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In this note we investigated some conditions related to the commutativity of a prime ring R that satisfies certain identities and possesses a generalized (α, β)-reverse derivation. A few examples and counterexamples are also studied.

Let R be a prime ring with center Z(R). A map G : R →R is called a multiplicative (generalized) (α, β)-derivation if G(xy)= G(x)α(y)+β(x)g(y) is fulfilled for all x; y ∈ R, where g : R → R is any map (not necessarily derivation) and α; β : R → R are automorphisms. Suppose that G and H are two multiplicative (generalized) (α, β)-derivations associated with the mappings g and h, respectively, on R and α, β are automorphisms of R. The main objective of the present paper is to investigate the following algebraic identities: (i) G(xy) + α(xy) = 0, (ii) G(xy) + α(yx) = 0, (iii) G(xy) + G(x)G(y) = 0, (iv) G(xy) = α(y) ○ H(x) and (v) G(xy) = [α(y), H(x)] for all x, y in an appropriate subset of R.

Let $$(X, *, 0)$$ be a d-algebra and $$\alpha , \beta $$ are endomorphisms on X. Motivated by some results on derivations, $$(\alpha ,\beta )$$ -derivation in rings, and the generalizations of BCK and BCI-algebras, in this paper, we introduce the notion of $$(\alpha ,\beta )$$ -derivations on d-algebras, construct several examples and investigate some simple and important results.

In the present note, we use a pair of rings, which are the ingredients of a Morita context, and obtain that if one of the ring is prime with the generalized (α, β)-derivations that satisfy certain conditions on the trace ideal of the ring, which by default is a Lie ideal, and the other ring is reduced, then the trace ideal of the reduced ring is contained in the center of the ring. As an outcome, in case of a semi-projective Morita context, the reduced ring becomes commutative.

Let (R, ∗) be a 2-torsion free ∗-prime ring with involution ∗, and L � 0b e a ∗-Lie ideal of R. An additive mapping F: R → R is called a generalized derivation on R if there exists a derivation d: R → R such that F( xy ) = F(x)y + xd(y) holds for all x, y ∈ R. In the present paper, we shall show that when L satisfies any of several identities involving F, then L is central.

A classical problem in ring theory is to study conditions under which a ring is forced to become commutative. Stimulated from Jacobson's famous result, several techniques are developed to achieve this goal. In the present note, we use a pair of rings, which are the ingredients of a Morita context, and obtain that if one of the ring is prime with the generalized (α, β)-derivations that satisfy certain conditions on the trace ideal of the ring, which by default is a Lie ideal, and the other ring is reduced, then the trace ideal of the reduced ring is contained in the center of the ring. As an outcome, in case of a semi-projective Morita context, the reduced ring becomes commutative.

Let (R,∗) be a 2-torsion free ∗-prime ring with involution ∗, L= 0 be a nonzero square closed ∗-Lie ideal of R and Z the center of R. An additive mapping F: R −→ R is called a generalized derivation on R if there exists a derivation d: R−→Rcommutes with ∗ such that F(xy) = F(x)y +xd(y) holds for all x,y ∈ R. In the present paper, we shall show that L is contained in the center of R such that R admits a generalized derivations F and G with associated derivations d and g commute with ∗ satisfying several conditions.

Let (R, *) be a 2-torsion free *-prime ring with involution *, L ≠ 0 be a square closed *-Lie ideal of R and α, β automorphisms of R commuting with *. An additive mapping F: R → R is called a generalized (α, β)-derivation on R if there exists an (α, β)-derivation d such that F(xy) = F(x)α(y) + β(x)d(y) holds for all \({x, y \in R}\). In the present paper, we shall show that \({L\subseteq Z(R)}\) such that R is a *-prime ring admits a generalized (α, β)-derivation satisfying several conditions, but associated with an (α, β)-derivation commuting with *.

Let R be a ring and , be automorphisms of R. An additive mapping F: R → R is called a generalized ( \alpha,\beta )-derivation on R if there exists an ( \alpha,\beta )- derivation d: R → R such that F(xy) = F(x) \alpha(y) + \beta (x)d(y) holds for all x, y ∈ R. For any x, y ∈ R, set [x, y] _{\alpha,\beta} = x\alpha (y) − \beta (y)x and (x o y) _{\alpha,\beta} = x\alpha (y) + \beta (y)x. In the present paper, we shall discuss the commutativity of a prime ring R admitting generalized ( \alpha,\beta )-derivations F and G satisfying any one of the following properties: (i) F([x, y]) = (xoy) _{\alpha,\beta} , (ii) F(xoy) = [x, y] _{\alpha,\beta}, (iii) [F(x), y] _{\alpha,\beta} = (F(x)o y) _{\alpha,\beta} , (iv) F([x, y]) = [F(x), y] _{\alpha,\beta} , (v) F(xoy) = (F(x) o y)_{\alpha,\beta} , (vi) F([x, y] =[ \alpha(x),G(y)] and (vii) F(xoy) = ( \alpha(x)o G(y)) for all x, y in some appropriate subset of R. Finally, obtain some results on semi-projective Morita context with generalized (\alpha ,\beta )-derivations.

Let $R$ and $S$ be rings of a semi-projective Morita context, and $\alpha, \beta$ be automorphisms of $R$. An additive mapping $F$: $R\to R$ is called a generalized $(\alpha,\beta)$-derivation on $R$ if there exists an $(\alpha,\beta)$-derivation $d$: $R\to R$ such that $F(xy)=F(x)\alpha(y)+\beta(x)d(y)$ holds for all $x,y \in R$. For any $x,y \in R$, set $[x, y]_{\alpha, \beta} = x \alpha(y) - \beta(y) x$ and $(x \circ y)_{\alpha, \beta} = x \alpha(y) + \beta(y) x$. In the present paper, we shall show that if the ring $S$ is reduced then it is a commutative, in a compatible way with the ring $R$ . Also, we obtain some results on bialgebras via Cauchy modules.