SOME ALGEBRAIC IDENTITIES IN RINGS AND RINGS WITH INVOLUTION

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1 Palestie Joural of Mathematics Vol. 607, Palestie Polytechic Uiversity-PPU 07 SOME ALGEBRAIC IDENTITIES IN RINGS AND RINGS WITH INVOLUTION Chirag Garg ad R. K. Sharma Commuicated by Ayma Badawi MSC 00 Classicatios: 6W 5, 6N60, 6U80. Keywords ad phrases: Geeralized Jorda θ, ϕ-derivatio, Geeralized Jorda left θ, ϕ -derivatio, Geeralized Jorda θ, ϕ -derivatio. Abstract. I this paper, we study algebraic idetities which are i F x + = F xθx + ϕxdx +F x θx+ϕx Dx ii F x + = F xθx + i= ϕxi Dxθx i iii F x + = θx F x + i= θx i ϕx i Dx, where F ad D are additive mappigs o rig ad rig with ivolutio. Itroductio Throughout this paper R deotes a associative rig with idetity e ad ZR deotes the ceter of R. A additive mappig x x satisfyig x = x ad xy = y x is called a ivolutio. A rig equipped with a ivolutio is called -rig or rig with ivolutio. A rig R is said to be prime if for ay a, b R, arb = {0} implies either a = 0 or b = 0 ad ad is said to be semiprime if for ay a R, ara = 0 implies a = 0. Give a iteger >, a rig R is said to be -torsio free if for ay x R, x = 0 implies x = 0. A additive mappig D from R to R is said to be a derivatio if Dxy = Dxy + xdy for all x, y R ad is said to be a Jorda derivatio if Dx = Dxx + xdx for all x R. We otice that every derivatio is a Jorda derivatio but the coverse eed ot be true. Herstei [9] proved a mile stoe result which states that a Jorda derivatio o a prime rig R with characteristic differet from two is a derivatio. A brief proof ca be foud i Cusack [6]. Cusack [6] geeralized Herstei's result ad proved that if R is a semi prime rig which is -torsio free the every Jorda derivatio o R is a derivatio. We have divided this paper i two sectios. I Sectio, R is ay associative rig where as i Sectio, R is ay associative rig with ivolutio. Bre sar [5] itroduced the cocept of geeralized derivatio mappig. A additive mappig F o R is said to be geeralized derivatio if there exists a derivatio D o R such that F xy = F xy+xdy for all x, y R. A additive mappig F o R is said to be a geeralized Jorda derivatio if there exists a Jorda derivatio D o R such that F x = F xx + xdx for all x R. Vukma [] proved that if R is a -torsio free semi prime rig, the every geeralized Jorda derivatio o R is a geeralized derivatio. A additive mappig D : R R is called θ, ϕ-derivatio resp. Jorda θ, ϕ-derivatio if Dxy = Dxθy + ϕxdy resp. Dx = Dxθx + ϕxdx holds for all x, y R. A additive mappig F : R R is said to be geeralized θ, ϕ-derivatio resp. geeralized Jorda θ, ϕ-derivatio if there exists a θ, ϕ-derivatio resp. Jorda θ, ϕ-deviatio D : R R such that F xy = F xθy + ϕxdy resp. F x = F xθx + ϕxdx for all x, y R. Recetly, Dhara ad Sharma [7] proved a additive map satisfyig a idetity to be derivatio. I 03, Ashraf et al. [3] worked o additive mappigs satisfyig some algebraic idetities. I Sectio, we will prove a additive mappig satisfyig a algebraic idetity to be geeralized Jorda θ, ϕ-derivatio. I Sectio, we will study the results i rigs with ivolutio. Bresar ad Vukma [4] studied the otios of a -derivatio ad a Jorda -derivatio. Let R be a - rig. A addi-

2 ALGEBRAIC IDENTITIES IN RINGS 39 tive mappig D : R R is said to be a -derivatio resp. Jorda -derivatio if Dxy = Dxy + xdy resp. Dx = Dxx + xdx holds for all x, y R. Further, let θ, ϕ be the automorphisms o R. A additive mappig D : R R is said to be θ, ϕ - derivatio if Dxy = Dxθy + ϕxdy ad D is said to be a left θ, ϕ -derivatio if Dxy = θy Dx + ϕxdy holds for all x, y R. A additive mappig F : R R is said to be a geeralized -derivatio associated with -derivatio D if F xy = F xy + xdy holds for all x, y R. Further, let θ, ϕ be automorphisms of R. A additive mappig F : R R is said to be a geeralized θ, ϕ -derivatio resp. geeralized Jorda θ, ϕ -derivatio with associated θ, ϕ -derivatio D resp. Jorda θ, ϕ -derivatio if F xy = F xθy +ϕxdy resp. F x = F xθx +ϕxdx ad F is said to be a left geeralized θ, ϕ -derivatio resp. Jorda left geeralized θ, ϕ - derivatio with associated left θ, ϕ -derivatio D resp. Jorda left θ, ϕ -derivatio if F xy = θy F x+ϕxdy resp. F x = θx F x+ϕxdx holds for all x, y R. Vukma [] proved the followig result: Let R be a 6-torsio free semiprime -rig. Let D : R R be a additive mappig satisfyig the relatio Dxyx = Dxy x +xdyx +xydx for all x, y R. The D is a Jorda -derivatio. Ali [] exteded this result to Jorda triple θ, ϕ -derivatio. Very recetly, N.Rehma et al. [0] cosidered additive mappigs satisfyig some algebraic idetities o rig with ivolutio. I Sectio, we will dee some algebraic idetities o rig with ivolutio. Algebraic Idetity o Rig Dhara ad Sharma [8] proved a additive map satisfyig a idetity to be geeralized Jorda derivatio. Motivated by [8], we dee a idetity o a rig R ad prove the followig: Theorem.. Let be ay xed iteger, R be a +!-torsio free ay rig with idetity elemet ad θ, ϕ be two automorphisms o R. If F : R R ad D : R R are additive mappigs such that F x + = F xθx + ϕxdx + F x θx + ϕx Dx for all x R, the D is a Jorda θ, ϕ-derivatio ad F is a geeralized Jorda θ, ϕ-derivatio. Proof. We have the idetity F x + = F xθx + ϕxdx + F x θx + ϕx Dx. holds for all x R. Replacig x by e i., where e is a idetity of R, we get F e = F e + De which implies De = 0. Sice R is +!-torsio free, we get De = 0. Now replacig x by x + le i 5, where l is ay positive iteger, we get F { x + le +} = F x + leθx + le + ϕx + led {x + le } + F {x + le } θx + le + ϕx + le Dx + le Expadig the powers of x + le ad usig De = 0, we get { } + + F x l x + l x + l + e { } = F x + le θx + + l θx + l θx + l e { } + ϕx + led x + + l x + l x + l e { } + F x + + l x + l x + l e θx + le { } + ϕx + + l ϕx + l ϕx + l e Dx..3

3 40 Chirag Garg ad R. K. Sharma Usig., the above relatio ca be writte as lf θx, ϕx, e + l f θx, ϕx, e +...l f θx, ϕx, e = 0.4 for all x R. Now, replacig l by,,..., i.4 ad cosiderig the resultig system of homogeous equatios, we get that the resultig matrix of the system is a Va der Mode matrix Sice the determiet of the matrix is equal to a product of positive itegers, each of which is less tha ad R is +! torsio free. It follows that the system has oly a zero solutio. Thus f i θx, ϕx, e = 0 for all x R ad i =,,...,. Now, f θx, ϕx, e = 0 implies that + F x = + F eθx + + Dx.5 Agai sice R is +!-torsio free, we get F x = F eθx + Dx for all x R. Now, f θx, ϕx, e = 0 gives + F x = F xθx +! + F xθx + F eθx + ϕxdx +! F x + ϕxdx! Multiplyig both sides by i above equatio, we get Dx!.6 + F x = 4F xθx + 4ϕxDx + F x + F eθx + Dx.7 Usig +! torsio freeess of R ad F x = F eθx+dx, we get Dx = Dxθx+ ϕxdx, x R, hece D is a Jorda θ, ϕ-derivatio i R. Agai usig F x = F eθx+ Dx, we get F x = F eθx + Dxθx + ϕxdx = F xθx + ϕxdx, x R which implies that F is a geeralized Jorda θ, ϕ-derivatio i R. Thus the proof of theorem is completed. 3 Algebraic Idetities o Rig with Ivolutio I 04, N.Rehma et al. [0] cosidered the additive mappigs F : R R ad D : R R satisfyig the coditio F x + = F xx + i= xi Dxx i for all x R ad proved that if R is a +!-torsio free -rig with idetity, the D is a Jorda -derivatio ad F is a geeralized Jorda -derivatio o R. We will exted the results of A. Asari et al. [] to rig with ivolutio as follows: Theorem 3.. Let be ay xed iteger, R be a +!-torsio free ay rig with idetity elemet ad θ, ϕ be two automorphisms o R. If F : R R ad D : R R are additive mappigs such that F x + = F xθx + i= ϕxi Dxθx i for all x R, the D is a Jorda θ, ϕ -derivatio ad F is a geeralized Jorda θ, ϕ -derivatio. Proof. We have the idetity F x + = F xθx + ϕx i Dxθx i 3. i=

4 ALGEBRAIC IDENTITIES IN RINGS 4 for all x R. We replace x by e i 3.. Clearly e = e so that θe = ϕe = e. Hece, by -torsio freeess of R, De = 0 implies De = 0. Agai replacig x by x + le i 3., where l is ay positive iteger, we obtai F x + le + = F x + leθx + le + ϕx + le i Dxθx + le i i= = F x + lf eθx + le + ϕx + le i Dxθx + le i i= 3. for all x R. By expadig the powers of x + le, we get F x x l + = F x + lf e θx + + i= ϕx i + i ϕx i l + i i + θx i l + x l l + e θx l + i θx l l e ϕx i l l i e θx i l l i e Dx θx i 3.3 for all x R. 3.3 ca be rewritte as + + F x + + F x l + x l l + e = F x + lf eθx + F x + lf e θx l + θx l l e i i + ϕx i Dx θx i + θx i l + θx i l i= l i e + i= i + θx i l + i ϕx i l + i i ϕx i l l i e Dx θx i θx i l l i e 3.4

5 4 Chirag Garg ad R. K. Sharma for all x R. Usig 3., we have + + F x l + x l l + e = lf eθx + F x + lf e θx l + i i + ϕx i Dx θx i l + i= θx l l e θx i l l i e i i + ϕx i l + ϕx i l l i e Dx θx i i= i i + θx i l + θx i l l i e for all x R, where we deote k = 0 for k < 0 ad for k >. The above relatio ca be writte as 3.5 lf θx, ϕx, e + l f θx, ϕx, e +...l f θx, ϕx, e = for all x R. We proceed i similar way as i the proof of Theorem., we get f i θx, ϕx, e = 0, i =,,...,. Now, f θx, ϕx, e = 0 implies that + F x = F x + F eθx + Dx implies that Sice R is -torsio free, we obtai + F x = F x + F eθx + Dx 3.8 F x = F eθx + Dx 3.9 Agai f θx, ϕx, e = 0 implies that + F x = F xθx + F eθx + + Dxθx for all x R. 3.0 ca be rewritte as + ϕxdx 3.0 +F x = F xθx + F eθx ++ϕxdx+ Dxθx 3. for all x R. Sice R is -torsio free, we get + F x = F xθx + F eθx + + ϕxdx + Dxθx 3. Usig 3.9 i 3., we d + F x = + F eθx + + ϕxdx + + Dxθx 3.3 for all x R. Sice R is + -torsio free, so we have Replacig x by x i 3.9, we obtai F x = F eθx + ϕxdx + Dxθx 3.4 F x = F eθx + Dx 3.5

6 ALGEBRAIC IDENTITIES IN RINGS 43 Comparig 3.4 ad 3.5, we d that for all x R. Usig 3.6 i 3.5, we get Dx = Dxθx + ϕxdx 3.6 F x = F eθx + Dxθx + ϕxdx = {F eθx + Dx}θx + ϕxdx 3.7 for all x R. Agai, usig 3.9 i 3.7, we coclude F x = F xθx + ϕxdx. Thereby the proof of the theorem is completed. Corollary 3. [0], Theorem.. Let be ay xed iteger ad R be a +!-torsio free ay rig with idetity elemet. If F : R R ad D : R R are additive mappigs such that F x + = F xx + i= xi Dxx i for all x R, the D is a Jorda -derivatio ad F is a geeralized Jorda -derivatio. Proof. Take θ = ϕ = I, where I is the idetity map o R. Theorem 3.3. Let be ay xed iteger, R be a +!-torsio free ay rig with idetity elemet ad θ, ϕ be two automorphisms o R. If F : R R ad D : R R are additive mappigs such that F x + = θx F x + i= θx i ϕx i Dx for all x R, the D is a Jorda left θ, ϕ -derivatio ad F is a geeralized Jorda left θ, ϕ -derivatio. Proof. We have the idetity F x + = θx F x + θx i ϕx i Dx 3.8 for all x R. We replace x by e i 3.. Clearly e = e so that θe = ϕe = e. Hece, by -torsio freeess of R, De = 0 implies De = 0. Agai replacig x by x + le i 3.8, where l is ay positive iteger, we obtai i= F x + le + = θx + le F x + le + θx + le i ϕx + le i Dx i= = θx + le F x + lf e + θx + le i ϕx + le i Dx i= 3.9 for all x R. By expadig the powers of x + le, we get F x + + = + + x l + θx l + i θx + x l l + e + θx i + θx i l + i= i l i e ϕx i + ϕx i l + θx l l e F x + lf e i θx i l i ϕx i l l i e Dx 3.0

7 44 Chirag Garg ad R. K. Sharma for all x R. 3.0 ca be rewritte as + + F x + + F x l + x l l + e = θx F x + lf e + θx l + θx l l e F x + lf e i i + θx i + θx i l + θx i l i= i l i e ϕx i Dx + θx i + i= i i + θx i l l i e ϕx i l l i e Dx θx i l i ϕx i l 3. for all x R. Usig 3.8, we have + + F x l + x l l + e = θx lf e + θx l + θx l l e F x + lf e i i + θx i l + θx i l l i e ϕx i Dx i= i i + θx i + θx i l + i= i l i e ϕx i l + i ϕx i l l i e θx i l Dx 3. for all x R, where we deote k = 0 for k < 0 ad for k >. The above relatio ca be writte as lf θx, ϕx, e + l f θx, ϕx, e +...l f θx, ϕx, e = for all x R. We proceed i the similar way as i the proof of Theorem., we get f i θx, ϕx, e = 0, i =,,...,. Now, f θx, ϕx, e = 0 implies that + F x = F x + θx F e + Dx implies that Sice R is -torsio free, we obtai + F x = F x + θx F e + Dx 3.5 F x = θx F e + Dx 3.6 Agai f θx, ϕx, e = 0 implies that + F x = θx F x + θx F e + + θx Dx + ϕxdx 3.7

8 ALGEBRAIC IDENTITIES IN RINGS 45 for all x R. 3.7 ca be rewritte as +F x = θx F x+ θx F e++ϕxdx+ θx Dx 3.8 for all x R. Sice R is -torsio free, we get + F x = θx F x + θx F e + + ϕxdx + θx Dx 3.9 Usig 3.6, 3.9 becomes + F x = + θx F e + + ϕxdx + + θx Dx 3.30 for all x R. Sice R is + -torsio free, so we have Replacig x by x i 3.6, we obtai Comparig 3.3 ad 3.3, we d that for all x R. Usig 3.33 i 3.3, we get F x = θx F e + ϕxdx + θx Dx 3.3 F x = θx F e + Dx 3.3 Dx = θx Dx + ϕxdx 3.33 F x = θx F e + θx Dx + ϕxdx = θx {θx F e + Dx} + ϕxdx 3.34 for all x R. Agai, usig 3.6 i 3.34, we coclude F x = θx F x + ϕxdx. Thereby the proof of the theorem is completed. Corollary 3.4. Let be ay xed iteger, R be a +!-torsio free ay rig with idetity elemet ad ϕ be a automorphism o R. If F : R R ad D : R R are additive mappigs such that F x + = x F x + i= x i ϕx i Dx for all x R, the D is a Jorda skew left -derivatio ad F is a geeralized Jorda skew left -derivatio. Proof. Take θ = I, where I is the idetity map o R. Refereces [] S. Ali ad A. Foser, O Jorda α, β -derivatios i semiprime -rigs, It. J. Algebra 4, [] A. Asari ad F. Shujat, Additive mappigs satisfyig algebraic coditios i rigs, Red. Circ. Mat. Palermo 63, [3] M. Ashraf, N. Rehma ad A.Z. Asari, O additive mappigs satisfyig a algebraic coditios i rigs with idetity, J. Adv. Pure Math. 5, [4] M. Bre sar ad J. Vukma, O some additive mappigs i rig with ivolutio, Aequatioes Math. 38, [5] M. Bre sar, O the distace of the compositio of two derivatios to the geeralized derivatio, Glasgow Math. 33, [6] J.M. Cusack, Jorda derivatios o rigs, Proc. Amer. Math. Soc., [7] B. Dhara ad R.K. Sharma, O additive mappigs i semi prime rig with left idetity, Algebra Group ad Geom. 5, [8] B. Dhara ad R.K. Sharma, O additive mappigs i rigs with idetity, Iter Math. Forum 5, [9] I.N. Herstei, Jorda derivatios of prime rigs, Proc. Amer. Math. Soc. 8, [0] N. Rehma, A. Asari ad T. Bao, O geeralized Jorda -derivatio i rigs, J. Egy. Math. Soc., [] J. Vukma ad I. Kosi-Ulbl, Equatio related to cetralizers i semiprime rigs, Glasik Matematicki 38,

9 46 Chirag Garg ad R. K. Sharma [] J. Vukma, A ote o Jorda -derivatio i semiprime rigs with ivolutio, It. Math. Forum 3, Author iformatio Chirag Garg ad R. K. Sharma, Departmet of Mathematics, Idia Istitute of Techology Delhi, Hauz Khas, New Delhi-006, INDIA. garg88chirag@gmail.com Received: Jue 9, 05. Accepted: February 6, 06.