On Posner’s theorem with b-generalized skew derivations on Lie ideals

Let [Formula: see text] be a non-commutative prime ring of characteristic different from [Formula: see text] and [Formula: see text], [Formula: see text] its right Martindale quotient ring and [Formula: see text] its extended centroid. Suppose that [Formula: see text] is a non-central Lie ideal of [Formula: see text], [Formula: see text] a nonzero [Formula: see text]-generalized skew derivation of [Formula: see text]. If [Formula: see text] for all [Formula: see text], then one of the following holds: (a) there exists [Formula: see text] such that [Formula: see text], for all [Formula: see text]; (b) [Formula: see text], the ring of [Formula: see text] matrices over [Formula: see text], and there exist [Formula: see text] and [Formula: see text] such that [Formula: see text], for all [Formula: see text].

Differential identities in prime rings

Let R be a prime ring with center Z(R) and alpha,beta be automorphisms of R. This paper is divided into two parts. The first tackles the notions of (generalized) skew derivations on R, as the subject of the present study, several characterization theorems concerning commutativity of prime rings are obtained and an example proving the necessity of the primeness hypothesis of R is given. The second part of the paper tackles the notions of symmetric Jordan bi (alpha,beta)-derivations. In addition, the researchers illustrated that for a prime ring with char(R) different from 2, every symmetric Jordan bi (alpha,alpha)-derivation D of R is a symmetric bi (alpha,alpha)-derivation.

Some results on ∗-differential identities in prime rings

Let [Formula: see text] be a prime ring with involution ∗ of the second kind. An additive mapping [Formula: see text] is called generalized derivation if there exists a unique derivation [Formula: see text] such that [Formula: see text] for all [Formula: see text] In this paper, we investigate the structure of [Formula: see text] and describe the possible forms of generalized derivations of [Formula: see text] that satisfy specific ∗-differential identities. Precisely, we study the following situations: (i) [Formula: see text] (ii) [Formula: see text] (iii) [Formula: see text] (iv) [Formula: see text] for all [Formula: see text] Moreover, we construct some examples showing that the restrictions imposed in the hypotheses of our theorems are not redundant.

On Jordan ideals and Generalized (α, 1)- Reverse derivations in ∗-prime rings

Let R will be a 2- torsion free ∗-prime ring and α be an automorphisum of R. F be a nonzero generalized (α, 1)- reverse derivation of R with associated nonzero (α, 1)- reverse derivation d which commutes with ∗ and J be a nonzero ∗-Jordan ideal and a subring of R. In the present paper, we shall prove that R is commutative if any one of the following holds: (i) [F(u), u]α,1 = 0, (ii) F(u) α(u) = ud(u), (iii) F(u2) = ± α(u2), (iv) F(u2) = 2d(u) α(u), (v) d(u2) = 2F(u) α(u), for all u ∈ U.

Generalized g-derivations on prime rings

We introduce the definitions of [Formula: see text]-derivations and generalized [Formula: see text]-derivations on a ring [Formula: see text]. The main objective of the paper is to describe the structure of a prime ring [Formula: see text] in which [Formula: see text]-derivations and generalized [Formula: see text]-derivations satisfy certain algebraic identities with involution ⋆, anti-automorphism and automorphism. Some well-known results concerning derivations, generalized derivations, skew derivations and generalized skew derivations in prime rings, have been generalized to the case of [Formula: see text]-derivations and generalized [Formula: see text]-derivations.

On rings whose quasi-injective modules are injective or semisimple

Two obvious classes of quasi-injective modules are those of semisimples and injectives. In this paper, we study rings with no quasi-injective modules other than semisimples and injectives. We prove that such rings fall into three classes of rings, namely, (i) QI-rings, (ii) rings with no middle class, or (iii) rings that decompose into a direct product of a semisimple Artinian ring and a strongly prime ring. Thus, we restrict our attention to only strongly prime rings and consider hereditary Noetherian prime rings to shed some light on this mysterious case. In particular, we prove that among these rings, QIS-rings which are not of type (i) or (ii) above are precisely those hereditary Noetherian prime rings which are idealizer rings from non-simple QI-overrings.

Commutativity Results for Multiplicative (Generalized) (α,β) Reverse Derivations on Prime Rings

Let be a prime ring, be a non-zero ideal of and be automorphism on. A mapping is called a multiplicative (generalized) reverse derivation if where is any map (not necessarily additive). In this paper, we proved the commutativity of a prime ring R admitting a multiplicative (generalized) reverse derivation satisfying any one of the properties: for all x, y

COMMUTING SYMMETRIC BI-SEMIDERIVATIONS ON RINGS

In the present note, we discuss the notion of symmetric bi-semiderivations on rings and prove some commutativity results for commuting bi-semiderivations. Moreover, we obtain the characterization of symmetric bi-semiderivation on prime ring.

On n-generalized commutators and Lie ideals of rings

Let [Formula: see text] be an associative ring. Given a positive integer [Formula: see text], for [Formula: see text] we define [Formula: see text], the [Formula: see text]-generalized commutator of [Formula: see text]. By an [Formula: see text]-generalized Lie ideal of [Formula: see text] (at the [Formula: see text]th position with [Formula: see text]) we mean an additive subgroup [Formula: see text] of [Formula: see text] satisfying [Formula: see text] for all [Formula: see text] and all [Formula: see text], where [Formula: see text]. In the paper, we study [Formula: see text]-generalized commutators of rings and prove that if [Formula: see text] is a noncommutative prime ring and [Formula: see text], then every nonzero [Formula: see text]-generalized Lie ideal of [Formula: see text] contains a nonzero ideal. Therefore, if [Formula: see text] is a noncommutative simple ring, then [Formula: see text]. This extends a classical result due to Herstein [Generalized commutators in rings, Portugal. Math. 13 (1954) 137–139]. Some generalizations and related questions on [Formula: see text]-generalized commutators and their relationship with noncommutative polynomials are also discussed.

A result related to derivations on unital semiprime rings

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.