Open Math. 2017; 15: 1123–1131
Open Mathematics
Research Article
Sara Shafiq* and Muhammad Aslam On Jordan mappings of inverse semirings https://doi.org/10.1515/math-2017-0088 Received August 24, 2016; accepted June 20, 2017.
Abstract: In this paper, the notions of Jordan homomorphism and Jordan derivation of inverse semirings are introduced. A few results of Herstein and Bresar on Jordan homomorphisms and Jordan derivations of rings are L generalized in the setting of inverse semirings.
Keywords: Inverse semiring, Jordan homomorphism, Jordan triple homomorphism, Jordan derivation, Jordan triple derivation
MSC: 16Y60, 16W25
1 Introduction
Let .S; ; / be a semiring with commutative addition and an absorbing zero 0. A semiring S is called an inverse C semiring [1] if for every a S there exists a unique element a S such that a a a a and a a a a . 2 0 2 C 0 C D 0 C C 0 D 0 Throughout this paper, S will represent an inverse semiring which satisfies the condition that for every a S; a a 2 C 0 is in the center of S. This class of inverse semirings, known as MA-semiring [2], is useful in developing the theory of commutators and investigating certain additive mappings in semirings. In this connection, commuting maps [3], skew-commuting maps [4], centralizers [5,6], dependent elements and free actions [7] have been studied. However, the theory of Jordan homomorphism and Jordan derivation of inverse semirings has been unexplored. According to [2], a commutator [. , .] in inverse semiring is defined as Œx; y xy yx xy yx. We will make use of D CK D C K commutator identities Œxy; z xŒy; z Œx; zy and Œx; yz Œx; yz yŒx; z (see [2]). By [2], an additive map D C D C d S S is a derivation if d.ab/ d.a/b ad.b/; a; b S. S is prime if aSb .0/ implies that a 0 or W ! D C 8 2 D D b 0 and is semiprime if aSa .0/ implies that a 0. S is n-torsion free if nx 0; x S implies that x 0. D D D D 2 D The idea of Jordan homomorphism of rings arose initially in Ancochea’s study of semi-automorphisms [8,9]. Later on, Kaplansky [10], Hua [11] and Jacobson and Ricket [12] made contribution and took the subject up. In 1950’s, Herstein studied Jordan homomorphisms [13,14] and Jordan derivations [15] in prime rings. His results have a notable impact on the study of Jordan structure and Jordan mappings. Bresar [16,17,18], Baxer and Martindale L [19] generalized Herstein’s work on semiprime rings. In this paper, we introduce Jordan homomorphism and Jordan derivation in inverse semirings and generalize a few results of Herstein [13,15] and Bresar [16,17] in the setting of inverse semirings. We define Jordan L homomorphism between the two inverse semirings as follows. Let S and T be inverse semirings then an additive mapping ' S T is called a Jordan homomorphism if '.ab ba/ '.a/'.b/ '.b/'.a/ 0; a; b S: W ! C C 0 C 0 D 8 2 In section 2, we generalize remarkable result of Herstein [13] for inverse semirings as follows: every Jordan homomorphism of inverse semiring onto prime inverse semiring is either homomorphism or anti-homomorphism. In section 3, we define Jordan triple homomorphism between inverse semirings S and T as an additive mapping such that .aba/ .a/.b/.a/ 0; a; b S. Bresar [17] showed that a Jordan triple homomorphism of a C 0 D 8 2 L ring R onto a prime ring R of characteristic different from 2 is of the form , where is a homomorphism N D ˙
*Corresponding Author: Sara Shafiq: Mathematics Department, GC University Lahore, 54000, Pakistan Muhammad Aslam: Mathematics Department, GC University Lahore, 54000, Pakistan
Open Access. © 2017 Shafiq and Aslam, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 License. 1124 S. Shafiq, M. Aslam or an anti-homomorphism of R onto R. We generalize this result for inverse semirings. In section 4, we introduce N the notion of Jordan derivation of inverse semirings thereby extend a classical result of Bresar [16] as follows. Every L Jordan derivation of a 2-torsion free inverse semiring is a derivation. In the last section, Jordan triple derivation of inverse semiring is considered and a Bresar’s result [17] is extended in semirings. L We need the following lemmas in our arguments.
Lemma 1.1 (Lemma 1.1, [5]). Let S be an inverse semiring and a; b S: If a b 0 then a b . 2 C D D 0 Lemma 1.2. Let S be a 2-torsion free semiprime inverse semiring. If a; b S are such that axb bxa 0; x S 2 C D 8 2 then axb bxa 0: D D Proof. Using Lemma 1.1 in axb bxa 0 we have axb bxa ; x S: Thus .bxa/y.bxa/ b.xby/a xa C D D 0 2 D 0 D axbybxa: Hence, 2.bxa/y.bxa/ .axb bxa/ybxa 0; since S is 2-torsion free, bxa 0: Similarly, we can D C D D show axb 0; x S: D 8 2 The following lemma is an extension of Lemma 3.10 of [20], in a canonical fashion.
Lemma 1.3. Let S be a 2-torsion free prime inverse semiring. If a; b S are such that axb bxa 0; x S 2 C D 8 2 then either a 0 or b 0. D D Lemma 1.4 (Lemma 1.5 , [5]). Let S be a semiprime inverse semiring and f; g S S S biadditive mappings. W ! If f .x; y/wg.x; y/ 0; x; y; w S then f .x; y/wg.s; t/ 0; x; y; s; t; w S. D 8 2 D 8 2
2 Jordan homomorphism
We begin this section by introducing the notion of Jordan homomorphism of inverse semirings. Let S and T be inverse semirings, an additive mapping ' S T is called Jordan homomorphism if W ! '.ab ba/ '.a/'.b/ '.b/'.a/ 0; a; b S (1) C C 0 C 0 D 8 2 Lemma 2.1. Let ' be a Jordan homomorphism of inverse semiring S into 2-torsion free inverse semiring T: Then for all a; b; c S the following statements are true: 2 (i) '.a2/ '.a/2 D (ii) '.aba/ '.a/'.b/'.a/ D (iii) '.cba abc/ '.c/'.b/'.a/ '.a/'.b/'.c/ C D C Proof. Replacing b by a in (1) and then using Lemma 1.1 we obtain (i). For (ii); put b ab ba in (1), we have D C '.a.ab ba/ .ab ba/a/ '.a/'.ab ba/ '.ab ba/'.a/ 0: In the view of Lemma 1.1 and (1), we C C C C C 0 C C 0 D can replace '.ab ba/ by '.a/'.b/ '.b/'.a/: Thus we have C C '.a2b ba2/ 2'.aba/ '.a/'.a/'.b/ '.a/'.b/'.a/ '.a/'.b/'.a/ '.b/'.a/'.a/ 0 C C C 0 C 0 C 0 C 0 D By (i) we have '.aba/ '.a/'.b/'.a/ 0 C 0 D and hence by Lemma 1.1, we conclude (ii). Linearizing the above relation we obtain
'.cba abc/ '.c/'.b/'.a/ '.a/'.b/'.c/ 0 (2) C C 0 C 0 D thus by Lemma 1.1, we arrive at (iii).
b Following [13], we fix some notations; a '.ab/ '.a/'.b/ and ab '.ab/ '.b/'.a/ . Thus (1) can be D C 0 D C 0 written as ab ba 0 C D On Jordan mappings of inverse semirings 1125
Lemma 2.2. If ' is a Jordan homomorphism from an inverse semiring S to 2-torsion free inverse semiring T then
b b a ab 0 aba D D Proof. We have,
b a ab '.ab/'.ab/ '.ab/'.b/'.a/ '.a/'.b/'.ab/ '.a/'.b/'.b/'.a/ D C 0 C 0 C By Lemma 2.1 (i) we get
b 2 a ab '.abab/ '.ab/'.b/'.a/ '.a/'.b/'.ab/ '.a/'.b /'.a/ D C 0 C 0 C b b Using Lemma 2.1 (ii) and (2) we arrive at a ab 0, as desired. Similarly, we can compute aba 0: D D Lemma 2.3. Let ' be a Jordan homomorphism from inverse semiring S to 2-torsion free inverse semiring T: Then for any a; b; r S 2 b b b a '.r/a a '.Œa; br/ and ab'.r/ab '.Œa; br/ab: D D Proof. Consider, '.r/ab '.r/'.ab/ '.r/'.a/'.b/ : D C 0 Applying Lemma 1.1 in (2) and then using it in the above relation, we obtain
'.r/ab '.r/'.ab/ '.b/'.a/'.r/ '.rab bar/ D C C C 0 But '.rab bar/ '.rab .r r /ab bar/ '.rab ab.r r / bar/ '.rab abr/ '.abr bar / C 0 D C C 0 C 0 D C C 0 C 0 D C 0C C 0 D '.r/'.ab/ '.ab/'.r/ '.abr bar /: Thus we have, '.r/ab '.r/.'.ab/ '.ab/ / '.b/'.a/'.r/ 0 C 0 C C 0 D C 0 C C '.ab/'.r/0 '.Œa; br/ .'.ab/ '.ab/0/'.r/ '.b/'.a/'.r/ '.ab/'.r/0 '.Œa; br/ ab'.r/ '.Œa; br/. C D b Cb b C C C D C By Lemma 2.2 we obtain a '.r/a a '.Œa; br/ and ab'.r/ ab '.Œa; br/ab 0: D 0 C D Lemma 2.4. Let ' be a Jordan homomorphism from inverse semiring S to 2-torsion free inverse semiring T: If a; b; r S then 2 b b '.Œa; br/ '.r/a ab'.r/ and '.rŒa; b/ a '.r/ '.r/ab: D C D C Proof. From (1), we can replace '.ab ba/ by '.a/'.b/ '.b/'.a/ a; b S: Using this together with b C C 8 2 Lemma 2.1 (iii), we have a '.r/ '.r/ab '.ab/'.r/ '.a/'.b/'.r/ '.r/'.ab/ '.r/'.b/'.a/ C D C 0 C C 0 D '.abr0 rba0 abr rab/ '.ab.r r0/ rba0 rab/ '..r r0/ab rba rab/ '.rŒa; b/. Similarly, C C b C D C C C D C C K C D '.Œa; br/ '.r/a ab'.r/. D C Theorem 2.5. Let ' be a Jordan homomorphism from inverse semiring S to 2-torsion free inverse semiring T: Then for all a; b; r S 2 b b ab'.r/a a '.r/ab 0: C D b Proof. Replacing r by Œa; br in '.rŒa; b/ a '.r/ '.r/ab we have D C b '.Œa; brŒa; b/ a '.Œa; br/ '.Œa; br/ab D C By Lemma 2.1 (ii) and Lemma 2.4, we get
b b '.Œa; b/'.r/'.Œa; b/ a '.r/a ab'.r/ab D C b Now, '.Œa; b/ '.ab ab ab b a/ 2'.ab/ '.ab ba/ 2'.ab/ '.a/'.b/ '.b/'.a/ ab a : D C C 0 C 0 D C C 0 D C C D C Thus we have b b b b .ab a /'.r/.ab a / a '.r/a ab'.r/ab C C D C which implies that
b b b b b b ab'.r/ab a '.r/ab a '.r/a ab'.r/a a '.r/a ab'.r/ab C C C D C a b b a Adding ba'.r/ab b '.r/a on both sides of the above equation and using the fact that ab ba 0 a b ; C C D D C we get the required result. 1126 S. Shafiq, M. Aslam
Theorem 2.6. Every Jordan homomorphism ' of S onto 2-torsion free prime inverse semiring T is either a homomorphism or an anti-homomorphism.
b Proof. By Theorem 2.5 and Lemma 1.3 we have either ab 0 or a 0: Thus by Lemma 1.1 either ' is a D D homomorphism or an anti- homomorphism.
3 Jordan triple homomorphism
Let S and T be inverse semirings. An additive map S T is called Jordan triple homomorphism if W ! .aba/ .a/.b/.a/ 0; a; b S (3) C 0 D 8 2 For the sake of convenience, we fix some notations
G1.a; b; c/ .abc/ .a/.b/.c/ and D C 0
G2.a; b; c/ .abc/ .c/.b/.a/ D C 0 Linearization of (3) gives
.abc cba/ .a/.b/.c/ .c/.b/.a/ 0 (4) C C 0 C 0 D or
G1.a; b; c/ G1.c; b; a/ 0 C D Lemma 3.1. If is a Jordan homomorphism of inverse semiring S onto 2-torsion free prime inverse semiring T then
G1.a; b; c/.x/G2.a; b; c/ G2.a; b; c/.x/G1.a; b; c/ 0; a; b; c; x S: C D 8 2 Proof. Replacing a by abc, b by x and c by cba in (4), we obtain
.abcxcba cbaxabc/ .abc/.x/.cba/ .cba/.x/.abc/ 0 (5) C C 0 C 0 D According to Lemma 1.1 and (4) we have
.cba/ .abc/ .a/.b/.c/ .c/.b/.a/ D 0 C C thus from (5) we get
.abcxcba cbaxabc/ .abc/.x/.abc/ .abc/.x/.c/.b/.a/ C C C 0C .abc/.x/.a/.b/.c/ .abc/.x/.abc/ .c/.b/.a/.x/.abc/ 0 C C 0C .a/.b/.c/.x/.abc/ 0 (6) 0 D Also, by definition of triple homomorphism and Lemma 1.1, we have
.abcxcba cbaxabc/ .a/.b/.c/.x/.c/.b/.a/ .c/.b/.a/.x/.a/.b/.c/ (7) C D C From (6) and (7) we obtain
G1.a; b; c/.x/G2.a; b; c/ G2.a; b; c/.x/G1.a; b; c/ 0 C D as desired.
Theorem 3.2. Let be a Jordan triple homomorphism of an inverse semiring onto 2-torsion free prime inverse semiring. Then or ; where is either a homomorphism or an anti-homomorphism. D D 0 On Jordan mappings of inverse semirings 1127
Proof. By Lemmas 3.1 and 1.3, we have either G1.a; b; c/ 0 or G2.a; b; c/ 0: If G1.a; b; c/ 0 then D D D .abc/ .a/.b/.c/: Replacing b by bxa and c by b we get D .abxab/ .a/.b/.x/.a/.b/ (8) D Also, by definition of triple homomorphism we have
.abxab/ .ab/.x/.ab/ 0 (9) C 0 D From (8) and (9) we get .a/.b/.x/.a/.b/ .ab/.x/.ab/ 0 (10) C 0 D a If b .ab/ .a/.b/ and ba .ab/ .a/.b/ then using (10) we have D C 0 D C a a b .x/ba ba.x/b .a/.b/ .x/ .a/.b/.x/ .ab/ .ab/ .x/.a/.b/ .x/.a/.b/ C D f 0 C g C f C 0g But .x/.a/.b/ .xab/; thus we get D a a b .x/ba ba.x/b .abx/ .a/.b/.x/ .ab/ .ab/ .xab/ .x/.a/.b/ 0: C D f 0 C g C f C 0g D a Hence, by Lemma 1.3 either b 0 or ba 0: Therefore, either .ab/ .a/.b/ or .ab/ .a/.b/ for all D D D D 0 a; b S. 2 On similar lines, we can show that if G2.a; b; c/ 0 then .ab/ .b/.a/ or .ab/ .b/.a/ for all D D D 0 a; b S: Thus or , where is either a homomorphism or an anti-homomorphism. 2 D D 0
4 Jordan derivation
We define Jordan derivation of an inverse semiring S as an additive map d S S such that W ! d.x2/ d.x/x x d.x/ 0; x S (11) C 0 C 0 D 8 2 holds. For example, if R is a commutative ring and I.R/ is semiring of all two sided ideals of R with respect to ordinary addition and product of ideals, and T is subsemiring of I.R/ then S .r; I / r R;I T is an inverse D f W 2 2 g semiring with respect to and defined as .r1;I/ .r2;J/ .r1 r2;I J/ and .r1;I/ .r2;J/ .r1r2;IJ/ ˚ ˇ ˚ D C C ˇ D (see [4]). If we fix a .r; 0 / S then d.x/ Œa; x; x S is a Jordan derivation. D f g 2 D 8 2 Lemma 4.1. If d is a Jordan derivation of S then for all x; y S the following statements hold: 2 (i) d.xyx/ d.x/yx xd.y/x xyd.x/ D C C (ii) d.xyz zyx/ d.x/yz xd.y/z xyd.z/ d.z/yx zd.y/x zyd.x/: C D C C C C C Proof. Linearization of (11) gives
d.xy yx/ d.y/x d.x/y x d.y/ y d.x/ 0 (12) C C 0 C 0 C 0 C 0 D Replacing y by xy yx in .12/ we get C d.x.xy yx/ .xy yx/x/ d.xy yx/x d.x/.xy yx / C C C C C 0 C 0 C 0 C x d.xy yx/ .xy yx /d.x/ 0 (13) 0 C C 0 C 0 D By (12),(13) and Lemma 1.1 we have
d.x2y yx2/ 2d.xyx/ 2d.x/y x 2x d.y/x 2x yd.x/ d.y/ x2 C C C 0 C 0 C 0 C 0 C yd.x/x d.x/xy x d.x/y x2d.y/ yx d.x/ 0 (14) 0 C 0 C 0 C 0 C 0 D 1128 S. Shafiq, M. Aslam
Replacing x by x2 in .12/ we have
d.x2y yx2/ d.y/ x2 d.x2/y x2d.y/ y d.x2/ 0 C C 0 C 0 C 0 C 0 D In the view of (11) and Lemma 1.1, we can replace d.x2/ by d.x/x xd.x/. Thus C d.x2y yx2/ d.y/ x2 d.x/xy xd.x/y x2d.y/ y d.x/x y xd.x/ 0 (15) C C 0 C 0 C 0 C 0 C 0 C 0 D From (14) and (15) we have d.xyx/ d.x/y x xd.y/x x yd.x/ 0 (16) C 0 C 0 C 0 D which gives (i). To obtain (ii), linearize (16) we have
d.xyz zyx/ d.x/yz xd.y/ xy d.z/ d.z/yx zd.y/x zy d.x/ 0 (17) C C C 0 C 0 C 0 C 0 C 0 D and hence (ii) follows by Lemma 1.1. This completes the proof. If we write xy for d.xy/ d.x/y x d.y/ then by .12/ and Lemma 1.1, we have C 0 C 0 y x x y 0 (18) D Moreover, xy z xy xz (19) C D C holds for all x; y S. 2 Lemma 4.2. Let S be a 2-torsion free inverse semiring and d S S be a Jordan derivation. Then W ! xy sŒx; y Œx; ysxy 0: (20) C D x; y; s S. 8 2 Proof. Replacing y by ysy in (16) we get
d.xysyx/ d.x/ysy x xd.ysy/x xys yd.x/ 0 (21) C 0 C 0 C 0 D Replacing x by y and y by xsx in (16) gives
d.yxsxy/ d.y/xsx y yd.xsx/y yxs xd.y/ 0 (22) C 0 C 0 C 0 D Adding .21/ and .22/, we have
d.xysyx yxsxy/ d.x/ysy x xd.ysy/x xys yd.x/ d.y/xsx y yd.xsx/y yxs xd.y/ 0 C C 0 C 0 C 0 C 0 C 0 C 0 D By (i) of Lemma 4.1, we have
d.xysyx yxsxy/ d.x/ysy x xd.y/sy x xyd.s/y x xysd.y/x xys yd.x/ C C 0 C 0 C 0 C 0 C 0 C d.y/xsx y yd.x/sx y yxd.s/x y yxsd.x/y yxs xd.y/ 0 (23) 0 C 0 C 0 C 0 C 0 D Replacing x by xy; y by s and z by yx in (17) we get
d.xysyx yxsxy/ d.xy/sy x xyd.s/y x xy sd.yx/ d.yx/sx y yxd.s/x y yx sd.xy/ 0 C C 0 C 0 C 0 C 0 C 0 C 0 D From this, we have
d.xysyx yxsxy/ xyd.s/y x yxd.s/x y d.xy/syx xysd.yx/ d.yx/sxy yxsd.xy/ (24) C C 0 C 0 D C C C (23) and (24) give
d.xy/syx xysd.yx/ d.yx/sxy yxsd.xy/ d.x/ysy x xd.y/sy x xysd.y/x C C C C 0 C 0 C 0C xys yd.x/ d.y/xsx y yd.x/sx y yxd.d/x y yxs xd.y/ 0 0 C 0 C 0 C 0 C 0 D y x y x y y y y which can be written as 0 x syx xysy yxsx y sxy x syx xysx 0 yxsx x 0sxy: D C C C D C C C Hence, xy sŒx; y Œx; ysxy 0: C D On Jordan mappings of inverse semirings 1129
Theorem 4.3. A Jordan derivation d on 2-torsion free inverse semiring S is a derivation.
Proof. To prove the theorem, we have to show that xy 0 x; y S: D 8 2 By Lemmas 1.2 and 4.2, we get xy sŒx; y 0 Œx; ysxy (25) D D Replacing y by y z in xy sŒx; y 0 and using it again, we have C D xy sŒx; z xzsŒx; y 0 (26) C D which gives xy sŒx; z xzsŒx; y: Thus .xy sŒx; z/t.xy sŒx; z/ xzsŒx; ytxy sŒx; z 0; t S. D K D K D 8 2 Semiprimeness of S gives xy sŒx; z 0; x; y; z; s S (27) D 8 2 Similarly, Œx; zsxy 0: D By using the above technique on (27), we get
xy sŒw; z 0; x; y; z; s; w S (28) D 8 2 Pre multiplying (28) by Œw; z and post multiplying it by xy , we get
Œw; zxy 0; x; y; w; z S (29) D 2 Therefore, Œxy ; trŒxy ; t xy trŒxy ; t t xy rŒxy ; t 0; r S D C 0 D 8 2 which implies that Œxy ; t 0 or xy t t xy 0: Adding txy on both sides of the last expression we have D C 0 D xy t .t t /xy xy t or xy t xy .t t / txy ; t S: Thus xy is in the center of S: C C 0 D C C 0 D 8 2 From (18), we have
y 2 y y x y 2.x / x .x y 0/ x d.xy/ d.yx/ Œy; d.x/ Œd.y/; x (30) D C D f C 0 C C g or 2.xy /2 xy dŒx; y Œy; d.x/ Œd.y/; x D f C C g Using (29) and the fact that xy is in center, we get
2.xy /2 xy dŒx; y (31) D Also, xy Œx; y Œx; yxy 0 which gives C D d.xy Œx; y Œx; yxy / 0 (32) C D Put x xy and y Œx; y in (12) and using (32), we obtain D D d.xy /Œx; y xy dŒx; y dŒx; yxy Œx; yd.xy / 0 C C C D By (31), we have d.xy /Œx; y Œx; yd.xy / 4.xy /2 0. Post multiplying last equation by xy , we have 4.xy /3 C C D D 0: Thus .xy /3 0: D Since xy is in the center of S so for t S; .xy /2r.xy /2 .xy /3.xy /r 0; r S. Thus .xy /2 0: This 2 D D 2 D implies that xy 0; x; y S, as required. D 8 2
5 Jordan triple derivation
An additive mapping d S S is Jordan triple derivation if W ! d.aba/ d.a/ba ad.b/a abd.a/ 0 (33) C 0 C 0 C 0 D 1130 S. Shafiq, M. Aslam
Linearization of (33) gives
d.abc cba/ d.a/bc ad.b/c abd.c/ d.c/ba cd.b/a cbd.a/ 0 (34) C C 0 C 0 C 0 C 0 C 0 C 0 D
Put F1.a; b; c/ d.abc/ d.a/bc ad.b/c abd.c/ and F2.a; b; c/ abc cba then D C 0 C 0 C 0 D C 0
F1.a; b; c/ F1.c; b; a/ 0 (35) C D Lemma 5.1. Let S be an inverse semiring. Then
F1.a; b; c/xF2.a; b; c/ F2.a; b; c/xF1.a; b; c/ 0 C D for all a; b; c; x S: 2 Proof can be obtained by simple modification of the proof of Lemma 3.1.
Theorem 5.2. Every Jordan triple derivation d of a 2-torsion free semiprime inverse semiring S is a derivation.
Proof. By Lemmas 5.1 and 1.3, we have either F1.a; b; c/ 0 or F2.a; b; c/ 0: If F1.a; b; c/ 0 then D D D d.abc/ d.a/bc ad.b/c abd.c/ 0 (36) C 0 C 0 C 0 D which gives d.abc/ d.a/bc ad.b/c abd.c/ (37) D C C Replacing b by bxa and c by b in (37), we get
d.abxab/ d.a/bxab ad.b/xab abd.x/ab abxd.a/b abxad.b/ (38) D C C C C Also, by definition of triple derivation we have
d.abxab/ d.ab/xab abd.x/ab abxd.ab/ 0 (39) C 0 C 0 C 0 D From (38) and (39) we get
.d.ab/ d.a/b ad.b/ /xab abx.d.ab/ d.a/b ad.b/ / abd.x/ab abd.x/ab 0 (40) C 0 C 0 C C 0 C 0 C C 0 D Applying Lemma 1.1 in (39) we get
abd.x/ab d.abxab/ d.ab/xab abxd.ab/ D C 0 C 0 Adding abd.x/ab on both sides we have abd.x/ab abd.x/ab 0: Thus from (40) we obtain 0 C 0 D .d.ab/ d.a/b ad.b/ /xab abx.d.ab/ d.a/b ad.b/ / 0 (41) C 0 C 0 C C 0 C 0 D By Lemmas 1.2 and 1.4, we obtain
.d.ab/ d.a/b ad.b//xcd 0; a; b; c; d; x S C K C K D 8 2 Semiprimness of S implies that d.ab/ d.a/b ad.b/ 0, this gives that d is a derivation. C 0 C 0 D If F2.a; b; c/ 0 then by Lemma 1.1, we have D abc cba; a; b; c S D 8 2 Thus d.a/bc cbd.a/; ad.b/c cd.b/a; abd.c/ d.c/ba: Hence from (34) D D D 2d.abc/ 2d.a/bc 2ad.b/c 2abd.c/ 0 C 0 C 0 C 0 D As concluded above, if d.abc/ d.a/bc ad.b/c abd.c/ 0 then d is a derivation. This completes the C 0 C 0 C 0 D proof. On Jordan mappings of inverse semirings 1131
Acknowledgement: We are thankful to the referees for their useful comments and suggestions.
References
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