Let $\Re$ be an associative ring with involution $*$. An additive map $\lambda\rightarrow \lambda^{*}$ of $\Re$ into itself is called an involution if the following conditions are satisfied $(i) (\lambda\mu)^{*}=\mu^{*}\lambda^{*}$, $(ii) (\lambda^{*})^{*}=\lambda ~~ \mbox{for all}~ \lambda,\mu\in \Re$. A ring equipped with an involution is called an $*$-ring or ring with involution. The aim of the present paper is to establish some results on $*$-$\alpha$-derivations in $*$-rings and investigate the commutativity of Prime $*$-rings admitting $*$-$\alpha$-derivations on $\Re$ satisfying certain identities also prove that if $\Re$ admits a reverse $*$-$\alpha$-derivation $\delta$ of $\Re$, then $\alpha\in Z(\Re)$ and some related results have also been discussed.

A Prime ring ${S}$ with the centre ${Z}$ and generalised derivations that meet certain algebraic identities is considered. Let's assume that $\Psi$ and $\Phi$ are two generalised derivations associated with $\psi$ and $\phi$ on ${S},$ respectively. In this article, we examine the following identities: (i) $\Psi(a)b-a\Phi(b)\in {Z},$ (ii) $\Psi(a)b-b\Phi(a)\in {Z},$ (iii) $\Psi(a)a-b\Phi(b)\in {Z},$ (iv) $\Psi(a)a-a\Phi(b)\in {Z},$ (v) $\Psi(a)a-b\Phi(a)\in {Z},$ for every $a, b\in {J},$ where ${J}$ is a non-zero two sided ideal of ${S}.$ We also provide an example to show that the condition of Primeness imposed in the hypotheses of our results is essential.

Let R be a ring. The ring R is called weakly Prime center(WPC ring) if ab∈Z(R) implies that aRb is an ideal of R. In this paper, we prove that every left(right) duo ring is a WPC ring. Also we prove that some classes of rings with nilpotent Jacobson radical are WPC rings. Finally, we prove that a simple ring is a WPC ring if and only if it is a domain

This paper studies homoderivations satisfying certain conditions on semigroup ideals of near-rings. In addition, we include some examples of the necessity of the hypotheses used in our results.

Let $R$ be a Prime ring, $\alpha$ an automorphism of $R$ and $b$ an element of $Q$, the maximal right ring of quotients of $R$. The main purpose of this paper is to characterize skew $b$-derivations in Prime rings which satisfy various differential identities. Further, we provide an example to show that the assumed restrictions cannot be relaxed.

Let R be a Prime ring with its Utumi ring of quotients U, C=Z (U) the extended centroid of R, L a non-central Lie ideal of R and 0 ¹aÎR. If R admits a generalized derivation F such that a (F (u2)±F(u) 2) =0 for all uÎL, then one of the following holds: (1) there exists bÎU such that F (x) =bx for all xÎR, with ab=0;(2)F (x)=±x for all xÎR; (3) char (R)=2 and R satisfies s4; (4) char (R)¹2, R satisfies s4 and there exists bÎU such that F (x) = bx for all xÎR.We also study the situations (i)a (F (xmyn)±F (xm) F (yn)) =0 for all x, yÎR, and (ii) a (F (xmyn)±F (yn) F (xm))=0 for all x, y ÎR in Prime and semiPrime rings.

In this paper, we investigate the relationship between symmetric bi-semiderivations and Lie ideals of a Prime ring. Additionally, we extend some well-known results concerning symmetric biderivations of Prime rings to symmetric bi-semiderivations.

Let $R$ be a Prime ring and $Z(R)$ denotes the center of $R$. In this study, we expose the commutativity of $R$ as a consequence of specific differential identities involving derivations acting on left ideals of $R$. Finally, we give examples that demonstrate the necessity of hypotheses taken in the theorems.

Let R be a ring with derivation d, such that (d (xy)) n= (d (x)) n (d (y)) n for all x, y Î R and n³1 a fixed integer. In this paper, we show that if R is Prime, then d=0 or R is commutative. If R is semiPrime, thend maps R into its center. Moreover in semiPrime case let A=O (R) be the orthogonal completion of R and B=B (C) be the Boolian ring of C, where C is the extended centroid of R. Then there exists an idempotent e Î B such that e A is a commutative ring and d induces a zero derivation on (1-e) A.