(3s.) v. 35 1 (2017): 9–22. ISSN-00378712 in press doi:10.5269/bspm.v35i1.23568 Bol. Soc. Paran. Mat. c SPM –ISSN-2175-1188 on line SPM: www.spm.uem.br/bspm On symmetric biadditive mappings of semiprime rings Asma Ali, Khalid Ali Hamdin and Shahoor Khan abstract: Let R be a ring with centre Z(R). A mapping D(., .) : R × R −→ R is said to be symmetric if D(x, y) = D(y, x) for all x, y ∈ R. A mapping f : R −→ R defined by f (x) = D(x, x) for all x ∈ R, is called trace of D. It is obvious that in the case D(., .) : R × R −→ R is a symmetric mapping, which is also biadditive (i.e. additive in both arguments), the trace f of D satisfies the relation f (x + y) = f (x) + f (y) + 2D(x, y), for all x, y ∈ R. In this paper we prove that a nonzero left ideal L of a 2-torsion free semiprime ring R is central if it satisfies any one of the following properties: (i) f (xy) ∓ [x, y] ∈ Z(R), (ii) f (xy) ∓ [y, x] ∈ Z(R), (iii) f (xy) ∓ xy ∈ Z(R), (iv) f (xy) ∓ yx ∈ Z(R), (v) f ([x, y]) ∓ [x, y] ∈ Z(R), (vi) f ([x, y]) ∓ [y, x] ∈ Z(R), (vii) f ([x, y]) ∓ xy ∈ Z(R), (viii) f ([x, y]) ∓ yx ∈ Z(R), (ix) f (xy)∓f (x)∓[x, y] ∈ Z(R), (x) f (xy)∓f (y)∓[x, y] ∈ Z(R), (xi) f ([x, y])∓ f (x) ∓ [x, y] ∈ Z(R), (xii) f ([x, y]) ∓ f (y) ∓ [x, y] ∈ Z(R), (xiii) f ([x, y]) ∓ f (xy) ∓ [x, y] ∈ Z(R), (xiv) f ([x, y]) ∓ f (xy) ∓ [y, x] ∈ Z(R), (xv) f (x)f (y) ∓ [x, y] ∈ Z(R), (xvi) f (x)f (y)∓[y, x] ∈ Z(R), (xvii) f (x)f (y)∓xy ∈ Z(R), (xviii) f (x)f (y)∓ yx ∈ Z(R), (xix) f (x)◦f (y)∓[x, y] ∈ Z(R), (xx) f (x)◦f (y)∓xy ∈ Z(R), (xxi) f (x)◦ f (y)∓yx ∈ Z(R), (xxii) f (x)f (y)∓x◦y ∈ Z(R), (xxiii) [x, y]−f (xy)+f (yx) ∈ Z(R), for all x, y ∈ L, where f stands for the trace of a symmetric biadditive mapping D(., .) : R × R −→ R. Key Words: Semiprime rings, Left ideals, Symmetric biadditive mappings. Contents 1 Introduction 9 2 Preliminary result 10 3 Main Results 10 1. Introduction Throughout the paper R will denote an associative ring with centre Z(R). A ring R is said to be prime (resp. semiprime) if aRb = 0 implies that either a = 0 or b = 0 (resp. aRa = 0 implies that a = 0). We shall write for each pair of elements x, y ∈ R the commutator [x, y] = xy − yx and skew commutator x ◦ y = xy + yx. An additive mapping d : R −→ R is called a derivation if d(xy) = d(x)y + xd(y), for all x, y ∈ R. A derivation d is inner if there exists an element a ∈ R such that d(x) = [a, x] for all x ∈ R. A mapping D(., .) : R × R −→ R is said to be symmetric if D(x, y) = D(y, x) for all x, y ∈ R. A mapping f : R −→ R defined by f (x) = D(x, x) for all x ∈ R, is called trace of D. It is obvious that in the case D(., .) : R × R −→ R is a symmetric mapping, which is also biadditive (i.e. additive in both arguments), the trace f of 2000 Mathematics Subject Classification: 16W25, 16R50, 16N60. 9 Typeset by BSP style. M c Soc. Paran. de Mat. 10 Asma Ali, Khalid Ali Hamdin and Shahoor Khan D satisfies the relation f (x + y) = f (x) + f (y) + 2D(x, y), for all x, y ∈ R. A biadditive mapping D(., .) : R × R −→ R is said to be a biderivation on R if D(xy, z) = D(x, z)y + xD(y, z) and D(x, yz) = D(x, y)z + yD(x, z) for all x, y ∈ R. Gy. Maksa [6] introduced the concept of a symmetric biderivation. It was shown in [7] that symmetric biderivations are related to general solution of some functional equations. Some results on symmetric biderivation in prime and semiprime rings can be found in [8] and [9]. There has been ongoing interest concerning the relationship between the commutativity of a ring and the existence of certain specific types of derivations. Recently many authors viz. [1], [2], [3] and [4] have obtained commutativity of prime and semiprime rings with derivations satisfying certain polynomial identities. In this paper we prove that a nonzero left ideal of a semiprime ring admitting a biadditive map is central if it satisfies some polynomial identities. 2. Preliminary result We make extensive use of basic commutator identities [xy, z] = [x, z]y + x[y, z] and [x, yz] = [x, y]z + y[x, z]. Moreover, we shall require the following lemma: Lemma 2.1. [5, Lemma 1.1.5] If R is a semiprime ring, then the center of a nonzero one sided ideal is contained in the center of R. As an immediate consequence, any commutative one sided ideal is contained in the center of R. 3. Main Results Theorem 3.1. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f (xy) ∓ [x, y] ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Proof: Suppose f (xy) − [x, y] ∈ Z(R) for all x, y, ∈ L. (3.1) Replacing y by y + z in (3.1), we get f (xy) + f (xz) + 2D(xy, xz) − [x, y] − [x, z] ∈ Z(R) for all x, y, z ∈ L. (3.2) Since R is 2-torsion free, (3.2) yields that D(xy, xz) ∈ Z(R) for all x, y, z ∈ L. (3.3) Substituting y for z in (3.3), we get f (xy) = D(xy, xy) ∈ Z(R) for all x, y ∈ L. (3.4) On symmetric biadditive mappings of semiprime rings 11 In view of (3.1), (3.4) yields that [x, y] ∈ Z(R) for all x, y ∈ L. (3.5) [[x, y], r] = 0 for all x, y ∈ L, r ∈ R. (3.6) Then Replace x by xy in (3.6), to get [[x, y]y, r] = 0 for all x, y ∈ L, r ∈ R. (3.7) [x, y][y, r] = 0 for all x, y ∈ L, r ∈ R. (3.8) This implies that Replacing r by rx in (3.8), we get [x, y]r[y, x] = 0 for all x, y ∈ L, r ∈ R. (3.9) [x, y]R[x, y] = 0 for all x, y ∈ L. (3.10) This implies that Since R is semiprime, we get [x, y] = 0 for all x, y ∈ L, Hence L ⊆ Z(R) by Lemma 2.1. The proof is same for the case f (xy) + [x, y] ∈ Z(R) for all x, y ∈ L. ✷ Theorem 3.2. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f (xy) ∓ [y, x] ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Proof: The proof runs on the same parallel lines as of Theorem 3.1. ✷ Theorem 3.3. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f (xy) ∓ xy ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Proof: By hypothesis f (xy) − xy ∈ Z(R) for all x, y ∈ L. (3.11) Replacing y by y + z, we get f (xy) + f (xz) + 2D(xy, xz) − xy − xz ∈ Z(R) for all x, y, z ∈ L. (3.12) 12 Asma Ali, Khalid Ali Hamdin and Shahoor Khan Comparing (3.11) and (3.12) we obtain 2D(xy, xz) ∈ Z(R) for all x, y, z ∈ L. (3.13) Since R is 2-torsion free, we have D(xy, xz) ∈ Z(R) for all x, y, z ∈ L. (3.14) Substituting y for z in (3.14), we get f (xy) = D(xy, xy) ∈ Z(R) for all x, y ∈ L. (3.15) Using (3.11), we have xy ∈ Z(R) for all x, y ∈ L. This imples that [x, y] ∈ Z(R) for all x, y ∈ L. Arguing in the similar manner as in Theorem 3.1, we get the result . ✷ Theorem 3.4. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f (xy) ∓ yx ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Proof: The proof runs on the same parallel lines as of Theorem 3.3. ✷ Theorem 3.5. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f ([x, y]) ∓ [x, y] ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Proof: Let f ([x, y]) − [x, y] ∈ Z(R) for all x, y ∈ L. (3.16) Replacing y by y + z, we have f ([x, y] + [x, z]) − [x, y] − [x, z] ∈ Z(R) i.e. f ([x, y]) + f ([x, z]) + 2D([x, y], [x, z]) − [x, y] − [x, z] ∈ Z(R) for all x, y, z ∈ L. Using (3.16), we get 2D([x, y], [x, z]) ∈ Z(R) for all x, y, z ∈ L. (3.17) Substituting y for z in (3.17) and using the fact that R is 2-torsion free, we find f ([x, y]) ∈ Z(R) for all x, y ∈ L. (3.18) Using (3.16) and (3.18), we obtain [x, y] ∈ Z(R) for all x, y ∈ L. Arguing in the similar manner as in Theorem 3.1, we get the result . Similarly one can prove the result if f ([x, y]) + [x, y] ∈ Z(R) for all x, y ∈ L. ✷ Using similar arguments as we have done in the proof of Theorem 3.5, we can prove the following: 13 On symmetric biadditive mappings of semiprime rings Theorem 3.6. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f ([x, y]) ∓ [y, x] ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Theorem 3.7. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f ([x, y]) ∓ xy ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Proof: Let f ([x, y]) − xy ∈ Z(R) for all x, y ∈ L. (3.19) Replacing y by y+z in (3.19), we have f ([x, y]+[x, z])−xy−xz ∈ Z(R) for all x, y, z ∈ L. This implies that f ([x, y]) + f ([x, z]) + 2D([x, y], [x, z]) − xy − xz ∈ Z(R) for all x, y, z ∈ L. (3.20) Using (3.19), we obtain 2D([x, y], [x, z]) ∈ Z(R) for all x, y, z ∈ L. (3.21) Since R is 2-torsion free, (3.21) yields that D([x, y], [x, z]) ∈ Z(R) for all x, y, z ∈ L. (3.22) In particular, if we substitute y for z in (3.22), then we have f ([x, y]) ∈ Z(R) for all x, y ∈ L. Again using (3.19), we get xy ∈ Z(R) for all x, y ∈ L. This implies that [x, y] ∈ Z(R). Arguing in the similar manner as in Theorem 3.1, we get the result. Similarly we can prove the result if f ([x, y]) + xy ∈ Z(R) for all x, y ∈ L. ✷ Theorem 3.8. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f ([x, y]) ∓ yx ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Proof: The proof runs on the same parallel lines as of Theorem 3.7. ✷ Theorem 3.9. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f (xy) ∓ f (x) ∓ [x, y] ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Proof: Suppose f (xy) − f (x) − [x, y] ∈ Z(R) for all x, y ∈ L. (3.23) 14 Asma Ali, Khalid Ali Hamdin and Shahoor Khan Replacing y by y + z, we get f (xy) + f (xz) + 2D(xy, xz) − f (x) − [x, y] − [x, z] ∈ Z(R) for all x, y, z ∈ L. Using (3.23), we obtain f (xz) + 2D(xy, xz) − [x, z] ∈ Z(R) for all x, y, z ∈ L. (3.24) Substituting −z for z in (3.24), we get f (xz) − 2D(xy, xz) + [x, z] ∈ Z(R) for all x, y, z ∈ L. (3.25) Adding (3.24) and (3.25), we obtain 2f (xz) ∈ Z(R) for all x, y, z ∈ L. (3.26) Since R is 2-torsion free, we have f (xy) ∈ Z(R) for all x, y ∈ L. Using (3.23), we get f (x) − [x, y] ∈ Z(R) for all x, y ∈ L. (3.27) Replacing x by x + z, in (3.27), we have f (x) + f (z) + 2D(x, z) − [x, y] − [z, y] ∈ Z(R) for all x, z ∈ L. (3.28) Again using (3.27) and using 2-torsion freeness of R, we find D(x, z) ∈ Z(R). In particular f (x) = D(x, x) ∈ Z(R) for all x ∈ L. Since f (xz) ∈ Z(R) and f (x) ∈ Z(R), we have f (xy) − f (x) ∈ Z(R) for all x, y ∈ L. Using (3.23), we get [x, y] ∈ Z(R) for all x, y ∈ L. Arguing in the similar manner as in Theorem 3.1, we get the result. Similarly we can prove the result if f (xy)+f (x)+[x, y] ∈ Z(R) for all x, y ∈ L.✷ Theorem 3.10. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f (xy) ∓ f (y) ∓ [x, y] ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Proof: Let f (xy) − f (y) − [x, y] ∈ Z(R) for all x, y ∈ L. (3.29) Replacing y by y + z, we have f (xy) + f (xz) + 2D(xy, xz) − f (y) − f (z) − 2D(y, z) − [x, y] − [x, z] ∈ Z(R) for all x, y, z ∈ L. Using (3.29), we get 2(D(xy, xz) − D(y, z)) ∈ Z(R) for all x, y, z ∈ L. (3.30) Substituting y for z in (3.30) and using the fact that R is 2-torsion free, we find f (xy) − f (y) ∈ Z(R) for all x, y ∈ L. (3.31) This implies that [x, y] ∈ Z(R) for all x, y ∈ L. Arguing in the similar manner as in Theorem 3.1, we get the result. Similarly we can prove the result if f (xy)+f (y)+[x, y] ∈ Z(R) for all x, y ∈ L.✷ On symmetric biadditive mappings of semiprime rings 15 Theorem 3.11. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f ([x, y])∓f (x)∓[x, y] ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Proof: Suppose f ([x, y]) − f (x) − [x, y] ∈ Z(R) for all x, y ∈ L. (3.32) Replacing x by x + z in (3.32), we obtain f ([x, y]) + f ([z, y]) + 2D([x, y], [z, y]) − f (x) − f (z) −2D(x, z) − [x, y] − [z, y] ∈ Z(R) for all x, y, z ∈ L. (3.33) Using (3.32),we have 2(D([x, y], [z, y]) − D(x, z)) ∈ Z(R) for all x, y, z ∈ L. (3.34) Substituting x for z in (3.34) and using the fact that R is 2-torsion free, we obtain f ([x, y]) − f (x) ∈ Z(R) for all x, y ∈ L. (3.35) Again using (3.32) and (3.35), we have [x, y] ∈ Z(R) for all x, y ∈ L. Arguing in the similar manner as in Theorem 3.1, we get the result . Similarly we can prove the Theorem if f ([x, y]) + f (x) + [x, y] ∈ Z(R) for all x, y ∈ L. ✷ Theorem 3.12. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f ([x, y])∓ f (y)∓ [x, y] ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Proof: Suppose f ([x, y]) − f (y) − [x, y] ∈ Z(R) for all x, y ∈ L. (3.36) Replacing y by y + z, we get f ([x, y]) + f ([x, z]) + 2D([x, y], [x, z]) − f (y) − f (z) − 2D(y, z) −[x, y] − [x, z] ∈ Z(R) for all x, y, z ∈ L. (3.37) In view of (3.36), (3,37) yields that 2(D([x, y], [x, z]) − D(y, z)) ∈ Z(R) for all x, y, z ∈ L. (3.38) Substituting y for z in (3.38) and using the fact that R is 2-torsion free, we obtain f ([x, y]) − f (y) = D([x, y], [x, y]) − D(y, y)) ∈ Z(R) for all x, y ∈ L. (3.39) 16 Asma Ali, Khalid Ali Hamdin and Shahoor Khan Using (3.36) and (3.39), we have [x, y] ∈ Z(R) for all x, y ∈ L. Arguing in the similar manner as in Theorem 3.1, we get the result . Similarly we can prove the Theorem if f ([x, y]) + f (y) + [x, y] ∈ Z(R) for all x, y ∈ L. ✷ Using the similar techniques as we have used in the proof of Theorem 3.11 and 3.12, we can prove the following: Theorem 3.13. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f ([x, y])∓f (x)∓[y, x] ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Theorem 3.14. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f ([x, y])∓f (y)∓[y, x] ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Theorem 3.15. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f ([x, y])∓f (xy)∓[x, y] ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Proof: Let f ([x, y]) − f (xy) − [x, y] ∈ Z(R) for all x, y ∈ L. (3.40) Replacing y by y + z in (3.40), we get f ([x, y]) + f ([x, z]) + 2D([x, y], [x, z]) − f (xy) − f (xz) − 2D(xy, xz) −[x, y] − [x, z] ∈ Z(R) for all x, y, z ∈ L. (3.41) Using (3.40) and (3.41), we obtain 2(D([x, y], [x, z]) − D(xy, xz) ∈ Z(R) for all x, y, z ∈ L. (3.42) Since R is 2-torsion free, we have D([x, y], [x, z]) − D(xy, xz) ∈ Z(R) for all x, y, z ∈ L. (3.43) Substituting y for z in (3.43), we get f ([x, y]) − f (xy) ∈ Z(R) for all x, y ∈ L. (3.44) Using (3.40), we have [x, y] ∈ Z(R) for all x, y ∈ L. Arguing in the similar manner as in Theorem 3.1, we get the result. On symmetric biadditive mappings of semiprime rings The proof is same if f ([x, y]) + f (xy) + [x, y] ∈ Z(R) for all x, y ∈ L. 17 ✷ Similarly we can prove the following: Theorem 3.16. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f ([x, y])∓f (xy)∓[y, x] ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Theorem 3.17. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f (x)f (y) ∓ [x, y] ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Proof: Suppose f (x)f (y) − [x, y] ∈ Z(R) for all x, y ∈ L. (3.45) Substituting y + z for y in (3.45), we have f (x)f (y) + f (x)f (z) + 2f (x)D(y, z) − [x, y] − [x, z] ∈ Z(R) for all x, y, z ∈ L. (3.46) Using (3.45), we find 2f (x)D(y, z) ∈ Z(R) for all x, y, z ∈ L. (3.47) Since R is 2-torsion free, we have f (x)D(y, z) ∈ Z(R) for all x, y, z ∈ L. (3.48) In particular if we replace z by y in (3.48), then f (x)f (y) ∈ Z(R) for all x, y ∈ L. (3.49) Hence using (3.49) and (3.45), we obtain [x, y] ∈ Z(R) for all x, y ∈ L. Arguing in the similar manner as in Theorem 3.1, we get the result. Similarly we can prove the case if f (x)f (y) + [x, y] ∈ Z(R) for all x, y ∈ L. ✷ Theorem 3.18. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f (x)f (y) ∓ [y, x] ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Proof: The proof runs on the same parallel lines as that of Theorem 3.17. ✷ 18 Asma Ali, Khalid Ali Hamdin and Shahoor Khan Theorem 3.19. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f (x)f (y) ∓ xy ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Proof: Let f (x)f (y) − xy ∈ Z(R) for all x, y ∈ L. (3.50) Substituting y + z for y in (3.50), we have f (x)f (y) + f (x)f (z) + 2f (x)D(y, z) − xy − xz ∈ Z(R) for all x, y, z ∈ L. (3.51) Applying (3.50), we find 2f (x)D(y, z) ∈ Z(R) for all x, y, z ∈ L. (3.52) Since R is 2-torsion free, we have f (x)D(y, z) ∈ Z(R) for all x, y, z ∈ L (3.53) In particular replacing z by y in (3.53) and using (3.50), we find f (x)f (y) ∈ Z(R) for all x, y ∈ L. (3.54) This implies that xy ∈ Z(R) for all x, y ∈ L. Arguing in the similar manner as in Theorem 3.1, we get the result. Similarly we can prove the case if f (x)f (y) + xy ∈ Z(R) for all x, y ∈ L. ✷ Theorem 3.20. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f (x)f (y) ∓ yx ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Proof: The proof runs on the same parallel lines as that of Theorem 3.19. ✷ Theorem 3.21. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f (x) ◦ f (y) ∓ [x, y] ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Proof: Suppose f (x) ◦ f (y) − [x, y] ∈ Z(R) for all x, y ∈ L. (3.55) Replacing y by y + z in (3.55), we get f (x) ◦ f (y) + f (x) ◦ f (z) + 2(f (x) ◦ D(y, z)) − [x, y] − [x, z] ∈ Z(R) for all x, y ∈ L. (3.56) 19 On symmetric biadditive mappings of semiprime rings Comparing (3.55) and (3.56), we have 2(f (x) ◦ D(y, z)) ∈ Z(R) for all x, y, z ∈ L. Since R is 2-torsion free, we find f (x) ◦ D(y, z) ∈ Z(R) for all x, y, z ∈ L. (3.57) Replacing y by z in (3.57),we get f (x) ◦ f (y) ∈ Z(R) for all x, y, z ∈ L. (3.58) From (3.55) and (3.58), we have [x, y] ∈ Z(R) for all x, y ∈ L. Arguing in the similar manner as in Theorem 3.1, we get the result. The proof is same if f (x) ◦ f (y) + [x, y] ∈ Z(R) for all x, y ∈ L. ✷ Theorem 3.22. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f (x) ◦ f (y) ∓ xy ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Proof: Suppose f (x) ◦ f (y) − xy ∈ Z(R) for all x, y ∈ L. (3.59) Replacing y by y + z, in (3.59), we have f (x)◦f (y)+f (x)◦f (z)+2(f (x)◦D(y, z))−xy−xz ∈ Z(R) for all x, y, z ∈ L. (3.60) Comparing (3.59) and (3.60), we have 2(f (x) ◦ D(y, z)) ∈ Z(R) for all x, y, z ∈ L. (3.61) Substitute y for z in (3.61) and using 2-torsion freeness of R, we get f (x) ◦ f (y) ∈ Z(R) for all x, y ∈ L. (3.62) Using (3.59) and (3.62), we obtain xy ∈ Z(R) for all x, y ∈ L. (3.63) Interchanging the role of x and y in (3.63) and subtracting from (3.63), we find [x, y] ∈ Z(R) for all x, y ∈ L. (3.64) Arguing in the similar manner as in Theorem 3.1, we get the result. The prove is same for the case f (x) ◦ f (y) + xy ∈ Z(R) for all x, y ∈ L. ✷ 20 Asma Ali, Khalid Ali Hamdin and Shahoor Khan Theorem 3.23. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f (x) ◦ f (y) ∓ yx ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Proof: The proof runs on the same parallel lines as of Theorem 3.22. ✷ Theorem 3.24. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f (x)f (y) ∓ x ◦ y ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Proof: Suppose f (x)f (y) − x ◦ y = 0 for all x, y ∈ L. (3.65) Replacing y by y + z in (3.65), we get f (x)f (y) + f (x)f (z) + 2f (x)D(y, z) − x ◦ y − x ◦ z = 0 for all x, y, z ∈ L. (3.66) From (3.65) and (3.66), we have 2f (x)D(y, z) = 0 for all x, y, z ∈ L. (3.67) Using 2-torsion freeness of R and replacing y by z in (3.67), we get f (x)f (y) = 0 for all x, y ∈ L. (3.68) Using (3.68) and (3.65), we have xy + yx = 0 for all x, y ∈ L. (3.69) Replace y by ry in (3.69) and using (3.69), we get [x, r]y = 0 for all x, y ∈ L, r ∈ R. (3.70) A simple calculation yields that [x, r]R[x, r] = 0 for all x, y ∈ L, r ∈ R. Since R is semiprime, we have [x, r] = 0 for all x ∈ L, r ∈ R. Hence L ⊆ Z(R). Similarly we can prove if f (x)f (y) + x ◦ y ∈ Z(R) for all x, y ∈ L. ✷ Theorem 3.25. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If f (x) ◦ f (y) ∓ x ◦ y = 0 for all x, y ∈ L, then L ⊆ Z(R). Proof: Suppose f (x) ◦ f (y) − x ◦ y = 0 for all x, y ∈ L. (3.71) On symmetric biadditive mappings of semiprime rings 21 Replace y by y + z in (3.71), we have f (x) ◦ f (y) + f (x) ◦ f (z) + 2f (x) ◦ D(y, z) − x ◦ y − x ◦ z = 0 for all x, y, z ∈ L. (3.72) Comparing (3.71) and (3.72), we get 2f (x) ◦ D(y, z) = 0 for all x, y, z ∈ L. (3.73) Using 2-torsion freeness of R and replacing z by y in (3.73), we obtain f (x) ◦ f (y) = 0 for all x, y ∈ L. (3.74) Using (3.74) and (3.71), we have x ◦ y = 0 for all x, y ∈ L. (3.75) Using the same argument as we have done in the proof of Theorem 3.24, we get the result. ✷ Theorem 3.26. Let R be a 2-torsion free semiprime ring and L be a nonzero left ideal of R. Let D(., .) : R × R −→ R be a symmetric biadditive mapping and f be the trace of D. If [x, y] − f (xy) + f (yx) ∈ Z(R) for all x, y ∈ L, then L ⊆ Z(R). Proof: Suppose that [x, y] − f (xy) + f (yx) ∈ Z(R) for all x, y ∈ L. (3.76) Replacing y by y + z in (3.76), we get [x, y] + [x, z] − f (xy) − f (xz) − 2D(xy, xz) +f (yx) + f (zx) + 2D(yx, zx) ∈ Z(R) for all x, y, z ∈ L. (3.77) This implies that −2D(xy, xz) + 2D(yx, zx) ∈ Z(R) for all x, y, z ∈ L. (3.78) Since R is 2-torsion free, we have −D(xy, xz) + D(yx, zx) ∈ Z(R) for all x, y, z ∈ L. (3.79) Replacing z by y in (3.79), we get −f (xy) + f (yx) ∈ Z(R) for all x, y ∈ L. (3.80) Comparing (3.76) and (3.80), we get [x, y] ∈ Z(R) and arguing in the similar manner as we have done in the proof of Theorem 3.1, we get the result. ✷ 22 Asma Ali, Khalid Ali Hamdin and Shahoor Khan References 1. Ashraf, M., Ali, A. and Ali, S., Some commutativity theorems for rings with generalized derivations, Southeast Asian Bull. Math. 31 (2007), 415-421. 2. Bell, H. E. and Martindale, W. S., Centralizing mappings of semiprime rings, Canad. Math. Bull. 30 (1987), 92-101. 3. Bell, H. E. and Daif, M. N., On derivations and Commutativity in prime rings, Acta. Math. Hungar., 66 (4) (1995), 337-343. 4. Daif, M. N. and Bell, H. E., Remarks on derivations of semiprime rings, Int. J. Math. and Math. Sci. 15 (1) (1992), 205-206. 5. Herstein, I.N., Rings with Involution, University of chicago press, chicago (1976). 6. Maksa, G., A remark on symmetric biadditive functions having non-negative diagonalization, Glasnik. Mat. 15 (35) (1980), 279-282. 7. Maksa, G., On the trace of symmetric biderivations, C. R. Math. Rep. Acad. Sci. Canada 9 (1987),303-307. 8. Vukman, J., Symmetric biderivation on prime and semiprime rings, Aequationes Math. 38 (1989), 245-254. 9. Vukman, J., Two results concerning symmetric biderivation on prime rings, Aequationes Math. 40 (1990), 181-189. Asma Ali Department of Mathematics Aligarh Muslim University Aligarh-202002, India E-mail address: asma_ali2@rediffmail.com and Khalid Ali Hamdin Department of Mathematics Aligarh Muslim University Aligarh-202002, India E-mail address: hamdinkh@yahoo.com and Shahoor Khan Department of Mathematics Aligarh Muslim University Aligarh- 202002, India E-mail address: shahoor.khan@rediffmail.com