The purpose of this paper is to prove the following result. Let n≥3 be some fixed integer and let R be a (n+1)!2n-2-torsion free semiprime unital ring. Suppose there exists an additive mapping D: R→ ...R satisfying the relation for all x ∈ R. In this case D is a derivation. The history of this result goes back to a classical result of Herstein, which states that any Jordan derivation on a 2-torsion free prime ring is a derivation.
Let R be a ring having the property that every proper ideal of R is contained in a maximal ideal of R (in particular, if R is finitely generated as an ideal). Generalizing several known results, we ...characterize higher commutators V of R whenever R is generated by V (respectively, R,V) as an ideal. In particular, if V is a higher commutator of a unital ring R with 1∈V, then V is equal to either R or R,R, or R,R,R,R. Given a semiprime ring R, which is generated by R,R, all higher commutators of R are obtained if R possesses a central higher commutator. We also characterize all higher commutators of Mn(D) for n≥2 when D is a unital commutative ring. In addition, if D is 2-torsion free and 2D⊊D, then M2(D) has infinitely many higher commutators.
In this article, we prove the following result. Let
be some fixed integer and let
be a prime ring with
. Suppose there exists an additive mapping
satisfying the relation
for all
In this case,
is a ...derivation. This result is related to a classical result of Herstein, which states that any Jordan derivation on a prime ring with
is a derivation.
Let $ \mathcal{A} $ be a $ (p+q)! $-torsion free semiprime ring. We proved that if $ \mathcal{H}, \mathcal{D} : \mathcal{A}\to \mathcal{A} $ are two additive mappings fulfilling the algebraic ...identity $ 2\mathcal{H}(a^{p+q}) = \mathcal{H}(a^p) a^q+ a^p \mathcal{D}(a^q)+\mathcal{H}(a^q) a^p+ a^q \mathcal{D}(a^p) $ for all $ a\in \mathcal{A} $, then $ \mathcal{H} $ is a generalized derivation with $ \mathcal{D} $ as an associated derivation on $ \mathcal{A} $. In addition to that, it is also proved in this article that $ \mathcal{H}_1 $ is a generalized left derivation associated with a left derivation $ \delta $ on $ \mathcal{A} $ if they fulfilled the algebraic identity $ 2\mathcal{H}_1(a^{p+q}) = a^p \mathcal{H}_1(a^q)+ a^q \delta(a^p)+a^q \mathcal{H}_1(a^p)+ a^p \delta(a^q) $ for all $ a \in \mathcal{A} $. Further, the legitimacy of these hypotheses is eventually demonstrated by examples and at last, an application of Banach algebra is presented.
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