Let R be a noncommutative prime ring, U be the left Utumi quotient ring of R, and k, m, n, r be fixed positive integers. If there exist a generalized derivation G and a derivation g (which is independent of G) of R such that [G(x^m)xⁿ + xⁿg(x^m), x^r]_k = 0, for all x ∈ R, then there exists a ∈ U such that G(x) = ax, for all x ∈ R. As a consequence of the result in the present article, one may obtain Theorem 1 in Demir and Argaç [ 10 Demir, Ç. , Argaç, N. (2010). A result on generalized derivations with Engel conditions on one-sided ideals. J. Korean Math. Soc. 47(3):483–494. [Crossref], [Web of Science ®] [Google Scholar] ].

Herstein proved that a prime ring R of $\mathit{char}(R)\ne 2$ is commutative if there is a nonzero derivation d of R such that $[d(x),d(y)]=0$ for all $x,y\in R.$ The aim of this paper is to prove the $*$-version of Herstein’s result with a pair of derivations on prime ideals of a ring with involution. Precisely, we prove the following result: let R be a ring with involution $*$ of the second kind, P a prime ideal of R such that $S(R)\cap Z(R)\cancel{\subseteq }P$ and $\mathit{char}(R/P)\ne 2.$ If d₁ and d₂ are derivations of R satisfying the condition $[d_1(x),d_2(x^{*})]\in P$ for all $x\in R,$ then one of the following holds: (a) $d_1(R)\subseteq P$ (b) $d_2(R)\subseteq P$ (c) R/P is a commutative integral domain. Moreover, some related results are also discussed. As consequences of our main theorems, many known results can be either generalized or deduced.

A well-known result of Posner’s second theorem states that if the commutator of each element in a prime ring and its image under a nonzero derivation are central, then the ring is commutative. In the present paper, we extended this bluestocking theorem to an arbitrary ring with involution involving prime ideals. Further, apart from proving several other interesting and exciting results, we established the ∗-version of Vukman’s theorem. Precisely, we describe the structure of quotient ring $\mathfrak{A}/\mathfrak{L}$, where $\mathfrak{A}$ is an arbitrary ring and $\mathfrak{L}$ is a prime ideal of $\mathfrak{A}$. Further, by taking advantage of the ∗-version of Vukman’s theorem, we show that if a 2-torsion free semiprime $\mathfrak{A}$ with involution admits a nonzero ∗-centralizing derivation, then $\mathfrak{A}$ contains a nonzero central ideal. This result is in the spirit of the classical result due to Bell and Martindale (Theorem 3). As the applications, we extended and unified several classical theorems. Finally, we conclude our paper with a direction for further research.