The motivation for this paper lies in an attempt to extend in some way the famous results due to Posner [46], Vukman [48] and Ali-Dar [6]. A number of authors have generalized these theorems in several ways (see, for example, [16,20,33,35,39,42,50,51], where further references can be found). Throughout this article, $\mathfrak{A}$ will represent an associative ring with center $\mathfrak{Z}\left(\mathfrak{A}\right)$. The standard polynomial identity $s_4$ in four variables is defined as $s_4({\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4)={\sum }_{\sigma \in s_4}{(-1)}^{\sigma }{\ell }_{\sigma \left(1\right)}{\ell }_{\sigma \left(2\right)}{\ell }_{\sigma \left(3\right)}{\ell }_{\sigma \left(4\right)},$ where ${(-1)}^{\sigma }$ is $+1$ or $-1$ according to $\sigma$ being an even or odd permutation in symmetric group $s_4$. For any $s,t\in \mathfrak{A}$, the symbol $[s,t]=st-ts$ stands for commutator, while the symbol $s\circ t$ will stand for the anti-commutator $st+ts$. The higher order commutator is define as follows: for any $s,t\in \mathfrak{A}$, and inductively, we write ${[s,t]}_k=[{[s,t]}_{k-1},t]$, (where $k>1$ is a fixed integer), is called commutator of order k or simply $k^{th}$-commutator. It is also known as Engel condition in literature (viz.; [35]). Analogously, we define the higher order anit-commutator, and set and inductively, we set $s{\circ }_{k}t=\left(s{\circ }_{k-1}t\right)\circ t$ for $k>1,$ is called anti-commutator of order $k.$

“Recall that a ring $\mathfrak{A}$ is called prime if, for $a,b\in \mathfrak{A}$, $a\mathfrak{A}b=\left(0\right)$ implies $a=0$ or $b=0.$ By a prime ideal of a ring $\mathfrak{A}$, we mean a proper ideal $\mathfrak{L}$ and for $\ell ,\vartheta \in \mathfrak{A}$, $\ell \mathfrak{A}y\subseteq \in \mathfrak{L}$ implies that $\ell \in \mathfrak{L}$ or $\vartheta \in \mathfrak{L}$. We note that for a prime ring $\mathfrak{A}$, $\left(0\right)$ is the prime ideal of $\mathfrak{A}$ and $\mathfrak{A}/\mathfrak{L}$ is a prime ring. An ideal $\mathfrak{L}$ of a ring $\mathfrak{A}$ is called semiprime if it is the intersection of prime ideals or alternatively, if $a\mathfrak{A}a\subseteq \mathfrak{L}$ implies that $a\in \mathfrak{L}$ for any $a\in \mathfrak{A}.$ A ring $\mathfrak{A}$ is said to be n-torsion free if $n\ell =0$, $\ell \in \mathfrak{A}$ implies $\ell =0$. An additive mapping $\ell \mapsto {\ell }^{*}$ satisfying ${(\ell y)}^{*}={\left(\vartheta \right)}^{*}{\ell }^{*}$ and ${\left({\ell }^{*}\right)}^{*}=\ell$ is called an involution. A ring equipped with an involution is known as ring with involution or *-ring. An element ℓ in a ring with involution * is said to be hermitian if ${\ell }^{*}=\ell$ and skew-hermitian if ${\ell }^{*}=-\ell$. The sets of all hermitian and skew-hermitian elements of $\mathfrak{A}$ will be denoted by $\mathfrak{H}\left(\mathfrak{A}\right)$ and $S\left(\mathfrak{A}\right)$, respectively. If $\mathfrak{A}$ is 2-torsion free then every $\ell \in \mathfrak{A}$ can be uniquely represented in the form $2\ell =h+k$ where $h\in \mathfrak{H}\left(\mathfrak{A}\right)$ and $k\in S\left(\mathfrak{A}\right)$. The involution is said to be of the first kind if $\mathfrak{H}\left(\mathfrak{A}\right)\subseteq \mathfrak{Z}\left(\mathfrak{A}\right)$, otherwise it is said to be of the second kind. We refer the reader to [32] for justification and amplification for the above mentioned notations and key definitions.

A map $e:\mathfrak{A}\to \mathfrak{A}$ is a derivation of a ring $\mathfrak{A}$ if e is additive and satisfies $e(\ell \vartheta )=e(\ell )\vartheta +\ell e\left(\vartheta \right)$ for all $\ell ,\vartheta \in \mathfrak{A}$. A derivation e is called inner if there exists $a\in \mathfrak{A}$ such that $e(\ell )=[a,\ell ]$ for all $\ell \in \mathfrak{A}.$ An additive map $F:\mathfrak{A}\to \mathfrak{A}$ is called a generalized derivation if there exists a derivation e of $\mathfrak{A}$ such that $F(\ell \vartheta )=F(\ell )\vartheta +\ell e\left(\vartheta \right)$ for all $\ell ,\vartheta \in \mathfrak{A}$ (see [14] for details). For a nonempty subset S of $\mathfrak{A}$, a mapping $\xi :S\to \mathfrak{A}$ is called commuting (resp. centralizing) on S if $\left[\xi \right(\ell ),\ell ]=0$ (resp. $\left[\xi \right(\ell ),\ell ]\in \mathfrak{Z}\left(\mathfrak{A}\right))$ for all $\ell \in S$. The study of commuting and centralizing mappings goes back to 1955 when Divinsky [31] proved that a simple artinian ring is commutative if it has a commuting automorphism different from the identity mapping. Two years later Posner [46] showed that a prime ring must be commutative if it admits a nonzero centralizing derivation. In 1970, Luh [38] generalized Divinsky’s result for prime rings. Later Mayne [44] established the analogous result of Posner for nonidentity centralizing automorphisms. The culminating results in this series can be found in [19,20,21,33,35,43] and [48,49]. In [48], Vukman generalized Posner’s second theorem for second order commutator and established that if a prime ring of characteristic different from 2 admits a nonzero derivation e such that ${[e(\ell ),\ell ]}_2=0$ for all $\ell \in \mathfrak{A}$, then $\mathfrak{A}$ is commutative. Most classical and elegant generalization of Posner’s second theorem is due to Lanski [34]. Precisely, he proved that if a prime ring $\mathfrak{A}$ admits a nonzero derivation e such that ${[e(\ell ),\ell ]}_k=0$ for all $\ell \in L$, where L is a non-commutative Lie ideal of $\mathfrak{A}$ and $k>0$ a fixed integer, then $char\left(\mathfrak{A}\right)=2$ and $\mathfrak{A}\subseteq M_2\left(F\right)$ for a field F. These results have been extended in various ways (viz; [3,23,26,30,50,51] and references therein). The goal of this paper is to study these results in the setting of arbitrary rings with involution involving prime ideals and describe the structure of a quotient ring $\mathfrak{A}/\mathfrak{L},$ where $\mathfrak{A}$ is an arbitrary ring and $\mathfrak{L}$ is a prime ideal of $\mathfrak{A}$.

Let $\mathfrak{A}$ be a ring with involution * and S be a nonempty subset of $\mathfrak{A}$. Following [6,25], a mapping $\varphi$ of $\mathfrak{A}$ into itself is called *-centralizing on S if $\varphi (\ell ){\ell }^{*}-{\ell }^{*}\varphi (\ell )\in \mathfrak{Z}\left(\mathfrak{A}\right)$ for all $\ell \in S,$ in the special case where $\varphi (\ell ){\ell }^{*}-{\ell }^{*}\varphi (\ell )=0$ for all $\ell \in S,$ the mapping $\varphi$ is said to be *-commuting on S. In [6,25], the first author together with Dar initiated the study of these mappings and proved that the existence of a nonzero *-centralizing derivation of a prime ring with second kind involution forces the ring to be commutative. Apart from the characterizations of these mappings of prime and semiprime rings with involution, they also proved *-version of Posner’s second theorem and its related problems. Precisely, they established that : Let $\mathfrak{A}$ be a prime ring with involution * such that $char\left(\mathfrak{A}\right)\ne 2$. Let e be a nonzero derivation of $\mathfrak{A}$ such that $[e(\ell ),{\ell }^{*}]\in \mathfrak{Z}\left(\mathfrak{A}\right)$ for all $\ell \in \mathfrak{A}$ and $e\left(S\right(\mathfrak{A})\cap \mathfrak{Z}(\mathfrak{A}\left)\right)\ne \left(0\right).$ Then $\mathfrak{A}$ is commutative. Futher, they showed that every ${\hspace{0.17em}}^{*}$-commuting map $f:\mathfrak{A}\to \mathfrak{A}$ on semiprine ring with involution of characteristic different from two is of the form $f(\ell )=\lambda {\ell }^{*}+{\mu }^{\prime }(\ell )$ for all $\ell \in \mathfrak{A},\lambda \in C$ (the extended centroid of $\mathfrak{A})$ and ${\mu }^{\prime }:\mathfrak{A}\to C$ is an additive mapping. In the sequel, recently Nejjar et al. [42, Theorem 3.7] established that if a 2-torsion free prime ring with involution of the second kind admits a nonzero derivation e such that $e(\ell ){\ell }^{*}-{\ell }^{*}e(\ell )\in \mathfrak{Z}\left(\mathfrak{A}\right)$ for all $\ell \in \mathfrak{A},$ then $\mathfrak{A}$ is commutative. In 2020, Alahmadi et al. [2] extend the above mentioned result for generalized derivations. Over the last few years the interest on this topic has been increased and numerous papers concerning these mappings on prime rings have been published (see [1,2,7,8,9,10,13,39,42,45] and references therein). In [24], Creedon studied the action of derivations of prime ideals and proved that if e is a derivation of a ring $\mathfrak{A}$ and $\mathfrak{L}$ is a semiprime ideal of $\mathfrak{A}$ such that $\mathfrak{A}/\mathfrak{L}$ is characteristic-free and $e^k\left(\mathfrak{L}\right)\subseteq \mathfrak{L}$, then $e\left(\mathfrak{L}\right)\subseteq \mathfrak{L}$ for some positive integer k.

In view of the above observations and motivation, the aims of the present paper is to prove the following main theorems.

Theorem-A Let $\mathfrak{A}$ be a ring with involution * of the second kind and $\mathfrak{L}$ a prime ideal of $\mathfrak{A}$ such that $S\left(\mathfrak{A}\right)\cap \mathfrak{Z}\left(\mathfrak{A}\right)⊈\mathfrak{L}$. If $e_1$ and $e_2$ are derivations of $\mathfrak{A}$ such that $e_1(\ell ){\ell }^{*}-{\ell }^{*}{e}_2(\ell )\in \mathfrak{L}$ for all $\ell \in \mathfrak{A}$, then one of the following holds:

Theorem-B Let $\mathfrak{A}$ be a ring with involution * of the second kind and $\mathfrak{L}$ a prime ideal of $\mathfrak{A}$ such that $S\left(\mathfrak{A}\right)\cap \mathfrak{Z}\left(\mathfrak{A}\right)⊈\mathfrak{L}$. If $e_1$ and $e_2$ are derivations of $\mathfrak{A}$ such that $[e_1(\ell ),{\ell }^{*}]+[\ell ,e_2\left({\ell }^{*}\right)]+[\ell ,{\ell }^{*}]\in \mathfrak{L}$ for all $\ell \in \mathfrak{A}$, then one of the following holds:

Theorem-C Let $\mathfrak{A}$ be a ring with involution * of the second kind and $\mathfrak{L}$ a prime ideal of $\mathfrak{A}$ such that $S\left(\mathfrak{A}\right)\cap \mathfrak{Z}\left(\mathfrak{A}\right)⊈\mathfrak{L}$. If $\mathfrak{A}$ admits a derivation e such that $e(\ell {\ell }^{*})-e\left({\ell }^{*}\right)e(\ell )\in \mathfrak{L}$ for all $\ell \in \mathfrak{A},$ then one of the following holds:

Theorem-D Let $\mathfrak{A}$ be a ring with involution * of the second kind and $\mathfrak{L}$ a prime ideal of $\mathfrak{A}$ such that $S\left(\mathfrak{A}\right)\cap \mathfrak{Z}\left(\mathfrak{A}\right)⊈\mathfrak{L}$. If $\mathfrak{A}$ admits a derivation e such that $[[e(\ell ),{\ell }^{*}],{\ell }^{*}]\in \mathfrak{L}$ for all $\ell \in \mathfrak{A},$ then one of the following holds:

Theorem-E Let $\mathfrak{A}$ be a 2-torsion free semiprime ring with involution * of the second kind. If $\mathfrak{A}$ admits a nonzero *-centralizing derivation $e,i.e.,[e(\ell ),{\ell }^{*}]\in \mathfrak{Z}\left(\mathfrak{A}\right)$ for all $\ell \in \mathfrak{A},$ then $\mathfrak{A}$ contains a nonzero central ideal.

In view of ${\hspace{0.17em}}^{*}$-centralizing mappings, Theorem-e and Theorem-E recognized as the ${\hspace{0.17em}}^{*}$-versions of well known theorems due to Posner [46] and Vukman [48]. As the applications of Theorems A to E just mentioned above, we extends and unify several classical theorems proved in [6,7,25,42,46] and [48,49]. Since these results are in new direction, so there are various interesting open problems related to our work. Hence, we conclude our paper with a direction for further research in this new and exciting area of theory rings with involution.

We shall do a great deal of calculation with commutators and anti-commutators, routinely using the following basic identities: For all $s,t,w\in \mathfrak{A};$

Let $\mathfrak{A}$ be ${\hspace{0.17em}}^{*}$-ring. Following [8,47], an additive mapping $e:R\to R$ is called a *-derivation of $\mathfrak{A}$ if $e(\ell \vartheta )=e(\ell ){\vartheta }^{*}+\ell d\left(\vartheta \right)$ for all $\ell ,\vartheta \in \mathfrak{A}.$ An additive mapping $e:\mathfrak{A}\to \mathfrak{A}$ is called a Jordan *-derivation of $\mathfrak{A}$ if $e\left({\ell }^2\right)=e(\ell ){\ell }^{*}+\ell e(\ell )$ for all $\ell \in \mathfrak{A}.$ In [21], Brešar showed that if a prime ring $\mathfrak{A}$ admit nonzero derivations $e_1$ and $e_2$ of $\mathfrak{A}$ such that $e_1(\ell )\ell -\ell e_2(\ell )\in \mathfrak{Z}\left(\mathfrak{A}\right)$ for all $\ell \in I,$ where I is a nonzero left ideal of $\mathfrak{A}$, then $\mathfrak{A}$ is commutative. Further, this result was extended by Argac [12] as follows: Let $\mathfrak{A}$ be a semiprime ring and $e_1,e_2$ are derivations of $\mathfrak{A}$ such that at least one is nonzero. If $e_1(\ell )\ell -\ell e_2(\ell )\in \mathfrak{Z}\left(\mathfrak{A}\right)$ for all $\ell \in \mathfrak{A}$ , then $\mathfrak{A}$ contains a nonzero central ideal. Motivated by the above mentioned results, first author together with Alhazmi et al. [10] studied more general problem in the setting of rings with involution. Prescisely, they proved that if a $(m+n)!$-torsion free prime ring with involution of the second kind admit Jordan *-derivations e and g of $\mathfrak{A}$ such that $e\left({\ell }^m\right){\ell }^n\pm {\ell }^{n}g\left({\ell }^m\right)=0$ for all $\ell \in \mathfrak{A}$ (where m and n are fixed positive integers), then $e=g=0$ or $\mathfrak{A}$ is commutative. In the sequel, very recently Nejjar et al. [42, Theorem 3.7] established that if a 2-torsion free prime ring with involution of the second kind admits a nonzero derivation e such that $e(\ell ){(\ell )}^{*}-{(\ell )}^{*}e(\ell )\in \mathfrak{Z}\left(\mathfrak{A}\right)$ for all $\ell \in \mathfrak{A},$ then $\mathfrak{A}$ is commutative. The goal of this section is to initiate the study of a more general concept than *-centralizing mappings are; that is, we consider the situation when the mappings $\varphi$ and $\xi$ of a ring $\mathfrak{A}$ satisfy $\varphi (\ell ){(\ell )}^{*}-{(\ell )}^{*}\xi (\ell )\in \mathfrak{L}$ for all $\ell \in \mathfrak{A},$ where $\mathfrak{A}$ is an arbitrary ring and $\mathfrak{L}$ is a prime ideal of $\mathfrak{A}$. Precisely, we prove the following theorem.

Following are the immediate consequences of Theorem 1. In fact, Corollary 1 is in sprit of the result due to Posner’s second theorem.

For the proof of Theorem 1, we need the following lemmas, some of which are of independent interest. We begin our discussions with the following.

In view of Lemmas $2.6\&2.8$, we conclude the following result.

We are now ready to prove our first main theorem.

Proof of Theorem 1. We assume that $char(\mathfrak{A}/\mathfrak{L})\ne 2$. By the assumption, we have

Linearizing (8), we have

Replacing ℓ by $\ell h$ in (9), where $0\ne h\in \mathfrak{H}\left(\mathfrak{A}\right)\cap \mathfrak{Z}\left(\mathfrak{A}\right)$, we get

Application of (9) yields

This gives that

Replace h by $k^2$ in (10), where $0\ne k\in S\left(\mathfrak{A}\right)\cap \mathfrak{Z}\left(\mathfrak{A}\right)⊈\mathfrak{L}$, to get

Substituting $\ell k$ in place of ℓ in (9), where $0\ne k\in S\left(\mathfrak{A}\right)\cap \mathfrak{Z}\left(\mathfrak{A}\right)$, we arrive at for all $\ell ,\vartheta \in \mathfrak{A}$. From (9), we have

Adding (12) and (13), we obtain this impels for all $\ell ,\vartheta \in \mathfrak{A}$. Using (11) in (14), we have

Since $char(\mathfrak{A}/\mathfrak{L})\ne 2$ and $S\left(\mathfrak{A}\right)\cap \mathfrak{Z}\left(\mathfrak{A}\right)⊈\in \mathfrak{L}$, we have

In particular, for $\vartheta =\ell$, we get $e_1(\ell )\ell -\ell e_2(\ell )\in \mathfrak{L}$ for all $\ell \in \mathfrak{A}.$ Therefore, from Lemma 2, we conclude that ($e_1\left(\mathfrak{A}\right)\subseteq \mathfrak{L}$ and $e_2\left(\mathfrak{A}\right)\subseteq \mathfrak{L}$) or $\mathfrak{A}/\mathfrak{L}$ is a commutative integral domain.□

We now prove another theorem in this vein.

The following result is interesting in itself.

Using similar approach with necessary variations, one can establish the following result.

In view of Theorems 3 & 4, we have the following corollaries: