Huang, Shuliang

Rehman, Nadeem ur

English

Repository of the Academy's Library

Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 20 (2019), No. 2, pp. 1175–1183 DOI: 10.18514/MMN.2019.2450ON LIE IDEALS AND SYMMETRIC GENERALIZED.˛;ˇ/-BIDERIVATION IN PRIME RINGNADEEM UR REHMAN AND SHULIANG HUANGReceived 17 November, 2017Abstract. LetR be a prime ring with char.R/¤ 2. A biadditive symmetric map WRR!Ris called symmetric .˛;ˇ/-biderivation if, for any fixed y 2R, the map x 7!.x;y/ is a .˛;ˇ/-derivation. A symmetric biadditive map   W RR! R is a symmetric generalized .˛;ˇ/-biderivation if for any fixed y 2R, the map x 7!   .x;y/ is a generalized .˛;ˇ/-derivation of Rassociated with the .˛;ˇ/-derivation .:;y/. In the present paper, we investigate the commut-ativity of a ring having a generalized .˛;ˇ/-biderivation satisfying certain algebraic conditions.2010 Mathematics Subject Classification: 16N60; 16U80; 16W25Keywords: Lie ideals, symmetric biderivations, generalized .˛;ˇ/-biderivations1. INTRODUCTIONThroughout this article R denotes an associative ring with center Z. An additivesubgroup L of R is said to be a Lie ideal of R if Œl; r 2 L, 8 l 2 L and r 2 R. ALie ideal L is called square closed if u2 2 L 8 u 2 L. And it is easy to check that2uv 2 L 8 u;v 2 L.A derivation d W R! R is an additive map such that d.xy/ D d.x/yCxd.y/ 8x;y 2 R. An additive map F W R! R is a generalized derivation if there existsa derivation d W R! R such that F .xy/ D F .x/yC xd.y/ holds 8 x;y 2 R. If' W RR! R is a symmetric map .'.x;y/ D '.y;x/ 8 x;y 2 R) the map  WR! R defined by .x/ D '.x;x/ is the trace of '. If ' is also biadditive (i.e.,additive in both arguments), its trace  satisfies .xCy/D .x/C.y/C2'.x;y/, 8x;y 2R. A symmetric biadditive map ' WRR!R is a symmetric biderivation if'.xy;´/D'.x;´/yCx'.y;´/8 x;y;´2R. The concept of symmetric biderivationwas introduced by G. Maksa [9]. A symmetric biadditive map  W RR! R is asymmetric left bicentralizer if .xy;´/ D .x;´/y ( and consequently .x;y´/ D.x;y/´/ 8 x;y;´ 2R.Symmetric biderivations were proved to be related to the general solution of somefunctional equations (see [11]). The maps .u;v/ 7! 	Œu;v;	 2 C , are typical ex-amples of biderivations and they were called inner biderivations. Here C is the exten-ded centroid ofR, that is, the center of the two-sided Martindale quotient ringQ (wec 2019 Miskolc University Press1176 NADEEM UR REHMAN AND SHULIANG HUANGrefer the reader to [3] for more details). In [7], it is shown that every biderivation ofa noncommutative prime ring R is inner. In [5], this result is extended to semiprime.In [14], it is proved that if ' is a nonzero symmetric biderivation, where R is a primering of char.R/¤ 2, with the property:'.x;x/x D x'.x;x/;x 2R: (1.1)then R is commutative. He also proved that if '1;'2 are nonzero biderivations onR, D is a symmetric biadditive map and 1.2.x// D d.x/ holds 8 x 2 R, where1; 2, and d are the traces of '1;'2, and D, respectively, then either '1 D 0 or'2 D 0. Let’s mention two results proved in [15]. The first one states that if '1and '2 are symmetric biderivations on a prime ring R, char.R/ ¤ 2;3, such that'1.x;x/'2.x;x/ D 0 holds 8 x 2 R, then either '1 D 0 or '2 D 0. The secondresult says that if ŒŒ'.x;x/;x;x 2 Z 8 x 2R, then R is commutative. In [16] theauthors extended the results in [14] assuming condition (1.1) over a nonzero idealand a nonzero Lie ideal of a prime ring respectively.The notion of generalized biderivation was introduced in [6]. A biadditive map  WRR!R is a generalized biderivation associated with a biderivation  WRR!R if for every x;y 2R, the maps y 7!  .x;y/ and y 7!  .y;x/ are generalizedderivations ofR associated with.x; :/ and.:;x/. That is,   .xy;´/D   .x;´/yCx.y;´/ and   .x;y´/D  .x;y/´Cy.x;´/ hold8 x;y;´2R. BreLsar shown thatevery generalized biderivation   of an ideal I (  W I  I ! R) of a prime ring Rwith char.R/¤ 2, is of the form   .x;y/D xayCybx for some a;b 2Q, where Qis Martindale quotient ring ofR ( see [8] and [10] for details). In [1] authors extendedsome results of [14, 15] to generalized biderivations on prime and semiprime rings.Recently, in [2], symmetric generalized .;/-biderivations of a prime ring R withchar.R/ ¤ 2 have been considered. Notice that a symmetric left bicentralizer is asymmetric generalized biderivation associated with the biderivation T D 0.2. PRELIMINARIESLemma 1 ([12, Lemma 3]). If the prime ring R contains a commutative nonzeroright ideal I , then R is commutative.Lemma 2 ([13, Lemma 2.6]). Let R be a prime ring with char.R/¤ 2. If L is acommutative Lie ideal of R, then LZ.Lemma 3 ([4, Lemma 4]). Let R be a prime ring with char.R/¤ 2. If L 6 Z isa Lie ideal of R and aLb D .0/, then either aD 0 or b D 0.Lemma 4. Let R be a prime ring, char.R/ ¤ 2, L a Lie ideal of R and ˛is automorphism. If T W RR! R is a symmetric left ˛-centralizer such thatT .˛.x1/;˛.x1//D 0 8 x1 2 L, then either LZ or T D 0.Proof. Assume that L 6Z. We haveT .x1;x1/D 0 8 x1 2 L: (2.1)ON LIE IDEALS AND SYMMETRIC GENERALIZED ...... 1177Linearizing (2.1), we getT .˛.x1/;˛.x2//D 0 8 x1;x2 2 L: (2.2)Let us replace x1 by x1s  sx1 8 s 2R in (2.2), to obtainT .˛.s/;˛.x2//˛2.x1/D 0 8x1;x2 2 L; s 2R:That is, ˛ 2.T .˛.s/;˛.x2///L˛ 2.T .˛.r/;˛.x2/// D .0/ 8 x2 2 L, s 2 R. ByLemma 3, ˛ 2.T .˛.r/;˛.x2/// D 0 8 x2 2 L, r 2 R that is, T .˛.s/;˛.x2// D 0.Now we replace x2 by Œx2; r 8 r 2 R, we get T .˛.s/;˛.r//˛2.x2/ D 0, that is˛ 2.T .˛.s/;˛.r///L˛ 2.T .˛.s/;˛.r///D .0/ and hence again by Lemma 3 givesthat ˛ 2.T .˛.s/;˛.r/// D 0 8 r;s 2 R i.e., T .˛.s/;˛.r// D 0 8 r;s 2 R. Now,replacing s by ˛ 1.t1/ and r by ˛ 1.t2/, we get T .t1; t2/ D 0 8 t1; t2 2 R, that isT D 0. Lemma 5. Let R be a ring and ˛;ˇ automorphisms. If   is a symmetric general-ized .˛;ˇ/-biderivation associated with a symmetric .˛;ˇ/-biderivation , then themap     WRR!R is a symmetric left ˛-bicentralizer.Proof. The map ˝ D    , is clearly biadditive. For all x1;x2;x3 2R,  .x1x2;x3/D .   /.x1x2;x3/D   .x1x2;x3/ .x1x2;x3/D   .x1;x3/˛.x2/Cˇ.x1/.x2;x3/ .x1;x3/˛.x2/ ˇ.x1/.x2;x3/D   .x1;x3/˛.x2/ .x1;x3/˛.x2/D˝.x1;x3/˛.x2/:Therefore, ˝ is a symmetric left ˛-centralizer of R. 3. MAIN RESULTSProposition 1. Let R be a prime ring, char.R/¤ 2, L a nonzero Lie ideal of R,˛;ˇ automorphisms, and WRR!R a symmetric .˛;ˇ/-biderivation with traceı. If ı.x1/D 0 8 x1 2 L, then either LZ or D 0.Proof. By our hypothesisı.u/D 0 8 x1 2 L: (3.1)Linearizing (3.1) and using (3.1), we get2.x1;x2/D 0 8 x1;x2 2 L: (3.2)If we replace x1 by x1r   rx1 in (3.2), to getˇ.x1/.r;x2/  ı.r;x2/˛.x1/D 0: (3.3)1178 NADEEM UR REHMAN AND SHULIANG HUANGAgain replacing r by rx3 and using (3.2), we find thatˇ.x1/.r;x2/˛.x3/ .r;x2/˛.x3x1/D 0: (3.4)Multiplying (3.3) from left by x3, we getˇ.x1/.r;x2/˛.x3/ .r;x2/˛.x1x3/D 0: (3.5)From (3.4), (3.5) it follows that .r;x2/˛.Œx1;x3/D 0 8 x1;x2;x3 2 L and r 2R.If we replace r by rs, s 2 R, we obtain .r;x2/˛.s/˛.Œx1;x3/ D 0 and hence˛ 1..r;x2//RŒx1;x3D .0/ Thus by primeness ofR it follows that either Œx1;x3D0 for x1;x3 2 L or ˛ 1..r;x2//D 0. If Œx1;x3D 0 8 x1;x3 2 L, then L  Z byLemma 2. In other case, ˛ 1..r;x2//D 0 8 x2 2 L, r 2R, that is, .r;x2/D 0 .Replacing x2 by x2s  sx2, we getˇ.x2/.r;s/ .r;s/˛.x2/D 0: (3.6)Now, replacing s by sx3 in (3.6), we getˇ.x2/.r;s/˛.x3/ .r;s/˛.x3x2/D 0: (3.7)If we multiply (3.6) by x3 to the right, and subtract (3.7) we get.r;s/˛.Œx2;x3/D 0 8x2;x3 2 L; r; s 2R:Replace r by t r , to get .t;s/˛.t/˛.Œx2;x3/ D 0, that is ˛ 1..t; s//T Œx2;x3 D.0/. So by primeness of R eitherD 0 or L is commutative. Hence, Lemma 2 givesthe required conclusion. Theorem 1. LetR be a prime ring, char.R/¤ 2, L a nonzero Lie ideal ofR, ˛;ˇautomorphisms and  WRR!R a symmetric .˛;ˇ/-biderivation with trace ı. Ifı.x1/ 2Z 8 x1 2 L, then either LZ or D 0.Proof. By Assumption we haveı.x1/ 2Z 8 x1 2 L: (3.8)The linearizing (3.8), we find that2.x1;x2/ 2Z 8 x1;x2 2 L: (3.9)Replace x1 by 2x21 in (3.9), to get4.x1;x2/˛.x1/Cˇ.x1/.x1;x2/ 2Z: (3.10)Therefore, in particular ˛.x1/ı.x1/Cˇ.x1/ı.x1/ 2Z 8 x1 2 L. Then we have0D Œ˛.x1/ı.x1/Cˇ.x1/ı.x1/; rD Œ˛.x1/Cˇ.x1/; rı.x1/: (3.11)For every r;s 2R, we haveŒ˛.x1/Cˇ.x1/; rsı.x1/D 0D Œ˛.x1/Cˇ.x1/; rsı.x1/D 0:By primeness of R, given an arbitrary element x1 2 L, we have either ı.x1/D 0 or˛.x1/Cˇ.x1/ 2 Z: If .˛Cˇ/.x1/ 2 Z then x1 2 Z. If Z\LD 0, then ı.x1/D 0ON LIE IDEALS AND SYMMETRIC GENERALIZED ...... 11798 x1 2 L. Assume that Z \L ¤ 0. If L 6 Z, then there exists x2 2 L nZ. Then8 x1 2 Z \L, the element x1C x2;x1   x2 2 L nZ. Hence ı.x1C x2/ D 0 andı.x1 x2/D 0 and hence ı.x1/D 0. In conclusion we prove that ı.x1/D 0 8 x1 2Z\L and above we already know that ı.x1/D 0 8 x1 2 LnZ. That is, ı.x1/D 08 x1 2 L. Theorem 2. Let R be a prime ring, char.R/¤ 2, L a nonzero square closed Lieideal of R, ˛;ˇ automorphisms and  1; 2 W RR! R be two symmetric gener-alized .˛;ˇ/-biderivations associated with symmetric .˛;ˇ/-biderivations 1;2 WRR! R, respectively. If  1.x1;x1/˛.x1/ D ˇ.x1/ 2.x1;x1/ 8 x1 2 L, theneither LZ or 2 D 0.Proof. Assume thatL 6Z. Suppose that 1;2; ı1; ı2 are traces of  1; 2;1;2,respectively. We have1.x1/˛.x1/D ˇ.x1/2.x1/ 8 x1 2 L: (3.12)Replacing x1 by x1Cx2 in (3.12), we get1.x1/˛.x2/C1.x2/˛.x1/C2 1.x1;x2/˛.x1/C2 1.x1;x2/˛.x2/D ˇ.x1/2.x2/Cˇ.x2/2.x1/C2ˇ.x1/ 1.x1;x2/C2ˇ.x2/ 2.x1;x2/: (3.13)Substituting x1 by  x1 in (3.13), we get1.x1/˛.x2/ 1.x2/˛.x1/C2 1.x1;x2/˛.x1/ 2 1.x1;x2/˛.x2/D ˇ.x1/2.x2/Cˇ.x2/2.x1/C2ˇ.x1/ 1.x1;x2/ 2ˇ.x2/ 2.x1;x2/: (3.14)Comparing (3.13) and (3.14) and using the fact that char(R/¤ 2, we obtain1.x1/˛.x2/C2 1.x1;x2/˛.x1/D ˇ.x2/2.x1/C2ˇ.x1/ 2.x1;x2/: (3.15)Replacing x2 by 2x2x1 in (3.15), we have1.x1/˛.x2x1/C2 1.x1;x2/˛.x21/C2ˇ.x2/ı1.x1/˛.x1/D ˇ.x2x1/2.x1/C2ˇ.x1/ 2.x1;x2/˛.x1/C2ˇ.x1/ˇ.x2/ı2.x1/: (3.16)That is,.1.x1/˛.x2/C2 1.x1;x2/˛.x1/ 2ˇ.x1/ 2.x1;x2//˛.x1/C2ˇ.x2/ı1.x1/˛.x1/D ˇ.x2x1/2.x1/C2ˇ.x1/ˇ.x2/ı2.x1/: (3.17)Using (3.15) in (3.17), we find thatˇ.x2/2.x1/˛.x1/C2ˇ.x2/ı.x1/˛.x1/D ˇ.x2x1/2.x1/C2ˇ.x1x2/ı2.x1/:(3.18)Replacing x2 by 2x3x2 in (3.18) and using char.R/¤ 2, we getˇ.x2x3/2.x1/˛.x1/C2ˇ.x3x2/ı.x1/˛.x1/D ˇ.x3x2x1/2.x1/C2ˇ.x1x3x2/ı2.x1/: (3.19)1180 NADEEM UR REHMAN AND SHULIANG HUANGNow, subtracting (3.19) from (3.18) multiplied by x3 to the left, we getˇ.Œx3;x1/ˇ.x2/ı2.x1/ 8 x1;x2;x3 2 L: (3.20)This implies that Œx3;x1Lˇ 1.ı2.x1//D .0/. By Lemma 3, gives that for an arbit-rary element x1 2 L either x1 2 Z.L/ or ı2.x1/ D 0. If Z.L/ D 0, then ı2.x1/ D0. If L D Z.L/, then L  Z, by Lemma 2, a contradiction. Let us assume thatL ¤ Z.L/ ¤ 0. Then there exists x1 2 L nZ.L/. So ı2.x1/ D 0 since ı2.x2/ D 08 x2 2 L nZ.L/. Take 0 ¤ x3 2 Z.L/. Then x1C x3;x1  x3 2 L nZ.L/ and so.x1Cx3;x1Cx3/D 0D.x1 x3;x1 x3/, that is2.x1;x1/C22.x1;x3/C2.x3;x3/D 0and2.x1;x1/ 22.x1;x3/C2.x3;x3/D 0:Adding the above two expression, we find that 2ı2.x3/D 0. Since char(R/¤ 2, wehave ı2.x3/D 0 8 x3 2 L. Using Proposition 1, we get the required result. Using the same technique with necessary variation one can prove the followingtheorem.Theorem 3. Let R be a prime ring, char.R/¤ 2, L a nonzero square closed Lieideal of R, ˛;ˇ automorphisms and  1; 2 W RR! R be two symmetric gener-alized .˛;ˇ/-biderivations associated with symmetric .˛;ˇ/-biderivations 1;2 WRR!R, respectively. If  1.x1;x1/˛.x1/Cˇ.x1/ 2.x1;x1/D 0 8 x1 2 L, theneither LZ or 2 D 0.It is immediately to get the following corollaries from Theorems 2, 3 and Lemma1.Corollary 1. Let R be a prime ring, char.R/ ¤ 2, I a nonzero ideal of R,and  1; 2 W RR! R be two symmetric generalized .˛;ˇ/-biderivations asso-ciated with symmetric .˛;ˇ/-biderivations 1;2 W RR! R, respectively. If 1.x;x/˛.x/D˙ˇ.x/ 2.x;x/ 8 x 2 I , then R is commutative.Corollary 2. Let R be a prime ring, char.R/ ¤ 2, L a nonzero square closedLie ideal of R, ˛;ˇ automorphisms and   WRR!R be a symmetric generalized.˛;ˇ/-biderivation associated with a symmetric .˛;ˇ/-biderivation WRR!R.If Œ  .x;x/;x˛;ˇ D 0 8 x 2 I , then either LZ or D 0.Corollary 3. Let R be a prime ring, char.R/ ¤ 2, I a nonzero ideal of R, ˛;ˇautomorphisms and   WRR!R be a symmetric generalized .˛;ˇ/-biderivationassociated with a symmetric .˛;ˇ/-biderivation  WRR!R. If   .x;x/˛.x/D˙ˇ.x/.x;x/ 8 x 2 I , then R is commutative or   is a left ˛-bicentralizer.ON LIE IDEALS AND SYMMETRIC GENERALIZED ...... 1181Theorem 4. Let R be a prime ring, char.R/¤ 2;3, L a nonzero Lie ideal of R,  W RR! R a symmetric generalized biderivation associated with a symmetricbiderivation . If  .  .x1;x1/;  .x1;x1// ..x1;x1/;.x1;x1//D.  .x1;x1/;  .x1;x1//   ..x1;x1/;.x1;x1//8 x1 2 L, then either LZ or D   or D 0.Proof. Let  , ı be the traces of   and , respectively. By our hypothesis we have2.x1/  ı2.x1/D ı..x1// .ı.x1//: (3.21)The substitution of x1 by x1Cx2 in (3.21), gives that4.  .x1;x2//C2  ..x1/;.x2//C4  ..x1/;  .x1;x2//C4  ..x2/;  .x1;x2//  ı..x1;x2// 2.ı.x1/;ı.x2// 4.ı.x1/;.x1;x2// 4.ı.x2/;.x1;x2//D 4ı.  .x1;x2//C2..x1/;.x2//C4..x1/;  .x1;x2//C4..x2/;  .x1;x2// 4..x1;x2// 2  .ı.x1/;ı.x2// 4  .ı.x1/;.x1;x2// 4  .ı.x2/;.x1;x2//: (3.22)Now, substituting x2 by  x2 in the above expression, we find that4.  .x1;x2//C2  ..x1/;.x2// 4  ..x1/;  .x1;x2// 4  ..x2/;  .x1;x2//  ı..x1;x2// 2ı.ı.x1/;ı.x2//C4.ı.x1/;.x1;x2//C4.ı.x2/;.x1;x2//D 4ı.  .x1;x2//C2..x1/;.x2// 4..x1/;  .x1;x2// 4..x2/;  .x1;x2// 4..x1;x2// 2  .ı.x1/;ı.x2//C4  .ı.x1/;.x1;x2//C4  .ı.x2/;.x1;x2//: (3.23)Comparing (3.22) and (3.23) and using the fact that char.R/¤ 2, we get2.  .x1;x2//C  ..x1/;.x2// 2ı..x1;x2// .ı.x1/;ı.x2//D 2ı..x1;x2//C..x1/;.x2// 2..x1;x2//   .ı.x1/;ı.x2//: (3.24)Now, substituting x2 by x2Cx3 in(3.24), gives that2  .  .x1;x2/;  .x1;x3//C  ..x1/;  .x2;x3// 2..x1;x2/;.x1;x3// .ı.x1/;.x2;x3//D 2ı..x1;x2/;.x2;x3//C..x1/;  .x2;x3// 2..x1;x2//   .ı.x1/;.x2;x3//: (3.25)Let us take T D    , and denote k D C ı the trace of K D   C. By Lemma5 T is a left bicentralizer of R. Then (3.24), reduces to1182 NADEEM UR REHMAN AND SHULIANG HUANG2T .  .x1;x2/  .x1;x3//CT ..x1/;  .x2;x3//C2T ..x1;x2/;.x1;x3//CT .ı.x1/;.x2;x3//D 0: (3.26)Replacing x3 by 2x3´ in (3.26)and using (3.26), we obtain2T .  .x1;x2/;x3/.x1;´/CT ..x1/;x3/.x2;´/C2T ..x1;x2/;x3/.x1;´/CT .ı.x1/;x3/.x2;´/D 0: (3.27)That is,2T .K.x1;x2/;x3/.x1;´/CT .k.x1/;x3/.x2;´/D 0: (3.28)Choosing x2 D x1 in (3.28), and using char.R/¤ 3, we getT .k.x1/;x3/.x1;´/D 0: (3.29)Choosing ´D x1 in (3.28), we obtain2T .K.x1;x2/;x3/ı.x1/CT .k.x1/;x3/.x2;x1/D 0: (3.30)Comparing (3.29) and (3.30) and using the fact that char.R/¤ 2, gives thatT .K.x1;x2/;x3/ı.x1/D 0: (3.31)Replacing x2 by 2x2x in (3.31), we find thatT .K.x1;x2/;x3/xı.x1/C2T .x2;x3/.x1;x/ı.x1/D 0: (3.32)By replacing x3 by 2x3x in (3.31), we find thatT .K.x1;x2/;x3/xı.x1/D 0: (3.33)From (3.32) and (3.33) and using the fact that char.R/¤ 2, we find thatT .x2;x3/.x1;x/ı.x1/D 0: (3.34)Replace x3 by 2x3w1, to get .T .x2;x3/L.x1;x/.x1/ D 0 8 x1;x2;x;x3 2 L.Lemma 3, gives that either .T .x2;x3/ D 0 or .x1;x/ı.x1/ D 0. In the first caseT .x2;x3/ D 0 8 x2;x3 2 L and hence by Lemma 4, proves that T D 0. Therefore  D . On the other hand, if .x1;x/ı.x1/ D 0 8 x1;x 2 L, then replacing x by2x1x, we have ı.x1/xı.x1/ D 0, that is, ı.x1/Lı.x1/ D 0. Again, by Lemma 3,ı.x1/ D 0 and hence ı.x1/ D 0 for al x1 2 L. Proposition 1, gives that L  Z orD 0. Corollary 4. Let R be a prime ring, char.R/ ¤ 2, I a nonzero ideal of R, ˛an automorphism, and   WRR!R a symmetric generalized .˛;˛/-biderivationassociated with a symmetric .˛;˛/-biderivation . If  .  .x1;x1/;  .x1;x1// ..x1;x1/;.x1;x1//D.  .x1;x1/;  .x1;x1//   ..x1;x1/;.x1;x1//8 x1 2 I , then either R is commutative or D   or  is a left ˛-bicentralizer.ON LIE IDEALS AND SYMMETRIC GENERALIZED ...... 1183ACKNOWLEDGEMENTThe authors are greatly indebted to the referee for his/her valuable suggestionswhich have improved the paper immensely.REFERENCES[1] A. Ali, V. D. Filippis, and F. Shujat, “ Results concerning symmetric generalized biderivations ofprime and semiprime rings.” Matematicki Vesnik, vol. 66, no. 4, pp. 410–417, 2004.[2] M. Ashraf, N. Rehman, S. Ali, and M. R. Mozumder, “On generalized .;/-biderivationsin rings.” Asian-European Journal of Mathematics, vol. 4, no. 3, pp. 389–402, 2011, doi:10.1142/S1793557111000319.[3] K. Beidar, M. S. Martindale, and A. Mikhalev, Rings with Generalized Identities. New York:Marcel Dekker Inc., 1996.[4] J. Bergen, I. N. Herstein, and J. W. Kerr, “Lie ideals and derivations of prime rings.” J. Algebra,vol. 7, pp. 259–267, 1981, doi: 10.1016/0021-8693(81)90120-4.[5] M. BreLsar, “On certain pairs of functions of semiprime rings.” Proc. Amer. Math. Soc., vol. 3, no.120, pp. 709–713, 1994, doi: 10.2307/2160460.[6] M. BreLsar, “On generalized biderivations and related maps.” J.Algebra, vol. 172, pp. 764–786,1995, doi: 10.1006/jabr.1995.1069.[7] M. BreLsar, M. S. Martindale, and C. Miers, “Centralizing maps in prime rings with involution.”J.Algebra, vol. 161, no. 2, pp. 342–357, 1993, doi: 10.1006/jabr.1993.1223.[8] I. Herstein, “Rings with involution.” The Universtity Chicago Press,Chicago, 1976.[9] G. Maksa, “On the trace of symmetric biderivations.” C. R. Math. Rep. Acad. Sci. Canada, vol. 9,pp. 303–307, 1987.[10] W. S. Martindale III, “Prime rings satisfying a generalized polynomial identity.” J. of Algebra,vol. 12, pp. 576–584, 1969, doi: 10.1016/0021-8693(69)90029-5.[11] G. Maska, “A remark on symmetric biadditive functions having non-negative diagonalization.”Glasnik. Mat., vol. 15, no. 35, pp. 279–282, 1980.[12] J. H. Mayne, “Centralizing mappings of prime rings.” Canad. Math. Bull., vol. 27, no. 1, pp.122–126, 1984, doi: 10.4153/CMB-1984-018-2.[13] N. Rehman, “On Commutativity of prime rings with generalized derivations.” Math. J. OkayamaUniv., vol. 44, pp. 43–49, 2002.[14] J. Vukman, “Symmetric biderivations on prime and semiprime rings.” Aequationes Math., vol. 38,pp. 245–254, 1989, doi: 10.1007/BF01840009.[15] J. Vukman, “Two results concerning symmetric biderivations on prime rings.” Aequationes Math.,vol. 40, pp. 181–189, 1990, doi: 10.1007/BF02112294.[16] M. Yenigul and N. Argac, “Ideals and symmetric biderivations on prime and semiprime rings.”Math. J. Okayama Univ., vol. 35, no. 9, pp. 189–192, 1993.Authors’ addressesNadeem ur RehmanDepartment of Mathematics, Aligarh Muslim University, 202002, Aligarh, IndiaE-mail address: nu.rehman.mm@amu.ac.inShuliang HuangShuliang Huang, Department of Mathematics, Chuzhou University, Chuzhou 239012, ChinaE-mail address: shulianghuang@sina.com

On Lie ideals and symmetric generalized (α, β)-biderivation in prime ring

Abstract

Similar works

Full text

Available Versions

Repository of the Academy's Library