(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
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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