฀(x, y) = xD(y) for all x, y ∈ R. We have ฀(xy, z) ฀ ฀(x, yz) + ฀(x, y)z = xyD(z) ฀ xD(yz) + xD(y)z = xyD(z) ฀ xyD(z) ฀ xD(y)z + xD(y)z = 0. Hence, D(xy) = D(x)y + xD(y) = D(x)y + ฀(x, y) is a left centralizer associated with ฀. Example 2.4. Suppose D : R → R is a derivation. Then, every Dderivation f : R → R is a left centralizer associated with ฀, if ฀ is defined as above. Example 2.5. Suppose θ : R → R is an endomorphism. Then, h = θ ฀ id, where id is the identity mapping on R , is a (θ, id) ฀ derivation A GENERAL GENERAL CHARACTERIZATION CHARACTERIZATIONOF OFADDITIVE ADDITIVE...... 123 123 as following: following: h(xy) = θ(xy) θ(xy) ฀ ฀ xy xy = = θ(x)θ(y) θ(x)θ(y)฀฀θ(x)y θ(x)y++θ(x)y θ(x)y฀฀xy xy==h(x)y h(x)y++θ(x)h(y). θ(x)h(y). If we define define ฀(x, ฀(x,y) y) = = θ(x)h(y) θ(x)h(y) for for all all x,x,yy ∈∈ R, R,then then฀฀isisa al-semi l-semi Hochschild Hochschild 2-cocycle. 2-cocycle. It It means meansthat thathhisisaaleft leftcentralizer centralizerassociated associatedwith with ฀. Furthermore, Furthermore, suppose suppose that that TT ::R R→ →RRisisaaleft leftθθ฀฀centralizer, centralizer,i.e. i.e.T T is additive additive and and TT(xy) (xy) = = TT(x)θ(y) (x)θ(y) holds holdsfor forall allx,x,yy∈∈R. R.Considering Considering h = θ ฀ id, id, we we have have TT (xy) (xy) = = TT(x)θ(y) (x)θ(y)==TT(x)(h (x)(h++id)(y) id)(y)==TT(x)(h(y) (x)(h(y)++y)y) = = TT(x)y (x)y+ +TT(x)h(y). (x)h(y). Defining ฀(x, ฀(x,y) y) = = TT(x)h(y) (x)h(y) for for all all x,x,yy ∈∈ R, R,we weconclude concludethat that฀฀isis a l-semi Hochschild Hochschild 2-cocycle. 2-cocycle. Hence, Hence, TT isisaaleft leftcentralizer centralizerassociated associated with ฀. Example Example 2.6. 2.6. Let Let σ, σ,ττ :: R R→ →R Rbe betwo twoendomorphisms endomorphismsand andd d: R : R→→RR be a σ-derivation. σ-derivation. (i) Every (σ, (σ, ττ)-derivation )-derivation FF :: R R→ → RR isisaaleft leftcentralizer centralizerassociated associated with ฀, if ฀ ฀ is is defined defined by by ฀(x, ฀(x,y) y)==FF(x)(σ (x)(σ฀฀id)(y) id)(y)++τ τ(x)F (x)F(y). (y). (ii) Suppose Suppose δδ :: R R→ →R R isis aa generalized generalizedσ-derivation. σ-derivation. Put Put฀(x, ฀(x,y)y)== δ(x)(σ ฀ id)(y) id)(y) + + σ(x)d(y), σ(x)d(y), where wheredd: :RR→ →RRisisaaσ-derivation, σ-derivation,sosoδ δisis a left centralizer centralizer associated associated with with ฀. ฀. Let LetTT : :RR→ →RRbebea aJordan Jordanleft left centralizer centralizer associated associated with with ฀, ฀, and anddefine defineψψ: :RR××RR→→RRbybyψ(x, ψ(x,y)y)== T (xy) ฀ TT(x)y (x)y ฀ ฀ ฀(x, ฀(x,y) y) for for all all x, x,yy ∈∈ R. R. The Thefollowing followingproposition proposition demonstrates demonstrates several several properties propertiesofofψ. ψ. Proposition Proposition 2.7. 2.7. Let Let ψ, ψ, ฀฀ and and TT bebeas asabove. above. The Thebi-additive bi-additivemap mapψ ψ satisfies the the following: following: (1) ψ is anti anti symmetric, symmetric, i.e. i.e. ψ(x, ψ(x,y) y)==฀ψ(y, ฀ψ(y,x), x), (2) ψ is aa l-semi l-semi Hochschild Hochschild 2-cocycle, 2-cocycle, (3) 2ψ(xy, 2ψ(xy, z) z) + + ψ(x, ψ(x,y)z y)z฀ ฀ψ(y, ψ(y,z)x z)x++ψ(z, ψ(z,x)y x)y==0,0, (4) If R is is aa 2-torsion 2-torsion free free ring, ring, then thenψ(xy, ψ(xy,z)z)++ψ(xz, ψ(xz,y)y)==0,0, (5) If R is is aa 2-torsion 2-torsion free free ring, ring, then thenψ([x, ψ([x,y], y],z)z)==฀ψ(x, ฀ψ(x,y)z, y)z, (6) If R is is aa 2-torsion 2-torsion free free ring, ring, then thenψ(x, ψ(x,y)[z, y)[z,w] w]++ψ(z, ψ(z,w)[x, w)[x,y]y]==0,0, 124 124 124 A.A. HOSEINI A. HOSEINI HOSEINI (7) ψ([x, y]z, w)w) == 0, (7) R isisaa2-torsion 2-torsionfree freering, ring,then then ψ([x, y]z, w) =0,0, (7) IfIf If R 2-torsion free ring, then ψ([x, y]z, (8) ψ(x, y)[z, w]r == 0, (8) R isisaa2-torsion 2-torsionfree freering, ring,then then ψ(x, y)[z, w]r =0,0, (8) IfIf If R 2-torsion free ring, then ψ(x, y)[z, w]r for for all x,y,y,z,z,rr∈∈R. for all all x, R. Proof. Proof. (1)and and(2) arestraightforward. straightforward. Proof.(1) (2)are are straightforward. For proving (3), we have For proving (3), we have For proving have 2ψ(xy, z)x ++ ψ(z, x)y === 2ψ(xy, z)z) + ψ(x, y)zy)z ++ ψ(z, y)xy)x 2ψ(xy, z)++ψ(x, ψ(x,y)z ฀ψ(y, ψ(y, z)x + ψ(z, x)y 2ψ(xy, z)+ +ψ(x, ψ(x, y)z +ψ(z, ψ(z, y)x 2ψ(xy,z) y)z฀฀ ψ(y, z)x ψ(z, x)y 2ψ(xy, + ฀(xy, z)z) ฀฀฀(xy, z)z) ฀ T฀(x)yz ฀฀ ฀(x, y)zy)z + ψ(z, x)y==TT(xyz) ฀TTT(xy)z (xy)z฀฀ ฀ ฀(xy, z) ฀฀(xy, ฀(xy, z)฀ (x)yz ฀฀(x, ฀(x, y)z +ψ(z, ψ(z,x)y (xyz)฀฀ (xy)z ฀(xy, TT(x)yz + y)x ++ TT(zx)y ฀฀T (z)xy ฀฀ ฀(z, x)y ++ T+ (xyz) + (zy)x฀฀TT(z)yx ฀(z, y)x + T(zx)y (zx)y ฀T T(z)xy (z)xy ฀฀(z, ฀(z, x)y (xyz) +TTT(zy)x (z)yx฀฀฀(z, ฀(z, y)x x)y TT(xyz) = xy) ฀฀ ฀(xy, z)z) ++T+(yz)x ฀฀ T฀(yzx) ++ ฀(yz, x) x)x) = ฀T (zxy)++TT(z)xy +฀(z, ฀(z, xy) ฀ ฀(xy, z) (yz)x (yzx) +฀(yz, ฀(yz, =฀T ฀T(zxy) (z)xy++ ฀(z, xy) ฀(xy, TT(yz)x TT(yzx) + (zy)x ฀฀ T T(z)yx ฀฀฀(z, y)x ++ T+(zx)y ฀฀ T฀ (z)xy + ฀(x, yz)฀฀฀(x, +TTT (zy)x ฀ T(z)yx (z)yx ฀฀(z, ฀(z, y)x (zx)y (z)xy +฀(x, ฀(x,yz) ฀(x,y)z y)z++ (zy)x y)x TT(zx)y TT(z)xy ฀ ฀฀ ฀(y, zx) ฀฀฀(zx, y)y) + ฀(z, xy) ฀฀ ฀(xy, z) z)z) ฀ ฀(z, x)y==TT(yzx) ฀TTT(y)zx (y)zx ฀ ฀(y, zx) ฀฀(zx, ฀(zx, y)+ +฀(z, ฀(z, xy) ฀฀(xy, ฀(xy, ฀฀(z, ฀(z,x)y (yzx)฀฀ (y)zx ฀(y, zx) xy) + x)x) ++ ฀(x, yz) ฀฀฀(x, y)z ++ T+(zy)x ฀฀ T฀ (z)yx + (yz)x฀฀TT(yzx) ฀(yz, x) + ฀(x, yz) ฀฀(x, ฀(x, y)z (zy)x (z)yx +TTT(yz)x (yzx)++฀(yz, ฀(yz, ฀(x, yz) y)z TT(zy)x TT(z)yx ฀ ฀ ฀(z, y)x฀฀฀(z, ฀฀(z, ฀(z,y)x ฀(z,x)y x)y = TT(zy) ฀฀฀ TT (z)y ฀฀ ฀(z, y)]x = [T (yz)฀฀TT(y)z ฀(y,z)z) z)++ + T(zy) (zy) T(z)y (z)y ฀฀(z, ฀(z, y)]x =[T [T(yz) (y)z฀฀฀(y, ฀(y, y)]x ฀ x)y] ฀฀ [฀(xy, z)z) ฀฀ ฀(x, yz) ++ ฀(x, y)z] ฀ [฀(zx, y)฀฀฀(z, +฀(z, ฀(z, x)y] ฀ [฀(xy, z) ฀฀(x, ฀(x, yz) +฀(x, ฀(x, y)z] ฀[฀(zx, [฀(zx,y) ฀(z,xy) xy)++ ฀(z, x)y] [฀(xy, yz) y)z] = = 0. =0. 0. (4): z)z) == ฀ψ(x, y)z ++ ψ(y, z)x ฀฀ ψ(z, x)yx)y ฀ ฀฀ (4): By using using(3), getψ(xy, ψ(xy, z) = ฀ψ(x, y)z +ψ(y, ψ(y, z)x ฀ψ(z, ψ(z, x)y (4): By (3),we weget get ψ(xy, ฀ψ(x, y)z z)x ψ(xy, z)y ++ ψ(z, y)x ฀฀ ψ(y, x)z ฀฀ ψ(xz, y).y). Apψ(xy, z) and andψ(xz, ψ(xz,y)y)== =฀ψ(x, ฀ψ(x, z)y + ψ(z, y)x ฀ψ(y, ψ(y, x)z ฀ψ(xz, ψ(xz, y). Apψ(xy,z) ฀ψ(x, z)y ψ(z, y)x x)z Applying with the fact that ψ ψisψis anti symmetric, plying theprevious previoustwo twoequations equations with the fact that isanti anti symmetric, plyingthe two equations with the fact that symmetric, we z)z) ฀฀฀ ψ(xz, y),y), i.e.i.e. 2(ψ(xy, z) z) +z)++ we have ψ(xy,z)z)++ψ(xz, ψ(xz,y)y) y)== =฀ψ(xy, ฀ψ(xy, z) ψ(xz, y), i.e. 2(ψ(xy, we have have ψ(xy, ψ(xz, ฀ψ(xy, ψ(xz, 2(ψ(xy, ψ(xz, free ring, ψ(xy, z)z) + ψ(xz, y)y) =y)= 0.=0.0. ψ(xz, y))==0.0. Since Risis isaaa2-torsion 2-torsion free ring, ψ(xy, z)+ +ψ(xz, ψ(xz, ψ(xz,y)) SinceRR 2-torsion free ring, ψ(xy, (5): This part is achieved from (1), (2) and (4), immediately. (5): This part is achieved from (1), (2) and (4), immediately. (5): achieved from (1), (2) and (4), immediately. (6): and (5). (6): This part partisisobtained obtainedfrom from(1) (1) and (5). (6): This obtained from (1) and (5). (7): By using (2) and (5) our aim is achieved. (7): (5) our our aim aim isis achieved. achieved. (7): By using (2) and (5) (8): [z,[z, w]r) ++ ψ(x, y[z, w]r) (8): We have haveψ(x, y)[z,w]r w]r== =฀ψ(xy, ฀ψ(xy, [z, w]r) +ψ(x, ψ(x, y[z, w]r) (8): We ψ(x,y)[z, y)[z, w]r ฀ψ(xy, w]r) y[z, w]r) = w]zr) === 0฀ψ([yz, w]r, x)+ψ([y, w]zr, x) x)x) = ψ([z, w]r,xy)+ψ(x, xy)+ψ(x,[yz, [yz,w]r฀[y, w]r฀[y, w]zr) 0฀ψ([yz, w]r, x)+ψ([y, w]zr, =ψ([z, ψ([z,w]r, [yz, w]r฀[y, w]zr) 0฀ψ([yz, w]r, x)+ψ([y, w]zr, = = (see(7)). (7)). ฀฀ =000 (see The byby Zalar [11]. Now, wewe provide The followinglemma hasbeen beenproved proved by Zalar [11]. Now, we provide The following lemmahas has been proved Zalar [11]. Now, provide another another prooffor anotherproof forit.it. Lemma ring and a a∈a∈R element. Lemma 2.8.Let LetRRbebe beaaasemiprime semiprime ring and ∈Rbe Rbeabeafixed afixed fixed element. Lemma 2.8. semiprime ring and element. A GENERAL CHARACTERIZATION OF ADDITIVE ... 125 Then a[x, y] = 0 for all x, y ∈ R if and only if there exists an ideal I of R contained in Z(R) such that a ∈ I. Proof. First, note that [xy, z] = [x, z]y + x[y, z] and [x, yz] = [x, y]z + y[x, z] for all x, y, z ∈ R. Let a[x, y] = 0 for some fixed element a and for all x, y of R. We have [a, x]y[a, x] = ([ay, x] ฀ a[y, x])[a, x] = [ay, x][a, x] and further, [ay[a, x], x] = [ay, x][a, x] + ay[[a, x], x] for all x, y ∈ R. So, [a, x]y[a, x] = [ay, x][a, x] = [ay[a, x], x] ฀ ay[[a, x], x] = [a([ya, x] ฀ [y, x]a), x] ฀ a([y[a, x], x] ฀ [y, x][a, x]) = [a[ya, x] ฀ a[y, x]a, x] ฀ a[y[a, x], x] + a[y, x][a, x] = 0. Since R is semiprime, [a, x] = 0 for all x ∈ R, i.e. a ∈ Z(R). Suppose I = < a > denotes the ideal generated by a . Hence, we have I = {ra + as + na + Σm i=1 ri asi | r, s, ri , si ∈ R, n ∈ Z, i = 1, 2, ..., m}. Therefore, [ra + as + na + Σm i=1 ri asi , x] = 0 and thus, a ∈ I ⊆ Z(R). Conversely, assume that a ∈ I ⊆ Z(R), where I is a bi-ideal of R. We will show that a[x, y] = 0 for all x, y ∈ R. Note that (a[x, y])z(a[x, y]) = ([ax, y] ฀ [a, y]x)z([ax, y] ฀ [a, y]x) = [ax, y]z[ax, y] = 0. Since R is semiprime, a[x, y] = 0 for all x, y ∈ R and our purpose is achieved. ฀ We are now ready for the following main result. Theorem 2.9. Let R be a 2-torsion free semiprime ring and T : R → R be a Jordan left centralizer associated with a l-semi Hochschild 2-cocycle ฀ : R × R → R. Then, T is a left centralizer associated with ฀. Proof. By hypothesis, T (x2 ) = T (x)x + ฀(x, x) f or all x ∈ R. (1) Replacing x by x + y in (1), we get T (x2 ) + T (xy + yx) + T (y 2 ) = T (x)x + T (x)y + T (y)x + T (y)y + ฀(x, x) + ฀(x, y) + ฀(y, x) + ฀(y, y). This equation together with (1) imply that T (xy + yx) = T (x)y + T (y)x + ฀(x, y) + ฀(y, x) f or all x, y ∈ R. (2) 126 A. HOSEINI We replace y by xy + yx in (2) to get T (x(xy + yx) + (xy + yx)x) = T (x)xy + T (x)yx + T (x)yx + T (y)x2 + ฀(x, y)x + ฀(y, x)x + ฀(x, xy) + ฀(x, yx) + ฀(xy, x) + ฀(yx, x). (3) But this can also be calculated in a different way. T (x2 y + yx2 ) + 2T (xyx) = T (x)xy + ฀(x, x)y + T (y)x2 + ฀(x2 , y) + ฀(y, x2 ) + 2T (xyx) = T (x)xy + ฀(x, x)y + T (y)x2 + ฀(x, xy) ฀ ฀(x, x)y + ฀(yx, x) + ฀(y, x)x + 2T (xyx). It means that T (x2 y + yx2 ) + 2T (xyx) = T (x)xy + ฀(x, x)y + T (y)x2 + ฀(x, xy) ฀฀(x, x)y + ฀(yx, x) + ฀(y, x)x + 2T (xyx). (4) Comparing (3) and (4), it is obtained that 2T (x)yx+฀(x, y)x+฀(x, yx)+ ฀(xy, x) = 2T (xyx). Hence, 2T (x)yx + ฀(x, y)x + ฀(x, yx) + ฀(x, yx) ฀ ฀(x, y)x = 2T (xyx). It means that T (xyx) = T (x)yx + ฀(x, yx). (5) Putting x + z for x in (5), we have T (x)yx + T (x)yz + T (z)yx + T (z)yz + ฀(x, yx)+฀(x, yz)+฀(z, yx)+฀(z, yz) = T (xyx)+T (xyz+zyx)+T (zyz). The last equation together with (5) imply that T (xyz + zyx) = T (x)yz + T (z)yx + ฀(x, yz) + ฀(z, yx). (6) Put m = T (xyzyx + yxzxy). We compute m in two different methods. Using (5), we have m = T (x)yzyx + ฀(x, yzyx) + T (y)xzxy + ฀(y, xzxy) (7) 127 A GENERAL CHARACTERIZATION OF ADDITIVE ... and by applying (6), we have m = T (xy)zyx + T (yx)zxy + ฀(xy, zyx) + ฀(yx, zxy). (8) So, 0 = m ฀ m = T (xy)zyx + T (yx)zxy + ฀(xy, zyx) + ฀(yx, zxy) ฀ T (x)yzyx ฀ ฀(x, yzyx) ฀ T (y)xzxy ฀ ฀(y, xzxy) = T (xy)zyx + T (yx)zxy + ฀(xy, zyx) + ฀(yx, zxy) ฀ T (x)yzyx ฀ T (y)xzxy ฀ ฀(xy, zyx) ฀ ฀(x, y)zyx ฀ ฀(yx, zxy) ฀ ฀(y, x)zxy. Hence, (T (xy)฀T (x)y ฀฀(x, y))zyx+(T (yx)฀T (y)x฀฀(y, x))zxy = 0. By the last equation and introducing a bilinear map ψ(x, y) = T (xy) ฀ T (x)y ฀ ฀(x, y), it can be achieved that ψ(x, y)zyx + ψ(y, x)zxy = 0. (9) It follows from (2) that ψ(x, y) = ฀ψ(y, x). Using this fact and equality (9), we obtain ψ(x, y)z[x, y] = 0 f or all x, y, z ∈ R. (10) Replacing x by x + u in (10), we get 0 = ψ(x + u, y)z[x + u, y] = (ψ(x, y) + ψ(u, y))z([x, y] + [u, y]) = ψ(x, y)z[x, y] + ψ(x, y)z[u, y] + ψ(u, y)z[x, y] + ψ(u, y)z[u, y] = 0 + ψ(x, y)z[u, y] + ψ(u, y)z[x, y] + 0. Therefore, ψ(x, y)z[u, y] + ψ(u, y)z[x, y] = 0 f or all x, y, z, u ∈ R. Using (10) and (11), we find (ψ(x, y)z[u, y])ω(ψ(x, y)z[u, y]) = ψ(x, y)(z[u, y]ωψ(x, y)z)[u, y] = ฀ψ(u, y)z[u, y]ωψ(x, y)z[x, y] = 0. (11) 128 A. HOSEINI Since R is semiprime, we obtain ψ(x, y)z[u, y] = 0 f or all x, y, z, u ∈ R. (12) Replacing y by y + v in (12), we arrive at ψ(x, y)z[u, y] + ψ(x, y)z[u, v] + ψ(x, v)z[u, y] + ψ(x, v)z[u, v] = 0, this equation together with (12) imply that ψ(x, y)z[u, v] = ฀ψ(x, v)z[u, y] f or all x, y, z, u, v ∈ R. (13) From (13) and the fact that ψ(x, y)z[u, y] = 0, it can be concluded that (ψ(x, y)z[u, v])ω(ψ(x, y)z[u, v]) = ψ(x, y)(z[u, v]ωψ(x, y)z)[u, v] = ฀ψ(x, v)z[u, v]ωψ(x, y)z[u, y] = 0. Now, we have ψ(x, y)z[u, v]ωψ(x, y)z[u, v] = ψ(x, y)(z[u, v]ωψ(x, y)z)[u, v] = ฀ψ(x, v)(z[u, v]ωψ(x, y)z)[u, y] (see 13) = 0. (see 12) = ฀ψ(x, v)z[u, v]ω(ψ(x, y)z[u, y]) Reusing the fact that R is semiprime, it seen that ψ(x, y)z[u, v] = 0 f or all x, y, z, u, v ∈ R. (14) Hence, ψ(x, y)[u, v]zψ(x, y)[u, v] = ψ(x, y)([u, v]zψ(x, y))[u, v] = 0, (see 14) and it follows from semiprimeness of R that ψ(x, y)[u, v] = 0 f or all x, y, u, v ∈ R. (15) Now, let x and y be two fixed elements of R and for convenience write ψ instead of ψ(x, y). Using Lemma 1 we get the existence of an ideal I AAGENERAL OFOF ADDITIVE ... ... GENERALCHARACTERIZATION CHARACTERIZATION ADDITIVE 129129 such ψyψy ∈ Z(R) for for all all y ∈yR. such that that ψψ ∈∈II⊆⊆Z(R). Z(R).InInparticular, particular,yψ, yψ, ∈ Z(R) ∈ R. This gives us This gives us 2 2 2 2 2 2 2 yψyψ .x .x == y.ψy.ψ x x x.ψ x.ψy2 y==ψψy.x y.x== (16)(16) 2 2 2 and of of this equality willwill be be and so, so, 4T 4T(x.ψ (x.ψ 2y)y)==4T4T(y.ψ (y.ψx). x). Both Bothsides sides this equality computed using (2) and (16). Indeed, we have computed using (2) and (16). Indeed, we have 2 2 2 2 4T4T(x.ψ 4T4T (y.ψ (x.ψy) y)== (y.ψx). x). 2 2 2 2 2 2 ⇒ 2T2T (yψ xy). ⇒2T 2T(xψ (xψy2 y++ψ ψyx) yx)= = (yψx2+ xψ +ψ xy). 2 2 2 2 2 2 2 ⇒2T ++ 2T2T (ψ(ψ y)x + 2฀(ψ y, x) ⇒2T(x)ψ (x)ψy2 y++2฀(x, 2฀(x,ψψy) y) y)x + 2฀(ψ y, x) 22 2 2 2 2 2 2 ψ ψx)x) ++ 2T2T (ψ(ψx)yx)y + 2฀(ψ x, y). ==2T 2T(y)ψ (y)ψxx++2฀(y, 2฀(y, + 2฀(ψ x, y). 2 22 2 2 2 2 2 ⇒2T T (ψ yψyψ )x )x + 2฀(ψ y, x) ⇒2T(x)ψ (x)ψ 2yy++2฀(x, 2฀(x,ψψy) y)++ T (ψy2+ y+ + 2฀(ψ y, x) 22 2 2 2 2 2 2 2 2 ++ T (ψ xψxψ )y )y + 2฀(ψ x, y). ==2T 2T(y)ψ (y)ψxx++2฀(y, 2฀(y,ψψx)x) T (ψx + x+ + 2฀(ψ x, y). 2 22 2 2 2 2 2 2 ⇒2T (ψ(ψ )yx ++ ฀(ψ + T+(y)ψ x +x฀(y, ψ 2 )x ⇒2T(x)ψ (x)ψ 2yy++2฀(x, 2฀(x,ψψy) y)++T T )yx ฀(ψ, y)x , y)x T (y)ψ + ฀(y, ψ 2 )x 2 2 2 + T (ψ)ψxy + ฀(ψ, ψ)xy ψ 2ψx) + +2฀(ψ 2฀(ψ 2y,y,x) x)==2T 2T(y)ψ (y)ψx2 x++2฀(y, 2฀(y, x) + T (ψ)ψxy + ฀(ψ, ψ)xy 2 2 2 2 ฀(x, ψ 2ψ)y2 )y + 2฀(ψ x, y). ++฀(ψ ฀(ψ,2x)y , x)y++TT(x)ψ (x)ψy2 + y+ ฀(x, + 2฀(ψ x, y). 2 22 2 2 ⇒T (ψ)ψyx ++ ฀(ψ, ψ)yx + ฀(ψ , y)x + ฀(y, ψ 2 )x ⇒T(x)ψ (x)ψ 2yy++2฀(x, 2฀(x,ψψy) y)++T T (ψ)ψyx ฀(ψ, ψ)yx + ฀(ψ , y)x + ฀(y, ψ 2 )x 2 2 2฀(y, ψ 2ψx)2 x) ++ T (ψ)ψxy + ฀(ψ, ψ)xy + +2฀(ψ 2฀(ψ 2y,y,x) x)==TT(y)ψ (y)ψx2 x++ 2฀(y, T (ψ)ψxy + ฀(ψ, ψ)xy 2 2 2 ++฀(ψ ψ 2ψ)y2 )y ++ 2฀(ψ x, y). ฀(ψ,2x)y , x)y++฀(x, ฀(x, 2฀(ψ x, y). Since Since ψyx ψyx==xψy xψy==ψxy, ψxy,we weobtain obtain 2 22 2 2 2 2 ++ ฀(ψ + ฀(y, ψ 2 )x y, x)y,= TT(x)ψ (x)ψ 2yy++2฀(x, 2฀(x,ψψy) y)++฀(ψ, ฀(ψ,ψ)yx ψ)yx ฀(ψ, y)x , y)x + ฀(y, ψ 2+ )x2฀(ψ + 2฀(ψ x) = 2 22 2 2 2 2 ++ ฀(ψ + ฀(x, ψ 2 )y x, y). TT(y)ψ (y)ψ 2xx++2฀(y, 2฀(y,ψψx) x)++฀(ψ, ฀(ψ,ψ)xy ψ)xy ฀(ψ, x)y , x)y + ฀(x, ψ 2+ )y2฀(ψ + 2฀(ψ x, y). (17)(17) 130 130 130 130 130 130 A. A. HOSEINI HOSEINI A.A. HOSEINI A. HOSEINI HOSEINI A. HOSEINI Rearranging Rearranging(17) (17)wewegetget Rearranging (17) we get Rearranging Rearranging(17) (17)we weget get Rearranging (17) 2we get 2 2 2 2 2 + ฀(y, ψ 2 )x+2 TT(y)xψ y+ 2฀(x, ψ 2ψ y)22 y) + ฀(ψ, ψ)yx + ฀(ψ , y)x (y)xψ222 ==TT(x)ψ (x)ψ y+ 2฀(x, + ฀(ψ, ψ)yx + 2฀(ψ , y)x + ฀(y, 2 2 ψ2 )x+ 222y + 2฀(x, ψ22 y) 2 = TT(x)ψ (x)ψ +฀(ψ, ฀(ψ, ψ)yx +฀(ψ ฀(ψ , y)x ฀(y, )x+ TT(y)xψ (y)xψ = T y + 2฀(x, ψ y) + ψ)yx + , y)x ++฀(y, ψ 2ψ)x+ T (y)xψ = (x)ψ y + 2฀(x, ψ y) + ฀(ψ, ψ)yx + ฀(ψ ,2y)x + ฀(y, ψ )x+ 2 2 2 2 T (y)xψ = T (x)ψ y + 2฀(x, ψ y) + ฀(ψ, ψ)yx + ฀(ψ , y)x + ฀(y, ψ )x+ 2 2 ฀ ฀(ψ, ψ)xy ฀ ฀(ψ 2 , x)y 2 ฀ ฀(x, ψ 2 )y ฀ 2 ψ22ψ x) 2฀(ψ x, y). 2฀(ψ 2฀(ψ22222y,y,x)x)฀฀2฀(y, 2฀(y, x) ฀ ฀(ψ, ψ)xy ฀ ฀(ψ , x)y ฀ ฀(x, ψ 2 )y ฀ 2฀(ψ x, y). 2 2 , 2x)y ฀ ฀(x, ψ 2 2 )y2 ฀ 2฀(ψ 2 2 x,2y). 2 ฀ ฀(ψ, ψ)xy ฀ ฀(ψ y, x) ฀฀2฀(y, 2฀(y, ψψ2x) x) 2฀(ψ y, x) ฀ ψ ฀ ฀(ψ, ψ)xy ฀ ฀(ψ , x)y ฀ ฀(x, ψ )y ฀ 2฀(ψ x, y). 2฀(ψ 2฀(ψ y, x) 2฀(y, x) ฀ ฀(ψ, ψ)xy ฀ ฀(ψ , x)y ฀ ฀(x, ψ )y ฀ 2฀(ψ x, y). (18) 2 2 2 2 2 (18) 2฀(ψ y, x) ฀ 2฀(y, ψ x) ฀ ฀(ψ, ψ)xy ฀ ฀(ψ , x)y ฀ ฀(x, ψ )y ฀ 2฀(ψ x, y). (18) (18) (18) (18) On also have Onthe theother otherhand, hand,wewe also have On the other hand, we also have On have On the theother otherhand, hand,we wealso also have On the other hand, we also have 2 2 4T4T (xyψ ) =) 4T (xψ.yψ) (xyψ = 4T (xψ.yψ) 22 )2 = 4T (xψ.yψ) 4T (xyψ ) = 4T (xψ.yψ) 4T (xyψ 4T (xyψ ) = 4T (xψ.yψ) 4T (xyψ 2 ) = 4T (xψ.yψ) 2 2 2 2 ++ ψψ xy) = 2T + yψxψ) ⇒⇒2T2T (xyψ (xyψ xy) = (xψyψ 2T (xψyψ + yψxψ) 22 2+ ψ22 xy) 2 (xψyψ yψxψ) ⇒ 2T (xyψ ++ ψ ψxy) ==2T (xψyψ ++yψxψ) ⇒ 2T (xyψ ⇒ 2T (xyψ xy) =2T2T (xψyψ + yψxψ) 2 2 ⇒ 2T (xyψ + ψ xy) = 2T (xψyψ + yψxψ) 2 2 2 2 2 2฀(ψ 2 , xy) 2 ⇒2T ++ 2T2T (ψ(ψ )xy + 2฀(xy, ψ222)ψ+ ⇒2T(xy)ψ (xy)ψ )xy + 2฀(xy, ) + 2฀(ψ , xy) 222+ 2T (ψ22 )xy 2 2 , xy) 2 2 +2฀(xy, 2฀(xy, ψ)ψ)+2+ 2฀(ψ ⇒2T (xy)ψ + 2T (ψ )xy + ψ 2฀(ψ , xy) ⇒2T (xy)ψ ⇒2T (xy)ψ + 2T (ψ )xy + 2฀(xy, ) + 2฀(ψ , xy) 2 = 2T (ψx)ψy 2 2 2 2T2T (ψy)ψx 2฀(ψx, ψy)ψy) + 2฀(ψy, ψx) ψx) = 2T (ψx)ψy + (ψy)ψx + 2฀(ψx, + 2฀(ψy, (ψ )xy ++2฀(xy, ψ ) +++2฀(ψ , xy) ⇒2T (xy)ψ + ==2T 2T (ψx)ψy 2T (ψy)ψx ψy) 2฀(ψy, ψx) = 2T 2T (ψy)ψx + 2฀(ψx, ψy) ++2฀(ψy, ψx) 2T(ψx)ψy (ψx)ψy+++ 2T (ψy)ψx +2฀(ψx, 2฀(ψx, ψy) + 2฀(ψy, ψx) = 2T (ψx)ψy + 2T (ψy)ψx + 2฀(ψx, ψy) + 2฀(ψy, ψx) 22 2 2 ⇒2T ++2T2T (ψ)ψxy + 2฀(ψ, ψ)xy + 2฀(xy, ψ22 2) ψ +22฀(ψ , xy) ⇒2T(xy)ψ (xy)ψ (ψ)ψxy + 2฀(ψ, ψ)xy + 2฀(xy, ) + 2฀(ψ , xy) 2 222+ 2T (ψ)ψxy + 2฀(ψ, ψ)xy + 2฀(xy, ψ 2 2 , xy) ⇒2T (xy)ψ ) 2+2฀(ψ 2฀(ψ ⇒2T (xy)ψ ⇒2T (xy)ψ + 2T (ψ)ψxy + 2฀(ψ, ψ)xy + 2฀(xy, ψ ) + 2฀(ψ , xy) + 2T (ψ)ψxy + 2฀(ψ, ψ)xy + 2฀(xy, ψ ) + , xy) 2 2 2 ==2T ++ 2T2T (ψy)ψx 2฀(ψx, + 2฀(ψy, 2T(ψx)ψy (ψx)ψy (ψy)ψx + 2฀(ψx, + 2฀(ψy, ψx) ⇒2T (xy)ψ + 2T (ψ)ψxy + 2฀(ψ, ψ)xy ++ 2฀(xy, ψ ψy) )ψy) +ψy) 2฀(ψ , xy)ψx) = 2T (ψx)ψy 2T (ψy)ψx 2฀(ψy, ψx) =2T 2T(ψx)ψy (ψx)ψy+++ 2T (ψy)ψx +2฀(ψx, 2฀(ψx, ψy) + 2฀(ψy, ψx) = 2T (ψy)ψx ++2฀(ψx, ψy) ++2฀(ψy, ψx) = 2T (ψx)ψy + 2T (ψy)ψx + 2฀(ψx, ψy) + 2฀(ψy, ψx) 22 2 2 ++2T2T (ψ)ψxy + 2฀(ψ, ψ)xy + 2฀(xy, ψ22 2) ψ +22฀(ψ , xy) ⇒2T ⇒2T(xy)ψ (xy)ψ (ψ)ψxy + 2฀(ψ, ψ)xy + 2฀(xy, ) + 2฀(ψ , xy) 222+ 2T (ψ)ψxy + 2฀(ψ, ψ)xy + 2฀(xy, ψ 2 2 , xy) 2 ) 2+2฀(ψ 2฀(ψ ⇒2T (xy)ψ + 2T (ψ)ψxy + 2฀(ψ, ψ)xy + 2฀(xy, ψ ) + , xy) ⇒2T (xy)ψ + 2T (ψ)ψxy + 2฀(ψ, ψ)xy + 2฀(xy, ψ ) + 2฀(ψ , xy) ⇒2T (xy)ψ 2 ==2T TT(ψx ++ Tψ)xy (yψ ++ψy)ψx +ψ2฀(ψx, ψy)2 ,ψy) + 2฀(ψy, ψx) ψx) (ψx++xψ)ψy xψ)ψy T (yψ +2฀(xy, ψy)ψx + 2฀(ψx, + 2฀(ψy, (ψ)ψxy + 2฀(ψ, ) + 2฀(ψ xy) ⇒2T (xy)ψ 2 + = (ψx ++xψ)ψy xψ)ψy 2฀(ψx, ψy) 2฀(ψy, ψx)ψx) = TT(yψ ++ψy)ψx ++2฀(ψx, ψy) ++2฀(ψy, ψx) =TTT(ψx (ψx+ xψ)ψy+++ T(yψ (yψ +ψy)ψx ψy)ψx + 2฀(ψx, ψy) + 2฀(ψy, = T (ψx + xψ)ψy + T (yψ + ψy)ψx + 2฀(ψx, ψy) + 2฀(ψy, ψx) 22 2 2 ⇒2T ++2T2T (ψ)ψxy + 2฀(ψ, ψ)xy + 2฀(xy, ψ22 2) ψ +22฀(ψ , xy) ⇒2T(xy)ψ (xy)ψ (ψ)ψxy + 2฀(ψ, ψ)xy + 2฀(xy, ) + 2฀(ψ , xy) 2 222+ 2T (ψ)ψxy + 2฀(ψ, ψ)xy + 2฀(xy, ψ 2 2 , xy) ) 2+2฀(ψ 2฀(ψ ⇒2T (xy)ψ ⇒2T (xy)ψ ⇒2T (xy)ψ + 2T (ψ)ψxy + 2฀(ψ, ψ)xy + 2฀(xy, ψ ) + 2฀(ψ , xy) + 2T (ψ)ψxy + 2฀(ψ, ψ)xy + 2฀(xy, ψ ) + , xy) 2 2 2 + ฀(x, ψ)ψy + ฀(ψ, x)ψy + 2 T (ψ)ψxy2 + T (y)ψ x 2 y ==2T TT(x)ψ (x)ψ y + ฀(x, ψ)ψy + ฀(ψ, x)ψy + T (ψ)ψxy + T (y)ψ x ⇒2T (xy)ψ 2 + (ψ)ψxy + 2฀(ψ, ψ)xy 2฀(xy, ψ ) 2฀(ψ , xy) 2 2 2 x2 = (x)ψ ++฀(x, ฀(x, ψ)ψy ฀(ψ, x)ψy T(ψ)ψxy (ψ)ψxy T (y)ψ ψ)ψy +++ ฀(ψ, x)ψy ++T+ ++T + (y)ψ x x =TTT(x)ψ (x)ψ2 y2yy+ ฀(x, ψ)ψy ฀(ψ, x)ψy T (ψ)ψxy T (y)ψ = 2 ++฀(y, ψ)ψx +T + ฀(ψ, y)ψx ฀(y, ψ)ψx +(ψ)ψxy T (ψ)ψxy ++฀(ψ, y)ψx + ฀(ψ, x)ψy + T (ψ)ψxy T (y)ψ x = T (x)ψ 2 y + ฀(x, ψ)ψy + ฀(y, ψ)ψx + T (ψ)ψxy + ฀(ψ, y)ψx ++฀(y, ψ)ψx ++ T (ψ)ψxy + ฀(ψ, y)ψx ฀(y, ψ)ψx T (ψ)ψxy + ฀(ψ, y)ψx +(ψ)ψxy 2฀(ψx, ψy) + 2฀(ψy, ψx)ψx) + 2฀(ψx, ψy) + 2฀(ψy, + ฀(y, ψ)ψx + T + ฀(ψ, y)ψx 2฀(ψx, ψy) ψx) +++ 2฀(ψx, ψy) ++2฀(ψy, ψx) 2฀(ψx, ψy) +2฀(ψy, 2฀(ψy, ψx) + 2฀(ψx, ψy) + 2฀(ψy, ψx) 22 2 2฀(ψ 2 , xy) 2 = T (x)ψ 2 y +2฀(x, ψ)ψy ⇒2T ++2฀(ψ, ψ)xy ++ 2฀(xy, ψ222)ψ2+ ⇒2T(xy)ψ (xy)ψ 2฀(ψ, ψ)xy 2฀(xy, ) + 2฀(ψ , xy) = T (x)ψ y + ฀(x, ψ)ψy 222+ 2฀(ψ, ψ)xy + 2฀(xy, ψ 2 2, 2 2 2 y 2+ ฀(x, ψ)ψy )++ 2฀(ψ xy) T (x)ψ ⇒2T (xy)ψ + 2฀(ψ, ψ)xy + 2฀(xy, ψ ) 2฀(ψ , xy) ==T= (x)ψ y + ฀(x, ψ)ψyψ)ψy ⇒2T (xy)ψ ⇒2T (xy)ψ + 2฀(ψ, ψ)xy + 2฀(xy, ψ ) + 2฀(ψ , xy) T (x)ψ y + ฀(x, 2 2 2 2 ⇒2T (xy)ψ + 2฀(ψ, ψ)xy + 2฀(xy, ψ ) + 2฀(ψ , xy) = T (x)ψ y + ฀(x, ψ)ψy 2 2 ++ ฀(y, ψ)ψx + ฀(ψ, y)ψx + 2฀(ψx, ψy) ψy) + 2฀(ψy, ψx) ψx) +฀(ψ, (y)xψ +฀(ψ,x)ψy x)ψy++T T (y)xψ ฀(y, ψ)ψx + ฀(ψ, y)ψx + 2฀(ψx, + 2฀(ψy, 22 2+ ฀(y, ψ)ψx + ฀(ψ, y)ψx + 2฀(ψx, ψy) + 2฀(ψy, ψx) +฀(ψ, x)ψy ++TTT(y)xψ (y)xψ ++ ฀(y, ψ)ψx + ฀(ψ, y)ψx + 2฀(ψx, ψy) ψy) + 2฀(ψy, ψx) +฀(ψ, +฀(ψ,x)ψy x)ψy+ (y)xψ ฀(y, ψ)ψx + ฀(ψ, y)ψx + 2฀(ψx, + 2฀(ψy, ψx) (19) 2 (19) +฀(ψ, x)ψy + T (y)xψ + ฀(y, ψ)ψx + ฀(ψ, y)ψx + 2฀(ψx, ψy) + 2฀(ψy, ψx) (19) (19) (19) (19) A GENERAL CHARACTERIZATION OF ADDITIVE ... AA GENERAL GENERAL GENERAL CHARACTERIZATION CHARACTERIZATION CHARACTERIZATION OF OF OF ADDITIVE ADDITIVE ADDITIVE ......... 131 131 131 131 Using (18) and (19), we have Using Using (18) (18) and and and (19), (19), (19), wewe we have have have 2T (xy)ψ 22 + 2฀(ψ, ψ)xy + 2฀(xy, ψ 22 ) + 2฀(ψ 22 , xy) = T (x)ψ 22 y + ฀(x, ψ)ψy 2 22 2 22 2 22 22 2 2฀(ψ, ψ)xy 2฀(xy, ))++ 2฀(ψ , xy) ฀(x, ψ)ψy 2T2T (xy)ψ (xy)ψ ++ + 2฀(ψ, 2฀(ψ, ψ)xy ψ)xy ++ + 2฀(xy, 2฀(xy, ψψ ψ )+ 2฀(ψ 2฀(ψ , xy) , xy) === TT(x)ψ T(x)ψ (x)ψ yy+ y++ ฀(x, ฀(x, ψ)ψy ψ)ψy + ฀(ψ, x)ψy + ฀(y, ψ)ψx + ฀(ψ, y)ψx + 2฀(ψx, ψy) + 2฀(ψy, ψx) 2+ 2 ψy) 2ψx) ฀(ψ, x)ψy ฀(y, ψ)ψx ฀(ψ, y)ψx 2฀(ψx, 2฀(ψy, ++ ฀(ψ, x)ψy x)ψy +2 + + ฀(y, ฀(y, ψ)ψx ψ)ψx + + ฀(ψ, ฀(ψ, y)ψx y)ψx +++ 2฀(ψx, 2฀(ψx, ψy) ψy) +++ 2฀(ψy, 2฀(ψy, ψx) ψx) + T (x)ψ 2 y + 2฀(x, ψ 2 y) + ฀(ψ, ψ)yx + ฀(ψ 2 , y)x + ฀(y, ψ 2 )x 2 22 2 22 222 22 2 2 T ++ T (x)ψ (x)ψ y+ + 2฀(x, 2฀(x, y)y) y) +ψ+22+ ฀(ψ, ฀(ψ, ψ)yx + ฀(ψ ฀(ψ y)x , y)x + ฀(y, ฀(y, ψψψ )x y22yy, + 2฀(x, ψψ ฀(ψ, ψ)yx ++ ฀(ψ , y)x ฀(y, )x)x 2+ x) ฀ψ 2฀(y, x) ฀ψ)yx ฀(ψ, ψ)xy ฀,฀(ψ ,+ x)y +(x)ψ 2฀(ψ 2 22 2 22 22 2 2 x) 2ψ)xy 2฀(y, ฀(ψ, ฀(ψ x)y ++ 2฀(ψ 2฀(ψ y,y, x) y,x) x) ฀฀ ฀ 2฀(y, 2฀(y, ψ ψψ x) x) ฀฀฀ ฀(ψ, ฀(ψ, ψ)xy ψ)xy ฀฀฀ ฀(ψ ฀(ψ , ,x)y , x)y 2฀(ψ ฀ ฀(x, ψ 2 )y ฀ 2฀(ψ 2 x, y) 22 2 22 )y 2฀(ψ ฀(x, ฀฀ ฀ ฀(x, ฀(x, ψ 2ψψ )y )y ฀฀฀ 2฀(ψ 2฀(ψ x,x,x, y)y)y) Hence, Hence, Hence, 2 2T (xy)ψ 2 ฀ 2T (x)yψ 22 = ฀(x, ψ)ψy + ฀(ψ, x)ψy + ฀(y, ψ)ψx + ฀(ψ, y)ψx 2 22 2 22 2 (xy)ψ (x)yψ = ฀(x, ψ)ψy x)ψy ฀(y, ψ)ψx ฀(ψ, y)ψx 2T2T (xy)ψ ฀฀ ฀ 2T2T 2T (x)yψ (x)yψ =+ = ฀(x, ฀(x, ψ)ψy ψ)ψy +++ ฀(ψ, ฀(ψ, x)ψy x)ψy +++ ฀(y, ฀(y, ψ)ψx ψ)ψx ฀(ψ, ฀(ψ, y)ψx y)ψx 2+ y)+ 2฀(ψx, ψy) +฀(ψ, 2฀(ψy, ψx) + 2฀(x, ψ+ 22 2 2 ψx) 2 ψψψ 2฀(ψx, ψy) 2฀(ψy, ψx) 2฀(x, ++ + 2฀(ψx, 2฀(ψx, ψy) ψy) +++ 2฀(ψy, 2฀(ψy, ψx) +++ 2฀(x, 2฀(x, y) y)y) 2 + ฀(ψ, ψ)yx + ฀(ψ 2 , y)x + ฀(y, ψ 2 )x + 2฀(ψ 2 y, x) 2 22 22 2 22 2 2 , ψ)xy 2฀(ψ ++ + ฀(ψ, ฀(ψ, ψ)yx ψ)yx ++฀ + ฀(ψ ฀(ψ , y)x ,y)x y)x +++ ฀(y, ฀(y, ψψ22ψ )x )x ++฀ + 2฀(ψ 2฀(ψ y,y, y, ฀(ψ, ψ)yx ฀(ψ 2x) ฀ 2฀(y, ψ 22 x) ฀(ψ, ฀฀(y, ฀(ψ , )x x)y ฀(x, ψx) )yx) 2 22 22 2 22 2 2 ฀(ψ, ψ)xy x)y ฀(x, )y)y 2฀(y, ฀฀ ฀ 2฀(y, 2฀(y, ψ22ψx, ψ x)x) x) ฀฀ ฀(ψ, ฀(ψ, ψ)xy ψ)xy ฀฀฀ ฀(ψ ฀(ψ , ,x)y , x)y ฀(x, ฀(x, ψψψ )y 2 )฀ y) ฀฀ 2฀(ψ, ψ)xy ฀฀(ψ 2฀(xy, ψ฀฀ ฀ 2฀(ψ 2 22 22 2 2x, 2฀(ψ ฀฀ ฀ 2฀(ψ 2฀(ψ x, x, y)y) y) ฀฀฀ 2฀(ψ, 2฀(ψ, ψ)xy ψ)xy ฀฀฀ 2฀(xy, 2฀(xy, ψψψ )) ) 2฀(ψ, ψ)xy 2฀(xy, ฀ 2฀(ψ 2 , xy) () 2 22 ,,xy) () ฀฀ ฀ 2฀(ψ 2฀(ψ , xy) xy) () () 2฀(ψ Our next task is to prove the equation bellow: next task prove the equation bellow: Our Our next task task is is is toto to prove prove the the equation equation bellow: bellow: 2(T (xy) ฀ T (x)y ฀ ฀(x, y))ψ 22 = 0 2 22 2(T (xy) ฀(x, y))ψ 2(T 2(T (xy) (xy) ฀฀ ฀ TT (x)y T(x)y (x)y ฀฀฀ ฀(x, ฀(x, y))ψ y))ψ === 000 In order to prove the previous equation we need the following relations: order prove the previous equation we need the following relations: In In order toto prove prove the the previous previous equation equation we we need need the the following following relations: relations: ฀(ψ, x)ψy = ฀(ψ, xψ)y ฀ ฀(ψx, ψ)y = ฀(ψ, ψx)y ฀ ฀(ψx, ψ)y 2 ฀(ψ, x)ψy ฀(ψ, ฀฀ ฀(ψx, ψ)y == ฀(ψ, ψx)y ฀(ψx, ψ)y ฀(ψ, x)ψy x)ψy == = ฀(ψ, ฀(ψ, xψ)y ฀(ψx, ฀(ψx, ψ)y ψ)y =฀ ฀(ψ, ฀(ψ, ψx)y ψx)y ฀฀฀ ฀(ψx, ฀(ψx, ψ)y ψ)y 2 xψ)y = ฀(ψ ,xψ)y x)y฀+ ฀(ψ, ψ)xy ฀(ψx, ψ)y (i) 2 22 ,,x)y ฀(ψ, ψ)xy ฀(ψx, ψ)y (i)(i) ฀(ψ == = ฀(ψ ฀(ψ , x)y x)y +++ ฀(ψ, ฀(ψ, ψ)xy ψ)xy ฀฀฀ ฀(ψx, ฀(ψx, ψ)y ψ)y (i) ———————————————————————————————– ———————————————————————————————– ———————————————————————————————– ———————————————————————————————– 2฀(ψx, ψy) = ฀(ψx + xψ, ψy) = ฀(ψx, ψy) + ฀(xψ, ψy) 2 + ψy) ฀(ψx xψ, ψy) ฀(ψx, ψy) ฀(xψ, ψy) 2฀(ψx, 2฀(ψx, ψy) == = ฀(ψx ฀(ψx + + xψ, xψ, ψy) ψy) === ฀(ψx, ฀(ψx, ψy) ψy) +++ ฀(xψ, ฀(xψ, ψy) ψy) = ฀(ψ 2 x, y) + ฀(ψx, ψ)y + ฀(xψ, ψy) 2 22 2ψ)y ฀(ψ ฀(ψx, ψ)y ฀(xψ, ψy) == = ฀(ψ ฀(ψ x,2x, x, y)y) y) ++ + ฀(ψx, ฀(ψx, ψ)y +++ ฀(xψ, ฀(xψ, ψy) ψy) (ii) = ฀(ψ 2 x, y) + ฀(x, ψ 2 )y ฀ ฀(x, ψ)ψy + ฀(xψ, ψy) 2 22 2 22 ฀(x, ฀(x, ψ)ψy ฀(xψ, ψy) (ii) ฀(ψ == = ฀(ψ ฀(ψ x,x, x, y)y) y) ++ + ฀(x, ฀(x, ψ ψψ )y)y )y ฀฀฀ ฀(x, ฀(x, ψ)ψy ψ)ψy +++ ฀(xψ, ฀(xψ, ψy) ψy) (ii) (ii) ———————————————————————————————– ———————————————————————————————– ———————————————————————————————– ———————————————————————————————– ฀(ψ, y)ψx = ฀(ψ, yψx) ฀ ฀(ψy, ψx) = ฀(ψ, ψxy) ฀ ฀(ψy, ψx) 2 y)ψx ฀(ψ, ฀ ฀(ψy, ψx) == ฀(ψ, ψxy) ฀(ψy, ψx) ฀(ψ, ฀(ψ, y)ψx == = ฀(ψ, ฀(ψ, yψx) ฀ ฀(ψy, ฀(ψy, ψx) ψx) =฀ ฀(ψ, ฀(ψ, ψxy) ψxy) ฀฀฀ ฀(ψy, ฀(ψy, ψx) ψx) 2 yψx) ,yψx) xy)฀+ ฀(ψ, ψ)xy ฀(ψy, ψx) = ฀(ψ 2 22 ฀(ψ ,,xy) ฀(ψ, ψ)xy ฀(ψy, ψx) == = ฀(ψ ฀(ψ , xy) xy) ++ + ฀(ψ, ฀(ψ, ψ)xy ψ)xy ฀฀฀ ฀(ψy, ฀(ψy, ψx) ψx) (iii) (iii) (iii) (iii) 132 132 132 132 A. HOSEINI A. A. HOSEINI HOSEINI A. A.HOSEINI HOSEINI 2฀(ψy, ψx) ψx) = = 2฀(yψ, 2฀(yψ, ψx) = = 2฀(y,ψψ222x) x) ฀2฀(y, 2฀(y, ψ)ψx (iv) 2฀(ψy, (iv) 2฀(yψ, ψx) ψx) = 2฀(y, 2฀(y, ψ22x)฀ ฀ 2฀(y,ψ)ψx ψ)ψx (iv) 2฀(ψy, (iv) 2฀(ψy,ψx) ψx) = = 2฀(yψ, 2฀(yψ,ψx) ψx)==2฀(y, 2฀(y,ψψ x) x)฀฀2฀(y, 2฀(y,ψ)ψx ψ)ψx (iv) ———————————————————————————————– ———————————————————————————————– ———————————————————————————————– ———————————————————————————————– ———————————————————————————————– 2 ฀(ψ, ψ)yx ψ)yx = = ฀(ψ, ฀(ψ, ψyx) ฀ ฀ ฀(ψ222,,yx) yx) = ฀(ψ, ฀(ψ, ψxy)฀ ฀ ฀(ψ222,,yx) yx) ฀(ψ, ฀(ψ, ψyx) ψyx) ฀ ฀(ψ ฀(ψ22, yx) = = ฀(ψ,ψxy) ψxy) ฀฀(ψ ฀(ψ2 2, yx) 2ψyx) 2ψxy) , yx) = ฀(ψ, ฀ ฀(ψ , yx) ฀(ψ, ฀(ψ, ฀ ฀(ψ ฀(ψ,ψ)yx ψ)yx = = ฀(ψ, ψyx) ฀ ฀(ψ , yx) = ฀(ψ, ψxy) ฀ ฀(ψ , yx) xy) + ฀(ψ, ฀(ψ, ψ)xy ฀ ฀ ฀(ψ22,,yx) yx) = ฀(ψ ฀(ψ22,, xy) = , xy) + + ฀(ψ,ψ)xy ψ)xy ฀฀(ψ ฀(ψ2 2, yx) 222 = = ฀(ψ ฀(ψ22,,,xy) xy)+ +฀(ψ, ฀(ψ,ψ)xy ψ)xy฀ ฀฀(ψ ฀(ψ222,y, , yx) yx) xy) + ฀(ψ, ψ)xy ฀ ฀(ψ x)฀ ฀฀(ψ ฀(ψ222,,y)x y)x (v) = ฀(ψ , xy) + ฀(ψ, ψ)xy ฀ ฀(ψ y, x) (v) = ฀(ψ , xy) + ฀(ψ, ψ)xy ฀ ฀(ψ y, x) ฀ ฀(ψ , y)x (v) 22 22 22 (v) = ฀(ψ ฀(ψ ,,xy) xy)++฀(ψ, ฀(ψ,ψ)xy ψ)xy฀฀฀(ψ ฀(ψy,y,x)x)฀฀฀(ψ ฀(ψ, y)x , y)x (v) = ———————————————————————————————– ———————————————————————————————– ———————————————————————————————– ———————————————————————————————– ———————————————————————————————– 2 2 ฀(ψ22y, y, x) x) = = ฀(yψ ฀(yψ222,, x) x) = = ฀(y, ฀(y,ψψ222x) x) ฀ ฀฀(y, ฀(y,ψψ222)x )x (vi) ฀(ψ (vi) ฀(yψ , x) = ฀(y, ψ x) ฀ ฀(y, ψ2 2)x (vi) 22 22 22 (vi) ฀(ψ ฀(ψ y, y,x) x) = = ฀(yψ ฀(yψ ,,x) x)==฀(y, ฀(y,ψψ x) x)฀฀฀(y, ฀(y,ψψ)x)x (vi) ———————————————————————————————– ———————————————————————————————– ———————————————————————————————– ———————————————————————————————– ———————————————————————————————– 2 2 2 2 ฀2฀(xy, ψ ψ222)) = = ฀2฀(x, ฀2฀(x, yψ yψ222)) + + 2฀(x, 2฀(x,y)ψ y)ψ222 (vii) (vii) ฀2฀(xy, ฀2฀(x, yψ ) + 2฀(x, y)ψ (vii) 22 22 22 (vii) ฀2฀(xy, ฀2฀(xy,ψψ )) = = ฀2฀(x, ฀2฀(x,yψ yψ ))++2฀(x, 2฀(x,y)ψ y)ψ (vii) ———————————————————————————————– ———————————————————————————————– ———————————————————————————————– ———————————————————————————————– ———————————————————————————————– x, y) y) = = ฀฀(ψx, ฀฀(ψx, ψy) + + ฀(ψx,ψ)y ψ)y (viii) ฀฀(ψ22x, ฀฀(ψ (viii) ฀฀(ψx, ψy) ψy) + ฀(ψx, ฀(ψx, ψ)y (viii) 22 (viii) ฀฀(ψ ฀฀(ψ x, x,y) y) = = ฀฀(ψx, ฀฀(ψx,ψy) ψy)++฀(ψx, ฀(ψx,ψ)y ψ)y (viii) ———————————————————————————————– ———————————————————————————————– ———————————————————————————————– By using the the above above eight relations, relations, the equation equation () turns turns into ———————————————————————————————– ———————————————————————————————– By using above eight eight relations, the the equation () () turns into into By () By using using the the above above eight eightrelations, relations,the theequation equation ()turns turnsinto into 2 2(T(xy) (xy) ฀ ฀ TT(x)y (x)y ฀ ฀฀(x, ฀(x,y))ψ y))ψ22 = =0. 0. 2(T 2(T (xy) ฀ T (x)y ฀ ฀(x, y))ψ2 2 = 0. 2(T 2(T(xy) (xy)฀฀TT(x)y (x)y฀฀฀(x, ฀(x,y))ψ y))ψ ==0.0. Since R R is is aa 2-torsion 2-torsion free free ring, ring, the the above above equation equation reduces reduces to(T (T (xy)฀฀ Since 2-torsion free ring, the above equation reducesto to (T2(xy) (xy) ฀ 2 = free 3 =reduces 2 = 0, and this means that ψ 0. Hence, ψ Rψ T (x)y ฀ ฀(x, y))ψ Since R is a 2-torsion ring, the above equation to (T (xy) Since R is a 2-torsion free ring, the above equation reduces to (xy) ฀ 22 = 0, and this means that ψ 33 = 0. Hence, ψ 2(T 22฀ 2 T (x)y ฀ ฀(x, y))ψ Rψ = y))ψ22 = 0, and this means that ψ3 3 = 0. Hence, ψ2 2Rψ = 2 2 4 3 2 ψ R= =฀ ψψ Ry))ψ = {0}. {0}. semiprimeness R that ψRψ ==0.= T ฀(x, ==Thus, 0,0, and this means ψψRψ T(x)y ฀ψψ ฀(x, y))ψ andthe this meansthat thatψof ψof= =0.implies 0.Hence, Hence, 4(x)y 33R ψ R = {0}. Thus, Thus, the the semiprimeness semiprimeness ofR Rimplies impliesthat thatψψ2222==0.0. 4 3 4 3 2 ψ = ofofRRimplies that ψψR=is= ψR R= = ψψ ψψ R Rwe = {0}. {0}. Thus, theψsemiprimeness semiprimeness implies that Furthermore, we haveThus, ψRψthe = R = {0}. {0}. Reusing the fact that is0.a0. Furthermore, have have ψRψ ψRψ = = ψψ2222R R= = {0}. Reusing Reusing the the fact factthat thatR R isaa semiprime ring, it is concluded that ψ = 0. This is exactly what we had Furthermore, we have ψRψ = ψ Furthermore, we have ψRψ = ψ R = {0}. Reusing the fact that R isa a R = {0}. Reusing the fact that R is semiprime ring, it it is is concluded concluded that that ψψ = = 0. 0. This This isis exactly exactlywhat whatwe wehad had to prove. ring, ฀ itit isisconcluded semiprime ring, concludedthat thatψψ==0.0.This Thisisisexactly exactlywhat whatwewehad had semiprime to prove. ฀ to prove. prove. ฀ ฀ to The Previous Previous theorem theorem implies the the following corollaries. corollaries. The theorem implies implies the following following corollaries. The Previous Previous theorem theorem implies impliesthe thefollowing followingcorollaries. corollaries. The Corollary 2.10. 2.10. Let Let R R be be aa 2-torsion 2-torsion free free semiprime semiprime ring and and σ, T : Corollary 2.10. Let R be a 2-torsion 2free semiprime ring ring and σ,σ,TT : : R →R R be be additive additive maps such that T (x22free Corollary 2.10. Let Let R be be 2-torsion ring and σ, Corollary 2.10. R aa 2-torsion )free = semiprime Tsemiprime (x)σ(x) for ring alland x ∈σ,R. T TIf : : R → additive maps maps such such that that TT(x (x2 2)) = = TT(x)σ(x) (x)σ(x) for for all all xx∈∈R. R. IfIf R→ →R R be be additive additive maps maps such suchthat thatTT(x (x) )==TT(x)σ(x) (x)σ(x)for forallallx x∈∈R.R.IfIf R AAGENERAL GENERAL CHARACTERIZATION CHARACTERIZATION OF ADDITIVE OF ADDITIVE ... ... 133 133 T (xy)σ(z) (xy)σ(z)฀฀TT(xy)z (xy)z ฀฀ T (x)σ(yz) T (x)σ(yz) + T+ (x)σ(y)z T (x)σ(y)z =0= for0allforx, all y, z x, ∈ y, R,z ∈ R, then then TT(xy) (xy)==T T(x)σ(y) (x)σ(y) forfor all all x, yx,∈yR. ∈ R. 2 ) 2= Proof. Proof.Note Notethat thatT (x T (x )= T (x)σ(x) T (x)(x + σ(x) x)(x)x = T+(x)x + T (x)σ(x) = T= (x)(x + σ(x) ฀ x) ฀ =T (x)(σ(x)฀฀x)x)forforallall ∈ R. defining = T (x)(σ(y) ฀ y) and T (x)(σ(x) x ∈x R. By By defining ฀(x,฀(x, y) =y) T (x)(σ(y) ฀ y) and using the thehypothesis, hypothesis, have using wewe have ฀(xy,z) z)฀฀฀(x, ฀(x,yz) yz) ฀(x, T (xy)(σ(z) ฀ T (x)(σ(yz) ฀(xy, ++ ฀(x, y)zy)z = T=(xy)(σ(z) ฀ z)฀ ฀ z) T (x)(σ(yz) ฀ yz) ฀ yz) T (x)(σ(y) ฀ y)z + T+(x)(σ(y) ฀ y)z T (xy)σ(z) ฀ T (xy)z ฀ T (x)σ(yz) = T=(xy)σ(z) ฀ T (xy)z ฀ T (x)σ(yz) T (x)yz T (x)σ(y)z ฀ T (x)yz + T+(x)yz + T+ (x)σ(y)z ฀ T (x)yz T (xy)σ(z) ฀ T (xy)z ฀ T (x)σ(yz) + T (x)σ(y)z = T=(xy)σ(z) ฀ T (xy)z ฀ T (x)σ(yz) + T (x)σ(y)z = 0.= 0. It means meansthat that฀฀is isa l-semi a l-semi Hochschild 2-cocycle. By using Theorem It Hochschild 2-cocycle. By using Theorem 2.9, 2.9, it isis obtained obtainedthat that it (xy)==T (x)y T (x)y + ฀(x, y)T= T (x)y + T (x)(σ(y) TT(xy) + ฀(x, y) = (x)y + T (x)(σ(y) ฀ y) ฀ y) T (x)y T (x)σ(y) T (x)y ==T (x)y + T+(x)σ(y) ฀ T฀ (x)y T (x)σ(y) ==T (x)σ(y) for all all x,x,yy∈∈R.R. ฀฀ for Corollary2.11. 2.11.Let Let a 2-torsion semiprime ringδ and δ:R→ Corollary RR be be a 2-torsion free free semiprime ring and :R→ 2 2 R be be aaJordan Jordangeneralized generalized σ-derivation, ) = δ(x)σ(x)+σ(x)d(x) R σ-derivation, i.e. i.e. δ(x δ(x ) = δ(x)σ(x)+σ(x)d(x) for each eachxx∈∈RRand andsome some σ-derivation on Assume R. Assume for σ-derivation d ond R. that Tthat = δ T฀ = d δ฀d has the thefollowing followingproperty: property: has (xy)σ(z)฀฀TT(xy)z (xy)z T (x)σ(yz) T (x)σ(y)z = 0 all f orx,all TT(xy)σ(z) ฀฀ T (x)σ(yz) + T+ (x)σ(y)z = 0 for y, z x, ∈ y, R.z ∈ R. Then, δδisisaageneralized generalized σ-derivation. Then, σ-derivation. Proof.We Wehave have Proof. 22 2 2 2 2 δ(x ฀ d(x TT(x(x ) )==δ(x ) ฀) d(x ) ) δ(x)σ(x) + σ(x)d(x) ฀ d(x)σ(x) ฀ σ(x)d(x) ==δ(x)σ(x) + σ(x)d(x) ฀ d(x)σ(x) ฀ σ(x)d(x) (δ(x) ฀ d(x))σ(x) ==(δ(x) ฀ d(x))σ(x) T (x)σ(x) ==T (x)σ(x) 134 A. HOSEINI for all x ∈ R. According to the previous corollary, T (xy) = T (x)σ(y) for all x, y ∈ R. Hence, δ(xy) = T (xy) + d(xy) = T (x)σ(y) + d(x)σ(y) + σ(x)d(y) = δ(x)σ(y) + σ(x)d(y) for all x, y ∈ R. The following corollary has been proved by Vukman in [10]. Now, we present an alternative proof for it. Corollary 2.12. Let R be a 2-torsion free semiprime ring and δ : R → R be a Jordan generalized derivation. In this case δ is a generalized derivation. Proof. We have the relation δ(x2 ) = δ(x)x + xd(x) for all x ∈ R, where d is a Jordan derivation of R. It follows from Theorem 1 of [4] that d is a derivation. The proof is completed by substituting σ = id in Corollary 2.11. ฀ Acknowledgements The author is greatly indebted to the referee(s) for his/her valuable suggestions and careful reading of the paper. References [1] S. Ali and C. Haetinger, Jordan ฀-centralizers in rings and some applications, Bol. Soc. Paran. Mat., 26 (2008), 71-80. [2] M. Ashraf, N. Rehman, Sh. Ali, and M. R. Mozumder, On generalized (θ, φ)-derivations in semiprime rings with involution, Math. Slovaca, 62 (3) (2012), 451-460. [3] M. Ashraf, On left (θ, ϕ)-derivations of prime rings, Archivum Mathematicum, 41 (2) (2005), 157-166. [4] M. Brešar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc, 140 (4) (1988), 1003-1006. A GENERAL CHARACTERIZATION OF ADDITIVE ... 135 [5] M. Brešar and J. Vukman, Jordan derivations on prime rings, Bull. Austral. Math. Soc, 3 (1988), 321-322. [6] J. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc., 53 (1975), 1104-1110. [7] G. Dales, et al., Introduction to Banach Algebras, Operators and Harmonic Analysis, Cambridge University press, 2002. [8] I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc., 8 (1957), 1104-1110. [9] J. Vukman and I. Kosi-Ulbl, Centralizers on rings and algebras, Bull. Austral. Math. Soc., 71 (2005), 225-234. [10] J. Vukman, A note on generalized derivations of semiprime rings, Taiwanese Journal of Mathematics, 11 (2) (2007), 367-370. [11] B. Zalar, On centralizers of semiprime rings, Comment. Math. Univ. Carolin, 32 (4) (1991), 609-614. Amin Hosseini Department of Mathematics Assistant Professor of Mathematics Kashmar Higher Education Institute Kashmar, Iran E-mail: A.Hosseini@mshdiau.ac.ir