A Result Related to Herstein Theorem

A result related to derivations on unital semiprime rings

“…for all x ∈ R. Recently, M. Fošner and the authors ( [11]) proved the following result regarding the relation (4), which is related to the Vukman's conjecture mentioned above.…”

“…We see that in the above relations the coefficients of terms including D(x 2 ) follow as (1, 1), (1, 2, 1), (1,3,3,1), (1,4,6,4,1) and therefore form a Pascal triangle. The coefficients of terms including D(x) follow as (1, 0, 1), (3, 1, 1, 3), (7, 4, 2, 4, 7), (15,11,6,6,11,15). Is there any specific algorithm that can foretell the coefficients of terms including D(x) as we proceed with the above procedure?…”

“…Since the determinant of the matrix is different from zero, it follows that the system has only a trivial solution. In particular, if y is replaced with the identity element e and considering (11), we obtain…”

A Note on -Jordan Derivations of Rings and Banach Algebras

“…Cusack [9] generalised Herstein's theorem to 2-torsion free semiprime rings (see [2] for an alternative proof). For generalisations of Herstein's theorem, we refer to [1,8,11]. An additive mapping D : R → R is called a left derivation if D(xy) = yD(x) + xD(y) holds for all pairs x, y ∈ R, and a left Jordan derivation (or Jordan left derivation) if D(x 2 ) = 2xD(x) for all x ∈ R. The concepts of left derivation and left Jordan derivation were introduced by Brešar and Vukman [7].…”