In this paper we prove that on a 2-torsion free semiprime ring R every Jordan triple (resp. generalized Jordan triple) derivation on Lie ideal L is a derivation on L (resp. generalized derivation on L ).

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) = xF(y) + d(x)y holds for all x,y ∈ R. It is called a generalized (α,β)−derivation on R if there exists an (α,β)−derivation d: R → R such that the equation F(xy) = F(x)α(y)+β(x)d(y) holds for all x,y ∈ R. In the present paper, we investigate commutativity of a prime ring R, which satisfies certain differential identities on left ideals of R. Moreover some results on commutativity of rings with involutions that satisfy certain identities are proved.

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 .

The purpose of this article is to enhance the previous studies regarding the behavior of a quotient ring $\Im /\wp$, where ℘ is a prime ideal in a ring ℑ. In particular, we are going to explore more general scenarios whenever a ring ℑ admits a generalized $(\alpha ,\beta )$-derivation associated with an $(\alpha ,\beta )$-derivation ∂ that satisfies certain criteria involving ℘, where $\alpha$ and $\beta$ are automorphisms on ℑ. Moreover, we provide some examples to demonstrate the importance of the assumptions made in our results.

The current article focuses on studying the behavior of a ring $\Re /\Pi$ when ℜ admits generalized derivations $\Psi$ and $\Omega$ with associated derivations $\varphi$ and $\delta$, respectively. These derivations satisfy specific differential identities involving $\Pi$, where $\Pi$ is a prime ideal of an arbitrary ring ℜ, not necessarily prime or semiprime. Furthermore, we explore some consequences of our findings. To emphasize the necessity of the primeness of $\Pi$ in the hypotheses of our various theorems, we provide a list of examples.