On symmetric biadditive mappings of semiprime rings

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 ∈ R, where f stands for the trace of a
symmetric biadditive mapping D(., .) : R × R −→ R.