I Results in Mathematics

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1 Result.Matn. 46 (2004) /04/ DOII0.1007/s z Birkhauser Vertag, Basel, 2004 I Results in Mathematics Skew-commuting and Commuting Mappings in Rings with Left Identity R. K. Sharma and Basudeb Dhara Abstract. Our main object in this paper is to discuss some results on rings with left identity in which certain subsets satisfy some functional identities. Let R be an associative ring with centre Z = Z(R). Let f : R ---t R be a mapping and 5 be a subset of R. Then f is said to be commuting on 5 if [f(x), xj = 0 for all x E 5 and centralizing if [f(x), xl E Z for all x E S. Deng and Bell [3] extended these notions to n-commuting, by defining a mapping f : R ---t R to be n-commuting on 5 if [J(x), x n ] = 0 for all x E 5 and n-centralizing on 5 if [f(x), xnl E Z for all x E S. Analogously, call a mapping f : R ---t R. n-skew-commuting ( n-skew-centralizing) on 5 jf f(x)x n + x n f(x) = 0 (respectively f(x)x n + x n f(x) E Z) for all x E 5 We denote the Jordan product of x, y by (x, y) - i.e. (x, y) = xy + yx for all x, ye R. In [2], Bresar proved that if R is a prime ring of characteristic not 2 and f : R ---t R is an additive mapping on R such that f is skew-commuting on some ideal [ of R, then f(x) = 0 for all x E [. Bell and Lucier [lj proved some theorems in which primeness is replaced by the existence of a left identity. They investigated additive mappings which are skew-commuting or skew-centralizjng on subsets of certain rings, and obtained the following results: Theorem A. Let R be a 2-torsion free ring with left identity e, and let H be an additive subgroup of R containing e. If f is an additive map on R which is skew-comrnuting 011 H, then f(h) = {O}. In particular, if f is skew-cornmuting on R, then f = O. Theorem B. Let R be an arbitrary 2-torsion free ring with left identity. If f is any skew-centralizing additive mapping on R, then f is commuting on R. Moreover, f(x) = AX + g(x) for all x E R, where A E Z and 9 is an additive mapping from R into Z, "1991 Mathematics Subject Classifications: 16U80, 16W20, 16W25. Keywords: Skew-cornmuting and skew-centralizing!uaps. The second author is supported by a grant from Council of Scientific 8-Dd Industrial Research, India.

2 124 Sharma and Dhara Theorem C. Let n ;::: 2. Let R be an ni-torsion free ring with left identity e, and let H be an additive subgroup of R containing e. If f is an additive map on R which is n-skew commuting on If, then f(if) = {O} Our aim is to improve the above results in rings with left identity by introducing the multiadditive mapping. A mapping G : An -; R is said to be n-additive if G(X[, "', xn) is additive in each variable Xi, that is, G(X[,..., X, +Yi,..,xn) = G(Xl,..., Xi,.., xn)+g(xl,"., Yi,'.., x n ) for all Xi, Yi E A and ~ = 1,2,...,n. The mapping g : A -; R defined by g(x) = G(x, x,...,x) is called the trace of G. The mapping G : An -> R is said to be a symmetric mapping if G(SuCl)'.., SuCn») = G(SI.'., Sn), er E 3 n where 3 n denotes the symmetric group on n symbols. "Ve begin with a lemma Lemma 1.1 Let R be an n!-torsion free ring and If be an additive subgroup of R. Let G : Rn ---; R be an n-additive symmetric mapping and 9 be its tr?ce. If g(if) = 0 then G(XI..., x n ) = 0 for all Xt,- "lxn. E H. Proof. "Ve prove the lemma by induction on n. For n = 1, the lemma holds trivially. For n = 2. the relation (1 ) implies that 2G(XI,X2) = 0 and hence G(XI,X2) = 0 for all Xl,X2 EH. Thus the lemma holds true for n = 1,2. "Ve assume that the lemma is true for all n < m. Now we wish to prove that the result holds for n=m. For this, we suppose that G: R m -> R is an m-additive mapping. Then for Xj,X m E If, (2) where g1. = G(Xl l "" Xl, X tnr ; X m ). ~'-..--' m-i Since g(xl) = gm = go = 0, (2) reduces to (3) Now replacing Xl by Xj, 2XI, 3x[, (m - l)x[ in turn, and considering the resulting system of rn - 1 homogeneous equations, we see that the coefficient matrix of the system is a van der ~londe matrix

3 Sharma and Dhara 125 Since the determinant of the matrix is equal to a product of positive integers, each of which is less than m - I, and since R is m!-torsion free, it follows immediately that gl = 92 =... = gm-l = O. For a fixed X m E R,defineamappingG': Rm-l---. RbyG'(Xl,X2,...,X m _l) = G(XI,X2,...,X m -l,x Tn ). Then G' is a symmetric (m - I)-additive mapping whose trace G'(Xl,...,xI) = gtn-l = 0 for all Xl EH. Thus by our induction hypothesis G'(Xl,...,Xm-l) = 0 for ail XI"..,Xm-1 E Hand hence G(XI"'" x m ) = 0 for an X!,..., Xm E H. We prove the fonowing Theorem 1.2 Let R be an (n + m - 1)!-torsion free ring with left identity e, where m and n are some fixed natural numbers. Let H be an additive subgroup of R containing e and G : Rn ---. R be an n-additive mapping. F\uther, let 9 be the trace of G. If (4) for all X EH, where m = rni + ti = tk, i = 1,..,, k - 1; ti > 0, rn, ::>- 0 integers and Po, PI,...,Pk are integers such that Po =J 0 and Po + PI Pk =J 0, then g(h) = {O}, provided R is Po and (Po + PI Pk)-torsion free. In particular, if G is symmetric then G(Xl,"., Xn) = 0 for all Xl,.., Xn E H and if G is not symmetric then L G(Xo-(I),., Xo-(n)) = 0 for all Xl,., Xn E H. fjesn Proof. Suppose that G is a symmetric n-additive mapping. Putting X = e in (4) we get pog(e) + (PI + P Pk)g(e)e = O. Right multiplying by e we obtain (po + PI pk)g(e)e = O. Since R is (po + PI Pk)-torsion free, g(e)e = O. Then (5) implies that pog(e) = O. Again since R is po-torsion free, gee) = o. Now successive iinearization of (4) yields L { POXo-(I)'" Xo-(m)G(Xo-Cm+I)" ", Xo-(m+n)) JESm +n +... (6) Set gi = G(e,...,e,x,...,x) where X E H. Then 9n = gee) = O. '--...-"~ n-i

4 126 Sharma and Dhara Substituting Xl = X and Xz = X3 =... = X m +n = e in (6) and using gn = grey = 0 we get 1') (n + m-i).'.. n! m! {POgn-l -r- \Pl -r-... -r- Pk)gn-le} = O. ( \1 \ n-l Since R is (n + m - 1)!-torsion free, this implies Right multiplication of (7) by e gives (7) Since R is Po and (po + Pl Pk)-torsion free, gn-le = 0 and then from (7), gn-l = O. Now putting Xl = Xz = X and X3 =... = X m +n = e in (6) and using the fact gn = gn-l = 0 we get ( 2) (n + m- 2\) I '{ ( \} _ 2 n-2 n. m. pogn-z + PI Pk;gn-ze - O. Due to the torsion free restriction, this can be written as (8) Right multiplying (8) bye and again using (8) as before, we get gn-z = O. In a similar manner one can prove the general relation that (9) for i = 0,1,...,n For i = n, go = G(x,...,x) = g(x) = 0 for all X E H. Thus g(h) = {O} and then by Lemma 1.1, G(XI," _, Xn) = 0 for all Xl,..., Xn E H. Now suppose that G is not symmetric. The mapping G ' : Rn -+ R defined by GI(XI'.. _, xn) = L G(Xo-(l)"'" Xo-(n» is symmetric with trace n! G(x,..., x) i.e., n! g(x). Thus by replacing G' ryesrt for G, we obtain that L G(Xo-(l)"",x".(n» = 0 for all Xl,,Xn E H. ryest/. Corollary 1.3 Let R be a ring with left identity e and H be an additive subgroup containing e. Let G: Rn -+ R be an n-additive mapping with trace g, such that (10) for all X E H, where ml,..., mk are nonzero positive integers. Then g(h) = {O}, provided R is (ml +.. +mk +n-i)! and 2k-torsion free. In particular, if G is symmetric, then G(XI,'",xn) = o for all Xl,.. -,Xn E H; and if G is not symmetric, then L G(Xo-(l)"",Xo-(n» = 0 for all aes'n,

5 Sharma and Dhara 127 Proof. The identity (10) can be re-written as The number of terms In (11) is 2k. Since R is (ml mk + n - result follows from Theorem 1.2. (11) l)l and 2k-torsion free, the Theorem 1.4 Let R be an ni-torsion free ring with left identity e, where n > 0 is a fixed natural number. Let f : R -> R be an additive mapping. If for all x E R, where n = ni + ti = tk, i = 1,..., k - 1; ti > 0, ni :::: 0 integers and Po, Pl,...,Pk are integers such that Po cl 0 and Po + Pl Pk cl 0, then f is commuting on R, provided R is Po and (po + Pl Pk)-torsion free. Moreover, f(x) = >.x +!-L(x) for all x E R, where.\ E Z and /l- : R -> Z an additive mapping. Proof. For x = e, we have (12) pof(e) + (Pl Pk)f(e)e E Z. (13) Commuting with e we have po(f(e)e - J(e)) = 0 and then, since R is po-torsion free, J(e)e = fee). Then (13) implies (Po + Pl pdj(e) E Z. Again since R is (Po + Pl pd-torsion free, fee) E z. Linearization of (12) yields +... Putting Xl = X2 =... = X n = e, Xn+l = x and using fee) = J(e)e E Z in (14), we obtain that (14) Commuting this with e we have n l po(j(x)e f(x) = o. (16) Since R is Po and ni-torsion free, f(x)e = j(x). Then (15) reduces to n (n)! (po Pk)f(e)x + n! (Po Pk)f(x) E Z. \ '17),

6 128 Sharma and Dhara Since R is n! and (Po p,,)-torsion free, this can be written as n f(e)x + f(x) E Z. (18) It is now immediate that [J(x), x] = 0 for all x ER, i.e. f is commuting on R. Moreover, by re-writing (18) in the form j(x) = AX +!lex), where A = -nj(e) E Z and /-lex) = nj(e)x + j(x), we get the desired structure for f. Since j is additive, IL must be additive. Corollary 1.5 Let R be a ring with left identity e and j : R -+ R be an additive mapping. If for all x E R where ni,..., n" are nonzero positive integers, then j is commuting on Rand j(x) =.AI +!L(x) where A E Z and IL : R -+ Z is an additive mapping, provided R is (ni nk)! and 2k-torsion free. Proof. The identity (19) can be re-\vtitten as The number of terms in (20) is 2k. Then by Theorem 1.4, since R is (nl rq,y and 2k-torsion free, f is commuting on Rand f(x) = AX +!L(x), where A E Z and!l : R -+ Z is an additive mapping. Theorem 1.6 Let R be a ring with left identity e and H be an additive subgroup of R containing e. Suppose B : Rn -+ R is an n-additive mapping with trace b(x) = B(x,... x) such that [x m, b(x)] = 0 for all x E H (i.e., rn-commuting on H). For m = 2, jf R is (n + l)!-torsiorl free and for m> 2 if R is (n + m - 2)! and m-torsion free, then b(x) is commutmg on H. Proof. As earlier (in Theorem 1.2), we can assume that B is symmetric. \Ve define an (n + 1) additive mapping G : Rn+! -+ R by n+l G(Xj,...,xn+!) = I)Xi,B(Xl"",Xi-l,Xi+l,,xn+r)]. i=l Since B is symmetric, G is also symmetric and the trace of G is g(x) = (n + l)[x, B(x,..., x)). Now we have 0= [xm,b(x,,,.,x)j x m - 1 [x, B(x,.", x)] + x m - 2 [x, B(x,. ", x)lx [x, B(x,..., x)jx m - 1 (19) (20) This implies Then by Theorem 1.2, g(x) = 0 for all x EH. Since R is (n+ I)-torsion free, g(x) = 0 implies that [x, B(x,..., x)] = 0 for all x EH. Thus B is commuting on H.

7 Sharma and Dhara 129 References [1] H. E. Bell and J. Lucier, On additive maps and commutativity in rings, Result Math. 36 (1999) 1~8. [21 M. Bresar, On skew-commuting mappings of rings, Bull. Austral. Math. Soc. 47 (1993) [3] Q. Deng and H. E. Bell, On derivations and commutativity in semiprime rings, Comm. Algebra 23 (1995) R. K. Sharma Basudeb Dhara Department of Mathematics, Department of Mathematics, Indian Institute of Technology, Indian Institute of Technology, Hauz Khas, New Delhi, Kharagpur , INDIA , INDIA. Eingegangen am 30. Januar 2004