JORDAN *-DERIVATIONS ON PRIME AND SEMIPRIME *-RINGS د.عبد الرحمن حميد مجيد وعلي عبد عبيد الطائي كلية العلوم جامعة بغداد العراق.

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1 JORDAN *-DERIVATIONS ON PRIME AND SEMIPRIME *-RINGS A.H.Majeed Department of mathematics, college of science, University of Baghdad Mail: A.A.ALTAY Department of mathematics, college of science, University of Baghdad Mail: مشتقات-* جوردن يف احللقات-* االولية والشبه االولية د.عبد الرحمن حميد مجيد وعلي عبد عبيد الطائي كلية العلوم جامعة بغداد العراق Abstract Let R be a 2-torsion free *-ring, and d: R R be a Jordan *-derivation. In this paper we prove the following results: (1) If R is a non-commutative prime *-ring, and d(h) h + h d(h) Z(R) for all h H(R), then d(h) =0 for all h H(R).(2) If R be a noncommutative prime *-ring, and d([x,y])= [x,y] for all x, y R, then R is normal *- ring.(3) If R is a semiprime *-ring, then there is no d satisfies d(xy+yx)=xy+yx for all x, y R, where H(R)={x; x R s.t x*=x }. أملستخلص لتكه R حلقة-* طليقة االلتواء مه الىمط 2,و لتكهR R :d دالة مشتقة-* جوزدان في هرا البحث سىبسهه,H(R) فان h لكل Z(R) الىتائج التالية:) 1 ( اذا كان R حلقة-* اولية )غيس ابدالية( وان d(h) d(h) h + h في في d(h)=0 لكل h في )2).H(R) اذا كان R حلقة-* اولية )غيس ابدالية( وان [x,y] d([x,y])= لكل x,y في R, فان d(xy+yx)=xy+yx لكل d R تكون حلقة-* سوية.) 3 (اذا كان R حلقة-* شبه اولية,فاوه ال توجد دالة تحقق 1. Introduction. H(R)={x; x R s.t x*=x عىدما{.R x,yفي Throughout, R will represent an associative ring with center Z(R). A ring R is n- torsion free, if nx = 0, x R implies x = 0, where n is a positive integer. Recall that R is prime if arb = (0) implies a = 0 or b = 0, and semiprime if ara = (0) implies a =0. A mapping *: R R is called an involution if (x+y)*=x*+y*(additive), (xy)* = y* x* and (x)** = x for all x, y R. A ring equipped with an involution is called *-ring [1]. An element x in a *-ring R is said to be hermitian if x* = x and skew-hermitian if x* = -x. The 1

2 sets of all hermitian and skew-hermitian elements of R will be denoted by H(R) and S(R), respectively. If R is 2-torsion free then every x R can be uniquely represented in the form 2x = h + k where h H(R) and k S(R). An element x R is called normal element if xx* =x*x, and if all the elements of R are normal then R is called a normal ring (see [2]). As usual the commutator xy - yx will be denoted by [x, y]. We shall use basic commutator identities [xy, z] = [x, z]y + x[y, z] and [x, yz] = [x, y]z + y[x, z] for all x,y,z R. An additive mapping d: R R is called a derivation if d(xy) = d(x)y + xd(y) holds for all pairs x,y R, and is called a Jordan derivation in case d(x 2 ) = d(x)x + xd(x) is fulfilled for all x R. Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein [3] asserts that every Jordan derivation on a prime ring of characteristic different from 2 is a derivation. Cusack [4] generalized Herstein s theorem to 2-torsion free semiprime ring. An additive mapping d: R R is called a *-derivation if d(xy) = d(x)y* + xd(y) holds for all pairs x,y R and is called a Jordan *-derivation in case d(x 2 ) = d(x)x* + xd(x) is fulfilled for all x R, the concepts of *-derivation and Jordan*- derivation were first mentioned in [5] for more details see also ( [6] and [7]]). Every *- derivation is a Jordan *-derivation but the converse in general not true, for example let R be a 2-torsion free semiprime *-ring and let a R such that [a,x] 0, for some x R, define a map d: R R as follows, d(x)=ax*-xa for all x R, then d is a Jordan *-derivation but not a *-derivation. 2. The Main Results In the present note, we explore more about Jordan *-derivations on prime and semiprime *-rings. We will provide some properties for Jordan *-derivations on semiprime *-ring. Also we will study a normalization of a non-commutative prime *-ring. We begin with the following known results. Theorem 2.1. [5]. Let R be a non-commutative prime *-ring of characteristic different from 2, then R is normal ring if and only if there exists a nonzero commuting Jordan *- derivation. Lemma 2.2. [8]. Let R be a prime*-ring such that a H(R) b= 0, where either a H(R) or b H(R).Then either a=0 or b=0. 2

3 Lemma 2.3. [5]. Let R be a 2-torsion free non-commutative prime *-ring, and let d: R R be a Jordan *-derivation, then d(c)=0 for all c Z(R) H(R). In the following theorem we proved that, a Jordan *-derivation d on a noncommutative prime *-ring of characteristic different from 2, which satisfies d(h) h + h d(h) Z(R) for all h H(R), is finish on H(R). Theorem 2.4. Let R be a non-commutative prime *-ring of characteristic different from 2, and d: R R be a Jordan *-derivation which satisfies d(h) h + h d(h) Z(R) for all h H(R), then d(h) =0 for all h H(R). To prove above theorem we need the following lemmas Lemma 2.5. Let R be a 2-torsion free non-commutative prime *-ring, and d: R R be a Jordan *-derivation which satisfies d(h) h + h d(h) Z(R) for all h H(R), then d(h 2 ) =0 for all h H(R). Proof: We have d(h)h+hd(h)= d(h 2 ) Z(R) for all h H(R), (1) Replace h by h 2 in (1) we get d(h 2 )h 2 +h 2 d(h 2 )=2h 2 d(h 2 ) Z(R) for all h H(R), (2) Therefore, [2h 2 d(h 2 ),y]=0 for all h H(R), y R, (3) From the relation (1), and since R is a 2-torsion free we get [h 2,y] d(h 2 )=0 for all h H(R), y R, (4) Putting yz for y in (4) we obtain [h 2,y] z d(h 2 )=0 for all h H(R), y, z R, (5) By primness of a *-ring R, we get either d(h 2 )=0 for all h H(R) or h 2 Z(R) for all h H(R), If h 2 Z(R), then h 2 Z(R) H(R). Therefore by Lemma 2.3 we get d(h 2 ) =0 for all h H(R). Proof of Theorem2.4: By using Lemma2.5 d(h 2 ) =0 for all h H(R). (6) 3

4 Linearization the relation (6) we get d(hk+kh)=d(h)k+hd(k)+d(k)h+kd(h)=0 for all h, k H(R). (7) If we replace k by (hk+kh) H(R) in (7), and since is 2-torsion free we obtain d(hkh)=d(h)k h +hd(k)h +hkd(h)=0 for all h, k H(R), (8) Putting h for k and (h 1 k h 1 ) for h in (8), we get Now replace k by (h 1 k h 1 ) in (8) we obtain (h 1 kh 1 ) d(h) (h 1 kh 1 ) =0 for all h,h 1, k H(R). (9) d(h) (h 1 k h 1 ) h+ h (h 1 k h 1 ) d(h)=0 for all h,h 1, k H(R). (10) Left multiplication the relation (10) by (h 1 k h 1 ), and using (9) we get (h 1 k h 1 ) h (h 1 k h 1 ) d(h)=0 for all h,h 1, k H(R). (11) Linearization the relation (11) on h we get (h 1 k h 1 ) h (h 1 k h 1 ) d(l) + (h 1 k h 1 ) l (h 1 k h 1 ) d(h)=0 for all h,h 1, l, k H(R). (12) Replace l by (bab) in (12), we get Left and right multiplication (13) by b we get (h 1 k h 1 ) (bab) (h 1 k h 1 ) d(h)=0 for all h,h 1,a,b,k H(R), (13) b(h 1 k h 1 ) bab (h 1 k h 1 ) d(h) b=0 for all h,h 1,a,b,k H(R), (14) Setting b=h=h 1, then by using (6) we get (h 2 k h 2 ) a (h 2 k h 2 ) d(h)=0 for all h,h 1,a,b,k H(R), (15) Since (h 2 k h 2 ) H(R), then by using Lemma2.2, we get, either (h 2 k h 2 ) =0, or (h 2 k h 2 ) d(h)=0 for all h, k H(R). Therefore Then also by using Lemma 2.2, we obtain Linearization the above relation we get (h 2 k h 2 ) d(h)=0 for all h,k H(R). (16) h 2 d(h)=0 for all k H(R), (17) (h k+kh) d(k)+( h k+kh)d(h)+h 2 d(k)+k 2 d(h)=0 for all h,k H(R). (18) Replace k by - k in the above relation and comparing the relation so obtained with the relation (18) we get 4

5 ( h k+kh)d(h)+h 2 d(k)=0 for all h,k H(R). Putting ( h k+k h) for k in the above relation we obtain 2 h k h d(h)+ h 2 k d(h)=0 for all h,k H(R). (19) Right multiplication the above relation by h and using (6) we get h 2 k d(h) h =0 for all h,k H(R). By using Lemma2.2, we get if h 2 =0, then from relation (20), we get h k d(h) h =0 for all k H(R), therefore we obtain d(h) h =0 for all h H(R), let h=h +h 1 k h 1, then we get d(h) h 1 k h 1 =0 for all h,h 1, k H(R). Then from above relation and Lemma2.2 we get d(h)=0 for all h H(R). The proof of Theorem 2.4, is complete. In the following proposition we will give a condition on a Theorem2.4 to get R is a normal *-ring. Proposition 2.6. Let R be a non-commutative prime *-ring of characteristic different from 2, and d: R R be a Jordan *-derivation which satisfies d(h) h + h d(h) Z(R) for all h H(R), and [d(s),h] Z(R) for all h H(R), and s S(R), then R is normal *-ring. Proof: we have, [d(s),h] Z(R) for all h H(R), and s S(R), Since h 2 H(R), for all h H(R), [d(s), h 2 ] Z(R), for all s S(R), and h H(R). By assumption [d(h),s] Z(R) for all h H(R), s S(R), then we get Hence, 2h[d(s), h] Z(R), for all s S(R), and h H(R). 2[d(s), h[d(s), h]]=0 for all s S(R), and h H(R). Since [d(s),h] Z(R), and R is a 2-torsion free, then from above relation we get [d(s),h] 2 =0 for all s S(R), and h H(R). By the semiprimness of R, we get [d(s),h] =0 for all s S(R), and h H(R). To prove [d(x),x]=0, Since R be a 2-torsion free we only show, 4[d(x),x]=0 for all x R, we have for all x R then (2x=s+h for s S(R), and h H(R)), therefore 5

6 4[d(x),x]=[d(2x),2x]=[d(s+h), s+h] for s S(R), and h H(R). Hence, 4[d(x),x]=[d(s),s]+[d(s),h]+[d(h), h]+[d(h), s] From above relation and Theorem2.4, and characteristic of R not equal 2, we get [d(x),x] =0 for all x R. Then from Theorem2.1, we get R is normal *-ring. Daif and Bell[9] established that a semiprime ring R must be commutative if it admits a derivation d such that d([x,y])=[x,y] for all x, y R. In the following theorem we will prove if R be a 2-torsion free non-commutative prime *-ring, and d: R R be a Jordan *- derivation which satisfies d([x,y])= [x,y] for all x, y R, then R is normal *-ring, but under some conditions on a *-ring R. Theorem 2.7. Let R be a 2-torsion free non-commutative prime *-ring, and let d: R R be a Jordan *-derivation which satisfies d([x,y])= [x,y] for all x, y R, then R is normal *-ring. Proof: we have d([x,y])=[x,y] for all x, y R, (20) Since [x 2,y]=[x,y]x+x[x,y] then from (20) we get d([x,y]x+x[x,y])= [x,y] x*+[x,y]d(x)+d(x)[x,y]*+x[x,y] = [x,y]x+x[x,y] for all x, y R, (21) Replace x by [h,s] H(R), where h H(R), and s S(R), in (21) and using (20)we obtain, x[x,y]*+[x,y]x=0 for all y R, Replace y by xy, we get, x[x,y]*x+x[x,y]x=0 for all y R, hence we get [[x,y],x] x =0 for all y R, (22) Define an additive mapping, f x : R R by f x (y)=[x,y], then f x is inner derivation and from (22) we get Therefore, one can show from relation (23) that f x 2 (y) x =0 for all y R. (23) x f x 2 (y)=0 for all y R. (24) 6

7 Putting yw from y in (23) we get, f x ²(y)wx+2f x (y)f x (w)x=0 for all y, w R, left multiplication by x and using (24), therefore since R is a 2-torsion free we obtain xf x (y)f x (w)x=0 for all y, w R, (25) Putting yv for y in (25) we get, xf x (y)vf x (w)x+ xyf x (v)f x (w)x=0 for all y,w,v R, replace v by xv, and using (25) we get, xf x (y)x v f x (w)x =0 for all y,w,v R, Setting y=w, and putting vx for v, we obtain, xf x (y)x v x f x (y)x =0 for all y,v R, By primness of a *-ring R, we get, x f x (y) x =0 for all y R, Putting yw from y we get, x f x (y) w x+ x yf x (w) x=0 for all y, w R, since x f x (y) w x-xf x (y)xw-yxf x (w)x+ x yf x (w) x=0 for all y, w R, therefore f x (y) f x (w) x=xf x (y)f x (w) for all y, w R. Then from relation (25) we get, x 2 f x (y)f x (w)=0 for all y, w R, replace w by rx 2 y we obtain, x 2 f x (y)rx 2 f x (y)=0 for all y, w R, Since R is a *-prime ring we get, x 2 f x (y) = 0 for all y R, Putting wy from y in the above relation we get x 2 w f x (y)=0 for all y, w R, (26) Putting yw for w in the relation (26) we get x 2 y w f x (y)=0 for all y, w R, (27) Left multiplication the relation (27) by y we get y x 2 w f x (y)=0 for all y, w R, (28) Comparing the relations (27) and (28) we obtain [x 2,y] w [x,y]=0 for all y, w R, (29) Replace w by wx in (29) we get [x 2,y] w x [x,y]=0 for all y, w R, (30) Right multiplication the relation (29) by x we get [x 2,y] w [x,y] x=0 for all y, w R, (31) Comparing the relations (30) and (31) we obtain [x 2,y] w [x 2,y]=0 for all y, w R, (32) By primness of a *-ring R, x 2 Z(R), and hence x 2 Z(R) H(R). Therefore by Lemma 2.3 we get d(x 2 ) =0, then we obtain 0=d(x 2 )=2 x 2, therefore x 2 =0, from relations (21) one can obtain x [x,k]=0 for all k H(R), Therefore, x k x=0 for all for all k H(R), then by using Lemma 2.2 we get[s,h]=0 7

8 for all h H(R), s S(R), hence we obtain R is a normal *-ring. M. Hongan In [10] proved that, if R is a 2-torsion free ring with an identity element. Then there is no a derivation d: R R such that d(xy+yx)=xy+yx for all x, y R. In the following Proposition we will give a result similar to the result of M. Hongan [10], but in case Jordan*-derivation. Proposition 2.8. Let R be a 2-torsion free semiprime *-ring, then there is no Jordan*- derivation d: R R which satisfies d(xy+yx)=xy+yx for all x, y R. To prove above proposition we the following lemma Lemma 2.9. Let R be a semiprime *-ring, if there exist an element h H(R) which satisfied h x h=0 for all x H(R), then h=0. Proof: We have, h x h=0 for all x H(R), Since (y+y*) H(R), for all y R, hence h y h=- h y* h for all y R. Also since (yhy*) H(R), therefore h y h y* h=- h y h y h=0 for all y R, linearization we get, h y h z h+ h zh y h=0 for all z,y R, left multiplication by y h we get, h y h z h y h=0 for all z,y R. By the semiprimness of R, we get h=0. Proof of Proposition2.8: If d is a non-zero Jordan *-derivation, then we have d(xy+yx)=d(x)y*+xd(y)+d(y)x*+yd(x)=xy+yx for all x, y R. (33) Setting y=ab+ba, x=cd+dc where a,b,c,d H(R), then from (33) we get xy+yx=0 for all y=ab+ba, x=cd+dc where a,b,c,d H(R), (34) Now setting x= (cd+dc) 2, y=(ab+ba) in (33), we get x 2 y+yx 2 =0 for all y=ab+ba, x=cd+dc where a,b,c,d H(R), (35) Left multiplication the relation (34) by x we get x 2 y+xyx=0 for all y=ab+ba, x=cd+dc where a,b,c,d H(R), (36) Right multiplication the relation (34) by x we get y x 2 +xyx=0 for all y=ab+ba, x=cd+dc where a,b,c,d H(R), (37) According to (35), (36) and (37) we get x( ab+ba ) x=0 for all a,b H(R), (38) 8

9 Replace a by ab+ba in (49) we obtain Lift and right multiplying by a, we get x a b a x=0 for all a,b H(R), a x a b a x a=0 for all a,b H(R), By Lemma2.9 we get a (cd+dc) a=0 for all a,d,c H(R), Replace c by cd+dc we get H(R) =0, therefore x=-x* for all x R, hence R=0, which is contradiction. Then assume d=0, therefore xy+yx=0 for all x,y R, then also we get contradiction. Therefore d(x)=0 for all x R. References [1] I. N. Herstein: Topics in ring theory, University of Chicago Press, [2] F. J. Dyson, Quaternion determinants, Helvetica Physica Acta, 45 (1972), [3] I. N. Herstein: Jordan derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), [4] J. Cusack: Jordan derivations on rings, Proc. Amer. Math. Soc. 53 (1975), [5] M. BreŠar and J.Vukman, On some additive mappings in rings with involution, Aequationes Math. 38(1989), [6]M. N. Daif and M. S. Tammam, On Jordan and Jordan*-Generalized derivations in Semiprime rings with Involution. Int. J. Contemp. Math. Sciences, Vol. 2, (2007), no. 30, [7] J. Vukman: A note on Jordan*- derivations in semiprime rings with involution,international Mathematical Forum, no. 13,1,( 2006), [8] T.K. Lee. On nilpotent derivations of semiprime ring with involution, Chinese J. Math. Vol. 23, No. 2, pp , June (1995). [9] M. N. Daif and H. E. Bell, Remark on derivations on semiprime rings, Internat. J. Math. Sci. 15 (`1992), no. 1, [10] M. Hongan, A note on semiprime rings with derivations, Internat. J. Math. & Math. Sci. VOL. 20 (1997), no. 2,