## 1. Introduction

Through the 20th century, noncommutative rings have only been issues of systematic study quite recently. Commutative rings, on the contrary, have seemed, though in a covered way, much before, and as with countless theories, it all comes back up to Fermat’s Last Theorem. In 1847, the mathematician Lamé stated an optimal solution of Fermat’s Last Theorem. In dissimilarity to commutative ring theory, which increases from quantity theory, non-commutative ring theory progresses from a notion of Hamilton. He attempted to release the complex numbers as a two-dimensional algebra over the real to a tri-dimensional algebra. Other natural noncommutative objects that grow are matrices. In 1850, they were presented by Cayley, together with their rules of addition and multiplication and, in 1870, Pierce noted that the now commonplace ring axioms held for square matrices.

However, the origin of commutativity theorems for rings could be traced to the paper of Wedderburn (1905) which was under-titled “a finite division ring is necessarily a field” in theTransaction American Mathematical society. The study of derivation was initiated during the 1950s and 1960s. Despite the concept of derivation in rings being quite old and playing a significant role in various branches of mathematics, it developed tremendously when, in 1957, Posner [

1] founded two very striking results on derivations in prime rings. Additionally, there has been substantial interest in examining commutativity of rings, generally that of prime ring sand semiprime rings admitting suitably constrained the additive mappings a derivations. Over and above, Vukman [

2,

3] extended the above result for bi-derivations. Derivations in rings have been studied by several algebraists in various directions. It is very enjoyable and it is important that the analogous properties of derivation which is one of the requisite theory in analysis and applied mathematics are also satisfied in the ring theory.

Derivations of prime and semiprime rings were studied by several researchers? near-rings,

$BCI$-algebras, lattices and various algebraic structures [

4,

5,

6,

7,

8,

9]. Multiderivations which are covering (e.g., biderivation, 3-derivation, or

n-derivation, semiderivation and anti derivation in general) have been examined in (semi-) rings [

2,

10,

11,

12,

13,

14]. Some researchers have studied

n-derivations, (

n,

m)-derivations and higher derivations on various algebraic structures, such as triangular rings, von Neumann algebras, lattice ordered rings and

J-subspace lattice algebras [

15,

16,

17,

18,

19,

20,

21].

In 1976, I. N. Herstein [

22] depended on the composition of rings to find fundamental properties, where he established that, letting

R be a ring in which, given

$a,b\in R$, there exist integers

$m=m(a,b),n=n(a,b)$ greater than or equal to 1 such that

${a}^{m}{b}^{n}={b}^{n}{a}^{m}$. Then, the commutator ideal of

R is nil. Particularly, if

R has no non-zero nil ideals, then

R must be commutative. As a matter of fact, the theorems, especially the commutativity case for rings and near-rings with their applications, have been discussed by a lot of researchers. The core of that research is to encourage the pursuit of research on applications of ring theory in diverse areas, such as to emphasise the interdisciplinary efforts involved in the pursuit of information technology and coding theory. All types of rings collected so far contribute to their application in diverse sections of mathematics as well as in data communications, computer science, digital computing and so forth.

During the years, a lot of work has been finished in this context by a several of authors in different aspects. In 1980, G.Maksa [

23] pointed out to the concept of a symmetric biderivation on a ring

R. The concept of additive commuting mappings is closely connected to the concept of biderivations. Every commuting additive mapping

$d:R\to R$ gives rise to a biderivation on

R. Linearizing

$[d(x),x]=0$, for all

$x\in R$, we get

$[d(x),y]=[x,d(y)]$, for all

$x,y\in R$ and hence we note that the mapping

$(x,y)\to [d(x),y]$ is a biderivation on

R. Furthermore, all derivations appearing are inner. More details about biderivations and their applications can be found in Reference [

24].

Indeed, in Reference [

25] it was shown that every biderivation

D of a noncommutative prime ring

R is of the form

$D(x,y)=\lambda [x,y]$,

$x,y\in R$, where

$\lambda $ is a fixed element from the extended centroid of the ring

R. Using certain functional identities, Brešar [

24] extended this result to semiprime rings. Later, several authors have studied permuting 3-derivations in rings (see References [

26,

27,

28], where several references can be found). Nevertheless, some authors have done a great deal of work concerning commutativity of prime and semiprime rings admitting various types of maps which are centralizing (resp.commuting) on some appropriate subsets of a ring

R (see References [

29,

30,

31,

32,

33]).

The concept of a permuting tri-derivation has been introduced Öztürk in Reference [

34], while Ajda Fošner [

35] presented the notion of symmetric skew 3-derivations and made some basic observations. Taking into account the definitions of skew derivations, we would like to point out that in Reference [

35] Ajda Fošner introduced the notion of permuting skew 3-derivations in rings and extended the results given by Jung and Park [

10] for △ is a permuting skew 3-derivations and proved the commutativity of

R under certain identities, where

R is a 3!-torsion free prime ring and

$\u25b3\ne 0$. Meanwhile, Ajda Fošner [

36] also extended the notion of permuting skew 3-derivation to permuting skew

n-derivations in rings and proved several other results. In another contribution, the authors of Reference [

37] have obtained the commutativity of a ring satisfying certain identities involving the trace of permuting

n-derivation. Further, Mohammad Ashraf and Nazia Parveen [

38] introduced the notion of permuting generalized

$(\alpha ,\beta )-n$-derivations and permuting

$\alpha $-left

n-centralizers in rings and generalized the above results given by Ajda Fošner [

36] in a different setting under some suitable torsion restrictions imposed on the underlying ring. Notwithstanding, several authors have done a great deal of work concerning commutativity of prime and semiprime rings admitting different kinds of maps which are skew derivations on some appropriate subsets of R, then Xiaowei Xu, Yang Liu and Wei Zhang [

4] considered a skew

n-derivation (

$n\ge 3$) on a semiprime ring

R must map into the center of

R.

On the other hand, Badr Nejjar et al. [

39] proved that

n is a fixed positive integer and

R is a

$(n+1)!$-torsion free prime ring and

J a non-zero Jordan ideal of

R. If

R admits a non-zero permuting generalized

n-derivation

$\Omega $ with associated

n-derivation △ such that the trace of

$\Omega $ is centralizing on

J. Then

R is commutative, where an

n-additive mapping

$\Omega :{R}^{n}\u27f6R$ is called a generalized

n-derivation of

R with associated

n-derivation △ if

$\Omega ({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}}{x}_{i},...,{x}_{n})=\Omega ({x}_{1},{x}_{2},...,{x}_{i},...,{x}_{n})\stackrel{`}{{x}_{i}}+{x}_{i}\u25b3({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}},...,{x}_{n})$ for all

$\stackrel{`}{{x}_{i}},{x}_{i}\in R,$ and additive subgroup

J of

R is said to be a Jordan ideal of

R if

$u\circ r\in J$, for all

$u\in J$ and

$r\in R$. In the near ring the subject studied by some authors like A. Ali et al. [

40] assumed

N to be a 3!-torsion free 3-prime near ring and

U be a non-zero additive subgroup and a semigroup ideal of

N. If △ is a permuting 3-derivation with trace

$\delta $ and

$x\in N$ such that

$x\delta (y)=0$ for all

$y\in U$, then either

$x=0$ or

$\u25b3=0$ on

U. In addition to that, Mohammad Ashraf et al. [

41] came out with the notion of

$(\sigma ,\tau )-n$-derivation in near-ring

N and investigated some properties involving

$(\sigma ,\tau )-n$-derivations of a prime near-ring

N which force

N to be a commutative ring. Also, Mohammad Ashraf and Mohammad Aslam Siddeeque [

42] produced let

N be a 3-prime near-ring admitting a non-zero generalized

n-derivation F with associated

n-derivation

D of

N. Then

$F({U}_{1},{U}_{2},...,{U}_{n})\ne \left\{0\right\}$, where

${U}_{1},{U}_{2},...,{U}_{n}$ are non-zero semigroup left ideals or non-zero semigroup right ideals of

N.

Other authors had tried of a permuting

n-derivation of algebraic structure, for example D. Eremita [

43] who discussed that if functional identities of degree 2 in triangular rings and obtained some descriptions of commuting maps and generalized inner biderivations of triangular rings. Yao Wang et al. [

21] showed that if

$A=Tri(A,M,B)$ be a triangular algebra. Suppose that there exists

$m\in M$ such that

$[m,[A,A]]=0$. Set

${\Psi}_{n}({x}_{1},{x}_{2},...,{x}_{n})=[{x}_{1},[{x}_{2},...,[{x}_{n},m]...]]$ for all

${x}_{1},{x}_{2},...,{x}_{n}\in A$. Then

${\Psi}_{n}$ is a permuting

n-derivation of

A.

However, Skosyrskii [

44] who treated biderivations for different reasons, namely, in connection with noncommutative Jordan algebras.

K. H. Park [

12] initiated the notion of an

n-derivation and symmetric

n-derivation, where

n is any positive integer in rings and extended several known results, earlier in the setting of derivations in prime rings and semiprime rings as follows?suppose

$n\ge 2$ be a fixed positive integer and

${R}^{n}=R\times R\times ...\times R$. A map

$\Delta :{R}^{n}\to R$ is said to be symmetric (or permuting) if the equation

$\Delta ({x}_{1},{x}_{2},...,{x}_{n})=\Delta ({x}_{\pi (1)},{x}_{\pi (2)},\cdots ,{x}_{\pi (n)})$ holds for all

${x}_{i}\in R$ and for every permutation

$\{\pi (1),\pi (2),...,\pi (n)\}$. that is, for every permutation

$\pi \in {S}_{n}$ (permutation on

n symbol), where

${R}^{n}=R\times R\times R\times ...\times R$.

Let us consider the following map: let

$n\ge 2$ be a fixed positive integer. An

n-additive map

$\Delta :{R}^{n}\to R$ (that is, additive in each argument) will be called an

n-derivation if the relations

are valid for all

${x}_{i},\stackrel{`}{{x}_{i}}\in R$.

Also, in the same Reference [

12], a 1-derivation is a derivation and a 2-derivation is called a bi-derivation. As in the case of

$n=3$ we get the concept of tri-derivation. If

$\Delta $ is symmetric, then the above equalities are equivalent to each other. Let

$n\ge 2$ be a fixed positive integer and let a map

$\delta :R\to R$ defined by

$\delta (x)=\Delta ({x}_{1},{x}_{2},...,{x}_{n})$ for all

$x\in R$, where

$\Delta :{R}^{n}\to R$ is a symmetric map, be the trace of

$\Delta $. It is clear that, in the case when

$\Delta :{R}^{n}\to R$ is a symmetric map which is also

n-additive, the trace

$\delta $ of

$\Delta $ satiates the identity

$\delta (x+y)=\delta (x)+\delta (y)+{\sum}_{r=1}^{n-1}\left(\genfrac{}{}{0pt}{}{n}{r}\right)\Delta (x,x,...,x,y,y...,y)$ for all

$x,y\in R$ where

y appears

r times and

x appears

$n-r$ times.

Since we have $\Delta (0,{x}_{2},...,{x}_{n})=\Delta (0+0,{x}_{2},...,{x}_{n})=\Delta (0,{x}_{2},...,{x}_{n})+\Delta (0,{x}_{2},...,{x}_{n})$ for all ${x}_{i}\in R$, $i=2,3,...,n$, we obtain $\Delta (0,{x}_{2},...,{x}_{n})=0$ for all ${x}_{i}\in R$, $i=2,3,...,n$.

Hence, we get $0=\Delta (0,{x}_{2},...,{x}_{n})=\Delta ({x}_{1}-{x}_{1},{x}_{2},....,{x}_{n})=\Delta ({x}_{1},{x}_{2},...,{x}_{n})+\Delta (-{x}_{1},{x}_{2},...,{x}_{n})$ and so we see that $\Delta (-{x}_{1},{x}_{2},...,{x}_{n})=-\Delta ({x}_{1},{x}_{2},...,{x}_{n})$ for all ${x}_{i}\in R$, $i=1,2,...,n$. This tells us that $\delta $ is an odd function if n is odd and $\delta $ is an even function if n is even.

Yilmaz Çeven [

45] issued the definition which generalizes the notions of derivation, biderivation and 3-derivation on lattices, where the map

$\Delta :{L}^{n}\to L$ will be called an

n-derivation if

$\Delta $ is a derivation according to all components; that is,

are valid for all

${x}_{i}$ and

$a\in L.$In Reference [

46], Bell and Martindale have stated the following results. Specify

$f\ne 0$ be a semiderivation of a prime ring

R of characteristic not 2 with associated endomorphism

g of

R and

$U\ne 0$ be an ideal of

R. Suppose that

$a\in R$ such that

$af(U)=0$. Then

$a=0$.

Recently, Emine Koç and Nadeem ur Rehman [

32] studied symmetric

n-derivations on prime or semiprime rings with non-zero ideals. They proved that if a symmetric skew

n-derivation

$\u25b3:{R}^{n}\to R$ associated with an automorphis

T satisfies any one of the conditions

- (i)
$\delta (x)=0$,

- (ii)
$[\delta (x),T(x)]=0$ for all $x\in U$,

where $\delta $ is the trace of $\Delta $, then $\Delta =0$.

Furthermore, Basudeb Dhara and Faiza Shujat [

47], have obtained that let

R be a

n!-torsion free prime ring,

I a nonzero ideal of

R,

$\alpha $ an automorphism of

R and

$D:{R}^{n}\u27f6R$ be a symmetric skew

n-derivation associated with the automorphism

$\alpha $. If

$\tau $ is the trace of

D such that

$\tau (I)=0$, then

$D({x}_{1},{x}_{2},...,{x}_{n})=0$ for all

${x}_{1},{x}_{2},...,{x}_{n}\in R.$Throughout this paper, R represents an associative ring always. Denote by $Z(R)$ the center of R. Let $x,y,z\in R$. We write the notation $[y,x]$ for the commutator $yx-xy$ (the Lie product) and $x\circ y$ for anti-commutator $xy+yx$ (the Jordan product) also make use of the identities $[xy,z]=[x,z]y+x[y,z]$ and $[x,yz]=[x,y]z+y[x,z]$. The ring R is called semiprime if R satisfies the relation $aRa=0$ implies $a=0$ and R is called prime if R satisfies $aRb=0$ implies $a=0$ or $b=0$. The relation between the prime ring and semiprime ring said every prime ring is semiprime ring, but the converse is not true always. A map $d:R\to R$ is said to be commuting on R if d satisfies $[d(x),x]=0$ holds for all $x\in R$. If $[d(x),x]\in Z(R)$ is fulfilled for all $x\in R$ then a map $d:R\to R$ is said centralizing on R. If the Leibniz’s formula $D(xy)=D(x)y+xD(y)$ holds for all $x,y\in R$ then an additive map $D:R\u27f6R$ is called a derivation.

The concept of a generalized derivation was introduced in Reference [

48] as follows. An additive mapping

$D:R\u27f6R$ is called a generalized derivation if there exists an additive mapping

d on

R such that

$D(xy)=D(x)y+xd(y)$ for all

$x,y\in R$. Besides derivations and generalized inner derivations this also generalizes the concept of left multipliers, that is, additive mappings satisfying

$D(xy)=D(x)y$, for all

$x,y\in R$.

The inner derivation is fundamental example of derivation, that is, mappings of the form ${\delta}_{a}(x)=ax-xa$ where a is a fixed element in R. Generally, the mappings of the form $D(x)=ax+xb$ (with $a,b\in R$ fixed elements) are called generalized inner derivations. The additive map $d:R\u27f6R$ into itself which satisfies the rule $d(xy)=d(x)y+\alpha (x)d(y)$ for all $x,y\in R$ named a skew derivation of R. If $\alpha =1$ is the identity automorphism of R, then d is known as a derivation of R. If there exists a skew derivation d of R with associated automorphism $\alpha $ such that $D(xy)=D(x)y+\alpha (x)d(y)$ holds for all $x,y\in R$ then an additive mapping $D:R\u27f6R$ is said to be a (right) generalized skew-derivation of R.

In Reference [

49], J. Bergen introduced the concept of semiderivation of a ring

R as. An additive mapping

d of a ring

R into itself is called a semiderivation if there exists a function

$g:R\u27f6R$ such that

$d(xy)=d(x)g(y)+xd(y)=d(x)y+g(x)d(y)$ and

$d(g(x))=g(d(x))$ for all

$x,y\in R$. For

$g=1$ a semiderivation is of course a derivation. In Reference [

13], Mohammad Ashraf and Muzibur Rahman Mozumder generalized the concept of multiplicative (generalized)- derivation to multiplicative (generalized)- skew derivation. A mapping

$D:R\to R$ (not necessarily additive) is called a multiplicative (generalized)-skew derivation if

$D(xy)=D(x)y+\alpha (x)g(y)=D(x)\alpha (y)+xg(y))$ for all

$x,y\in R$, where

$g:R\to R$ is any mapping (not necessarily a skew derivation nor an additive map) and

$\alpha :R\to R$ is an automorphism of

R. Since the sum of two generalized derivations is a generalized derivation, every map of the form

$D(x)=cx+d(x)$ is a generalized derivation, where

c is a fixed element of

R and

d is a derivation of

R. Furthermore, Brešar and Vukman [

14] have introduced the notion of a reverse derivation (anti-derivation) as an additive mapping

d from a ring

R into itself satisfying

$d(xy)=d(y)x+yd(x)$,for all

$x,y\in R$. Obviously, if

R is commutative, then both derivation and reverse derivation are the same. The generalized reverse(anti) derivations were defined by [

50] Let

R be a ring and let

d be a reverse derivation of

R. An additive mapping

$F:R\to R$ is said to be a left generalized reverse derivation of

R associated with

d if

$D(xy)=D(y)x+yd(x)$ for all

$x,y\in R$. Also, the additive mapping

D is said to be a right generalized reverse derivation associated with

d if

$D(xy)=d(y)x+yD(x)$ for all

$x,y\in R$. However, there exists a controversial question here, which is whether we can find new generators of a permuting

n-derivations. So, the answer to this question is affirmative where the aim of this paper is to introduce the new types of a permuting

n-derivations for associative rings and to make some basic observations.

## 3. Δ Acts as a Permuting n-Semiderivations of Semiprime Ring

In this section, we want to study semiprime ring

R with a permuting

n-semiderivations

$\Delta $. In Reference [

56], W.D. Burgess, A. Lashgari and A. Mojiri introduced the concept of a weak zero-divisor of a ring

R. An element

$a\in R$ is called a weak zero-divisor if there are

$r,s\in R$ with

$ras=0$ and

$rs\ne 0$. The set of elements of

R which are not weak zero divisors is denoted by

${S}_{nw}$. In the following theorem, we obtain a semiprime ring

R has a weak zero-divisor.

For convenience, we suppose all the results of this section satisfy the identity $aRb\subset Z(R),a,b\in R$.

**Theorem** **1.** Let R be a semiprime ring and Δ be a permuting non-zero n-semiderivation with a trace δ such that Δ acts as right-multiplier. If R admits Δ satisfying the identity $[\Delta ({R}_{1}),\delta ({R}_{2})]\subseteq Z(R)$ then R has a weak zero-divisor.

**Proof.** From the main relation

$[\Delta ({R}_{1}),\delta ({R}_{2})]\subseteq Z(R)$, after

$({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}}{x}_{i},...,{x}_{n})\in R$ take place of

${R}_{1}$ with applying the Definition 1, we gain the following relation

for all

$\stackrel{`}{{x}_{i}}{x}_{i}\in R$.

By reason of

$\Delta $ is a right-multiplier mapping, then for all

$\stackrel{`}{{x}_{i}},{x}_{i},r\in R$ this relation becomes

In agreement with the main relation with using the fact that

$\Delta $ is a right-multiplier and replacing

${R}_{1}$ by

$(({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},...,{x}_{n})\lambda ({x}_{i}))$. That means Relation (1) reduces to

for all

$\stackrel{`}{{x}_{i}},{x}_{i},r\in R$.

Obviously, when we substitute this value of (1) with using the fact that

$\Delta $ is a right-multiplier and the main relation, we find that

Replacing

r by

$\delta ({R}_{2})$, we achieve that

Now replacing

${x}_{i}$ by

$D({R}_{1})$ of the above relation with the property

$D(\lambda )=\lambda (D)$, we show that

Without doubt, applying the main relation on this equation gives us

Substituting this relation in the Equation (

3), we find that

Moreover, for any arbitrary element

$t\in R$, left-multiplying by

$[[\lambda ({x}_{i}),\delta ({R}_{2})],\delta ({R}_{2})]t$ and right-multiplying by

$t\Delta ({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i+1}...,{x}_{n})$ with employing the fact from Lemma 2 yields that

Due to R being a semiprime, we acknowledge the set $\left\{{P}_{\alpha}\right\}$ of prime ideals of R such that $\cap {P}_{\alpha}=\left\{0\right\}.$ In agreement with Lemma 1, we gain the set $\left\{{P}_{\alpha}\right\}$ of prime ideals of R is semiprime ideal.

Let

$\cap {P}_{\alpha}=U$. We achieve that either

$t\Delta ({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i+1}...,{x}_{n})\in U$ that is,

which implies to a contradiction where

$\Delta \ne 0$. Actually, the above result is enough to achieve our proof after right-multiplying the Equation (

4) by any non-zero element of

R. □

Now let us introduce the definition of non-zero elements set of a semiprime ring R which is a nest in the collection of prime ideals of R. It is denoted by M-set respect to the name of author.

**Definition** **6.** A set of a non-zero elements which are located in the intersection of prime ideals $\cap {P}_{\alpha}$ of a semiprime ring R is said to be M-set if has the following property: For $a\in R$, then ${a}^{2}\in M-set$ while $a\notin M-set$. that is, ${a}^{2}\in M-set\subseteq \cap {P}_{\alpha}$ whilst $a\notin M-set\subseteq \cap {P}_{\alpha}$.

To make the previous definition closer for the readers, we list the following example.

**Example** **2.** Let $R={M}_{n}(\mathbb{F})$ be a ring of $n\times n$ matrices over a field $\mathbb{F}$, $n>1$ that is:for all ${x}_{j}\in \mathbb{F}$, $j=1,2...,n$. Then it is clear to be seen that Since $M-set\subseteq \cap {P}_{\alpha}$, we find thatwhile Clearly, any non-zero element of R has nilpotency index 2 belong to $M-set$.

**Corollary** **1.** Let R be a semiprime ring and Δ be a permuting non-zero n-semiderivation with a trace δ such that Δ and δ act as right-multiplier and surjective function respectively. If R satisfies the identity $[\Delta ({R}_{1}),\delta ({R}_{2})]\subseteq Z(R)$ then ${\Delta}^{2}\in M-set$.

**Proof.** Employing the same technique which is applied in the proof of Theorem 1 specific to the Equation (

4).

Hence, in the Equation (

4) replacing

t by

$t\Delta ({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})$ with consideration that

$\lambda $ acts as surjective function, we find that

Since R is semiprime that is, $\left\{{P}_{\alpha}\right\}$ of prime ideals of R such that $\cap {P}_{\alpha}=\left\{0\right\}.$ In agreement with Lemma 1, we obtain the set $\left\{{P}_{\alpha}\right\}$ of prime ideals of R is semiprime ideal.

Employing the same proceeding in the proof of Theorem 1, we conclude that either $[[D(\lambda ({R}_{1})),\delta ({R}_{2})],\delta ({R}_{2})]\in U$ for all ${x}_{i}\in R$ or $\Delta {({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})}^{2}\in U$.

Basically, from the main relation the first case yields $[[D(\lambda ({R}_{1})),\delta (y)],\delta (y]=0$ for all ${x}_{i},y\in R.$ Notwithstanding the second case proving $\Delta {({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})}^{2}\in U=\cap {P}_{\alpha}=\left\{0\right\}$ yields that $\Delta {({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})}^{2}=0$ for all $\stackrel{`}{{x}_{i}},{x}_{i}\in R.$

We utilize that $\Delta $ is non-zero permuting n-semiderivation of R which means $\Delta {({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})}^{2}\in M-set$ and coinciding with the relation $\Delta ({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})\notin U=\cap {P}_{\alpha}=\left\{0\right\}$

In fact, this result is meaningful, in which the $M-set$ collect a non-zero nilpotent element having degree 2 of semiprime ideal. □

**Theorem** **2.** Let R be a n-torsion free semiprime ring and Δ be a permuting non-zero n-semiderivation with a trace δ which is a homomorphism mapping. If Δ satisfies $\Delta ({R}_{1})\circ \delta ({R}_{2})\subseteq Z(R)$ then

- $(i)$
${\delta}^{2}(R)\subseteq Z(R)$.

- $(ii)$
either ${\delta}^{2}(R)\subseteq M-set$ or $\Delta (R)\subseteq Z(R)$.

**Proof.** Basically, from our hypothesis we have $\Delta ({R}_{1})\circ \delta ({R}_{2})\subseteq Z(R)$.

Replacing

${R}_{1}$ by

$({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})$ for all

$\stackrel{`}{{x}_{i}},{x}_{i}\in R,i=1,2,...,n$, we obtain

Obviously, we see that

for all

$\stackrel{`}{{x}_{i}},{x}_{i},y,r\in R$.

Taking

$r=\delta (y)$ this relation modifies into

After simple calculation, we find that

Consequently, we see that

According to our hypothesis that

$\delta $ is a homomorphism, this relation modifies to

In the main relation, we replace

${R}_{1}$ by

$({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},...,{x}_{n})$ and

${R}_{2}$ by

${y}^{2}$ showing that

Combining the Equations (5) and (6), we immediately obtain

For any arbitrary element such as

$t\in R$ and agreement with Lemma 4, we arrive to

for all

$\stackrel{`}{{x}_{i}},{x}_{i},y,t\in R$. In agreement with Lemma 5, we obtain

$\Delta ({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},...,{x}_{n})\delta ({y}^{2})\in Z(R)$.

Immediately, we obtain the result

Let

$y={y}_{1},{y}_{2},....,\stackrel{`}{{y}_{i}}{y}_{i},...,{y}_{n}$ and

$z={z}_{1},{z}_{2},....,\stackrel{`}{{z}_{i}}{z}_{i},...,{z}_{n}$ for all

$\stackrel{`}{{x}_{i}}{x}_{i},\stackrel{`}{{z}_{i}}{z}_{i}\in R$. Thus, in the Equation (

5), taking

$y=y+kz$, where we consider a positive integer

k,

$1\le k\le n-1$ and

$z\in R$, we deduce that

$[t,\Delta ({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},...,{x}_{n})\delta {(y+kz)}^{2}]=0.$Of course, by reason of

$\Delta :{R}^{n}\u27f6R$ is permuting and

n-additive mapping, then the trace

$\delta $ of

$\Delta $ satisfies the following relation

$\delta (x+y)=\delta (x)+\delta (y)+{\sum}_{i=1}^{n-1}\left(\genfrac{}{}{0pt}{}{n}{1}\right)\Delta (x,x,...,x,y...,y)$ where

x appears

$n-i$-times and

y appears

i-times, with a consequence being that the above relation can be rewritten as follows:

According to (7), the Equation (

8) reduces to

$[t,\Delta ({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},...,{x}_{n})(\delta (k(yz+zy))+{\sum}_{i=1}^{n-1}\Delta (({y}^{2}+{z}^{2}),({y}^{2}+{z}^{2}),...,({y}^{2}+{z}^{2}),(k(yz+zy)),...,(k(yz+zy)))]=0$ for all

$\stackrel{`}{{x}_{i}},{x}_{i},t\in R.$The element

y is used as a substitute for

z of this relation and, applying the Equation (

7), we achieve that

$[t,\Delta ({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},...,{x}_{n}){\sum}_{i=1}^{n-1}\Delta (({y}^{2}+{z}^{2}),({y}^{2}+{y}^{2}),...,({y}^{2}+{y}^{2}),(k({y}^{2}+{y}^{2})),...,(k({y}^{2}+{y}^{2}))]=0.$Applying Lemma 3, we see that

Utilization of our hypothesis that R is n-torsion free and putting ${x}_{1}={x}_{2}=....=\stackrel{`}{{x}_{i}}=...={x}_{n}={x}^{2}$, we find that $[t,\Delta ({x}^{2})\Delta ({y}^{2})]=[t,\delta {({y}^{2})}^{2}]=0.$

Clearly, we have that $\delta {({y}^{2})}^{2}\in Z(R)$.

Particularly, Lemma 4 can change this identity to $[[t,\delta {(y)}^{2}],\delta {(y)}^{2}]=0.$ Applying the same technique and using Lemma 4, we conclude that $2\delta {(y)}^{2}\in Z(R)$.

Due to R being n-torsion free, we see that $[t,\delta {(y)}^{2}]=0$ yields $\delta {(y)}^{2}\in Z(R)$.

(ii) From the first branch, we have the Equation (

5) which is

$[t,\Delta ({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},...,{x}_{n})\delta ({y}^{2})]=0$.

From this relation and using the result of the fist branch, we find that

Putting $t=tr,r\in R$, the last expression can be written as $[t,\Delta ({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},...,{x}_{n})]r\delta ({y}^{2})=0.$

The last expression is the same as the proof of Theorem 1, therefore applies the similar arguments as used in the proof of Theorem 1. Hence, we obtain two options, either $[t,\Delta ({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},...,{x}_{n})]\in U$ or $\delta ({y}^{2})\in U$. The first case proved $\Delta ({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},...,{x}_{n})\in Z(R)$ while the second $\delta ({y}^{2})=0$. Since $\delta \ne 0$ then $\delta ({y}^{2})\in M-set.$ □

**Theorem** **3.** Let R be a 2-torsion free semiprime ring and Δ be a permuting non-zero n-semiderivation. If Δ satisfies the identity $[\Delta ({R}_{1}),\Delta ({R}_{2})]\subseteq Z(R)$ then $\Delta (d)$ and $\Delta ({R}_{2})$ commute with R.

**Proof.** Putting

$y\Delta ({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})$ instead of for

${R}_{1}$ in the main relation, we arrive to

for all

$\stackrel{`}{{x}_{i}},{x}_{i},y\in R$For any arbitrary element of

R and using the property

$\Delta (\lambda )=\lambda (\Delta )$, we find that

Replacing ${R}_{1}$ by y in the main relation, we conclude that $[\Delta (y),\Delta ({R}_{2})]\in Z(R)$.

Putting $\lambda ({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})$ instead of ${R}_{1}$ give us $[\Delta ({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})),\Delta ({R}_{2})]\in Z(R)$.

Using these results for $[[\Delta (y)\Delta ({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}),\Delta ({R}_{2})],r]+[[yd(\Delta ({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})),\Delta ({R}_{2})],r]=0$. It reduces to

$[[yd(\Delta ({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})),\Delta ({R}_{2})],r]=0$. Again, applying the property

$\Delta (d)=d(\Delta )$ yields

In the main relation, replacing

${R}_{1}$ by

$d({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})$ and applying the result of this relation, it follows that

Taking

$\Delta ({R}_{2})$ for

r this relation modifies to

for all

$\stackrel{`}{{x}_{i}},{x}_{i},y\in R$.

Putting $\Delta ({R}_{1})$ instead of y, with applying the main relation and employing the fact that R is 2-torsion fee. In addition to that, replacing ${R}_{1}$ by $d({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})$ based on Lemma 2. From this equation, we arrive tot $\Delta (d)$ and $\Delta ({R}_{2})$ commute with R. We have completed the proof. □

We now state the consequence of Theorem 3.

**Corollary** **2.** Let R be a semiprime ring and Δ be a non-zero permuting n-semiderivation. If R admits Δ satisfying the identity $[\Delta ({R}_{1}),\Delta ({R}_{2})]\subseteq Z(R)$ then $\Delta (d(R))$ is central of R.

**Proof.** We begin with the identity $[\Delta ({R}_{1}),\Delta ({R}_{2})]\subseteq Z(R)$. According to Theorem 4, we find that $[\Delta (d({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})),\Delta ({R}_{2})]=0$.

Replacing

${R}_{2}$ by

$y\Delta ({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})$ and applying the property

$\Delta (d)=d(\Delta )$, it follows that

for all

$\stackrel{`}{{x}_{i}},{x}_{i},y\in R.$Furthermore, using the main relation with simple calculation, we see that

for all

$\stackrel{`}{{x}_{i}},{x}_{i},y\in R.$Replacing

$ty,t\in R$ with

y in this relation and using that result, we see that

Replacing

y by

$Ry$ in the Relation (9). This implies that

Again, in (9), substituting R for y and right-multiplying by y. Subtracting this result from (10), we arrive to $[\Delta (d({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})),y]=0.$ This implies to $\Delta (d(R))$ is central of R.

We obtain the required result. □

**Theorem** **4.** Let R be a semiprime ring and Δ be a permuting n-semiderivation with a trace δ such that Δ acts as a homomorphism. Suppose Δ satisfies the identity $\Delta ({R}_{1})\circ \delta ({R}_{2})\mp \phantom{\rule{4pt}{0ex}}[{R}_{1},{R}_{2}]\subseteq Z(R)$. Then

- $(i)$
either $\Delta {({R}_{1})}^{2}\subseteq M-set$ or $\delta ({R}_{2})$ and $\Delta ({R}_{1}^{2})$ commute with R.

- $(ii)$
R is commutative if $\delta (R)=0$ or $\Delta (R)=0$.

**Proof.** (i) As above, we have the relation $\Delta ({R}_{1})\circ \delta ({R}_{2})\mp \phantom{\rule{4pt}{0ex}}[{R}_{1},{R}_{2}]\subseteq Z(R)$. Putting ${R}_{1}={R}_{2}$, in the main relation yields that $\Delta ({R}_{1})\circ \delta ({R}_{2})\subseteq Z(R).$

Then the last expression of this relation can be written as $[\Delta ({R}_{1})\delta ({R}_{1}),r]+[\delta ({R}_{1})\Delta ({R}_{1}),r]=0$ for all $r\in R$.

Replacing

r by

$\Delta ({R}_{1})$ in this relation, we notice that

which implies that

$[\Delta {({R}_{1})}^{2},\delta ({R}_{1})]\subseteq Z(R)$. Moreover, in the relation

$\Delta ({R}_{1})\circ \delta ({R}_{1})\subseteq Z(R)$ putting

${R}_{1}^{2}$ instead of

${R}_{1}$ with using the fact that

$\Delta $ acts as homomorphism yields

Subtracting this result with Equation (

11), gives us

$2\Delta ({R}_{1}^{2})\delta ({R}_{1})\subseteq Z(R)$.

Application of the fact that

R has torsion restriction gives the following

for all

$r\in R.$ Hence, replacing

r by

$\Delta ({R}_{1}^{2})$ of this relation becomes

Left-multiplying by

$[\delta ({R}_{1}),\Delta ({R}_{1}^{2})]t$ and right-multiplying by

$t\Delta ({R}_{1}^{2})$,

t is any arbitrary element of

R with employed Lemma 2. Then it is easy to see that

Now repeating similar technique to those we applied in the final part of the proof of Theorem 1, we conclude that: either $\Delta ({R}_{1}^{2})\in U=\cap {P}_{\alpha}=\left\{0\right\}$ or $[\delta ({R}_{1}),\Delta ({R}_{1}^{2})]\in U=\cap {P}_{\alpha}=\left\{0\right\}$.

Actually, our hypothesis points out that $\Delta \ne 0$, in addition to the fact that $\Delta $ acts as a homomorphism mapping which means the first case implies $\Delta {({R}_{1})}^{2}\in M-set$. While the second case supplies that $\delta ({R}_{1})$ and $\Delta ({R}_{1}^{2})$ commute with R. (ii) If $\delta =0$ then the main identity reduces into $[{R}_{1},{R}_{2}]\subseteq Z(R)$ which implies that R is commutative ring. Hence, we get the required result. □

**Theorem** **5.** Let R be a 2-torsion free semiprime ring and Δ be a permuting n-semiderivation with a trace δ which acts as left-multiplier. Suppose Δ satisfies $\Delta ({R}_{1}\circ {R}_{2})\mp \delta ([{R}_{1},{R}_{2}])\mp \phantom{\rule{4pt}{0ex}}[{R}_{1},{R}_{2}]\subseteq Z(R)$. Then R is commutative.

**Proof.** First, we discuss the case $\Delta $ and $\delta $ are not equal to zero, so the main identity still $\Delta ({R}_{1}\circ {R}_{2})\mp \delta ([{R}_{1},{R}_{2}])\mp \phantom{\rule{4pt}{0ex}}[{R}_{1},{R}_{2}]\subseteq Z(R)$.

For any arbitrary element

$t\in R$, we conclude that

Putting

${R}_{1}$ instead of for

${R}_{2}$ in this relation and using the fact

R is 2-torsion free, we find that

Linearizing Equation (

13) with depending on the fact

R is 2-torsion free and using Equation (

13), we show that

Again, in (13), substituting ${R}_{2}+{R}_{1}$ in place of ${R}_{1}$ and using R has a 2-torsion free, we observe that $[\Delta ({R}_{2}{R}_{1}),t]=0.$

Combining Relation (14) with this relation, it follows that $[\Delta ({R}_{1}\circ {R}_{2}),t]=0.$

Substituting this Equation of (12), we gain that

Replacing

$[{R}_{1},{R}_{2}]$ with

t, we find that

Particularly, for

${R}_{1}={R}_{1}{R}_{3}$ this relation extends to

Replacing

${R}_{3}$ by

${R}_{2}$. Last expression implies that

By reason of that

$\delta $ is a left-multiplier and applying of Equation (

15), this relation arrives to

Furthermore, simplify this relation and using (15), this relation reduces to

Multiplying (16) from the right by

${R}_{2}t(\delta ([{R}_{1},{R}_{2}]){R}_{2}[{R}_{1},{R}_{2}]-[{R}_{1},{R}_{2}]{R}_{2}\delta ([{R}_{1},{R}_{2}]))$ and the left by

${R}_{2}^{2}t$,

t is any arbitrary element of

R with applying Lemma 2 and

$aRb\subset Z(R)$, we obtain

Now repeating similar technique we used in the final part of proof of Theorem 1, we arrive to two cases.

In first case, we have ${R}_{2}^{2}\in U=\cap {P}_{\alpha}=\left\{0\right\}$. This implies ${R}_{2}^{2}=0.$

Where

${R}_{2}\ne 0$, we see that

${R}_{2}^{2}\in M-set$. From the second case, we find that

Right-multiplying by

$[{R}_{1},{R}_{2}]$ this relation becomes

According to Relation (15) this equation modifies to

Subtracting Relations (17) and (18). Then, it is easy to see that

Left-multiplying by $[\delta ([{R}_{1},{R}_{2}]),[{R}_{1},{R}_{2}]]t$ and right-multiplying by $t[{R}_{1},{R}_{2}]{R}_{2}$, where t is any arbitrary element of R. In addition to that, applying similar method as we used in the proof of Theorem 1, we observe that; either $[{R}_{1},{R}_{2}]{R}_{2}\in U=\cap {P}_{\alpha}=\left\{0\right\}$ or $[\delta ([{R}_{1},{R}_{2}]),[{R}_{1},{R}_{2}]]\in U=\cap {P}_{\alpha}=\left\{0\right\}$.

Of course, the second case satisfied by work. That means we have the first case which $[{R}_{1},{R}_{2}]{R}_{2}\in U=\cap {P}_{\alpha}=\left\{0\right\}$ implies to $[{R}_{1},{R}_{2}]{R}_{2}=0.$

Replacing

${R}_{1}$ by

${R}_{1}t$, where

t is any arbitrary element of

R, we note that

Now in Equation (

19) putting

$t{R}_{1}$ for

t, we achieve that

Right-multiplying (19) by ${R}_{1}$ and subtracting this result from Relation (20) with using the semiprimeness of R, the R satisfy that

$[{R}_{1},{R}_{2}]=0.$ Obviously, R is commutative.

Now we take $\Delta =\delta =0$ yields $[{R}_{1},{R}_{2}]\subseteq Z(R)$. Without doubt, R is commutative. This finishes the proof. □

## 4. Permuting n-Generalized Semiderivation of Semiprime Rings

In this section, we study the behaviour of a permuting n-generalized semiderivation on semiprime rings R with $Z(R)$.

For more convenience, we suppose all the results of this section satisfies the relation $aRb\subset Z(R),a,b\in R$. Except Theorem 7.

**Theorem** **6.** Let R be a 2-torsion free semiprime ring, U be a non-zero ideal of R and D be a permuting n-generalized semiderivation. Suppose D satisfies the identity $D(R\circ U)\mp [R,U]\subseteq Z(R)$. If

- $(i)$
$D(R)\ne 0$ with the property ${x}_{i}^{2}={x}_{i}$ for all $x\in R$ then either $D(R)$ is central of R or $D(R)\subseteq M-set$ or $D({R}^{2})$ is commuting of R such that $i=1,2,...,n.$

- $(ii)$
$D=0$ then R contains a non-zero central ideal.

**Proof.** First, we observe that R satisfies the identity $D(R\circ U)\mp [R,U]\subseteq Z(R)$. Clearly, this relation implies $D(r\circ x)\mp [r,x]\in Z(R)$ for all $x\in U,r\in R$.

Moreover, we note that $D(rx)+D(xr)+[r,x]\in Z(R)$ for all $x\in U,r\in R$.

The last relation can be rewritten as

for all

$,\stackrel{`}{{x}_{i}},{x}_{i}\in U,\stackrel{`}{{r}_{i}},{r}_{i}\in R$ such that

$i=1,2,...,n.$Applying Definition 2 to the first term of this relation, we achieve that

$$D(({r}_{1},{r}_{2},....,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{2},{x}_{3},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})){x}_{1}\phantom{\rule{0ex}{0ex}}+\psi ({r}_{n})\Delta (({r}_{1},{r}_{2},....,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n-1})({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))\phantom{\rule{0ex}{0ex}}+D(({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})({r}_{1},{r}_{2},....,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n}))\phantom{\rule{0ex}{0ex}}+[({r}_{1},{r}_{2},....,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n}),({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})]\in Z(R).$$

Hence this relation can be rewritten as

For any arbitrary element of

R, we find that

Putting

$t={x}_{1}$ of this relation yields

In this relation replacing

${x}_{1}$ by

$\omega ({r}_{i},{x}_{i})$, we find that

Now left-multiplying by

$\omega ({r}_{i},{x}_{i})R$ and right-multiplying by

$R[D(({r}_{1},{r}_{2},....,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{2},{x}_{3},....,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})),\omega ({r}_{i},{x}_{i})]$ with applying Lemma 2 and

$aRb\subset Z(R),a,b\in R$, we arrive to

Due to R is a semiprime, we consider the set $\left\{{P}_{\alpha}\right\}$ of prime ideals of R such that $\cap {P}_{\alpha}=\left\{0\right\}.$ According to Lemma 1, we obtain $\left\{{P}_{\alpha}\right\}$ the set of prime ideals of R is semiprime ideal.

Let

$\cap {P}_{\alpha}=U$. Hence, we have either

$\omega ({r}_{i},{x}_{i})\in U$ that is,

We add the term

$-D(({r}_{1},{r}_{2},....,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))$ to both sided of this relation, we obtain that

$$\psi ({r}_{n})\Delta (({r}_{1},{r}_{2},....,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n-1})({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))-D(({r}_{1},{r}_{2},....,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))\phantom{\rule{0ex}{0ex}}=-D(({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})({r}_{1},{r}_{2},....,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n}))-D(({r}_{1},{r}_{2},....,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))\phantom{\rule{0ex}{0ex}}-[({r}_{1},{r}_{2},....,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n}),({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})].$$

Rewriting this relation as follows

$$\psi ({r}_{n})\Delta (({r}_{1},{r}_{2},....,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n-1})({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))-D(({r}_{1},{r}_{2},....,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})\phantom{\rule{0ex}{0ex}}=-D(({r}_{1},{r}_{2},....,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})\circ ({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))\phantom{\rule{0ex}{0ex}}-[({r}_{1},{r}_{2},....,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n}),({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})].$$

Corresponding to the main identity

$D(R\circ U)\mp [R,U]\subseteq Z(R)$, from the left side of this relation, we arrive to

Simplifying this expression, we find that

For any arbitrary element of

R from this relation, we show that

for all

$\stackrel{`}{{x}_{i}},{x}_{i}\in U,r,t\in R$Substituting

t for

$D(({r}_{1},{r}_{2},...,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))$ and

${x}_{1}$ by

$y{x}_{1}$, we notice that

Putting

$D(({r}_{1},{r}_{2},...,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))$ for

y of this relation, we see that

Left-multiplying by

$D(({r}_{1},{r}_{2},...,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))[{x}_{1},D(({r}_{1},{r}_{2},...,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))]\phantom{\rule{0ex}{0ex}}D(({r}_{1},{r}_{2},...,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))$ with applying Lemma 2 and

$aRb\subset Z(R),a,b\in R$, we find that

Now if we continue to carry out the same method as above, after right-multiplying this relation by

$[{x}_{1},D(({r}_{1},{r}_{2},...,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))]$ and employing Lemma 2 and

$aRb\subset Z(R),a,b\in R$, this relation reduces to

Using the action of Lemma 2 with right-multiplying

$y[{x}_{1},D(({r}_{1},{r}_{2},...,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n}))]$ and left-multiplying by

$D(({r}_{1},{r}_{2},...,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))y$,

$y\in R$, we satisfy that

Putting

${x}_{1}y$ for

y in this relation yields

Left-multiplying (21) by

${x}_{1}$ and subtracting this result from (22) with using

R is semiprime and

$D\ne 0$, we observe that

Hence, from the last equation, we see that $D(({r}_{1},{r}_{2},...,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))\in Z(R)$.

Replacing ${x}_{2}$ by ${x}_{1}{x}_{2}$, we achieve that $D(R)\subseteq Z(R)$.

Or

$[D(({r}_{1},{r}_{2},...,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{2},{x}_{3},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})),\omega ({r}_{i},{x}_{i})]\in U$ which implies to the relation

Now substituting the value of

$\omega ({r}_{i},{x}_{i})$ for this relation with simple calculation yields

$$[D(({r}_{1},{r}_{2},...,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{2},{x}_{3},...,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})),\psi ({r}_{n})\Delta (({r}_{1},{r}_{2},....,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n-1})({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))]\phantom{\rule{0ex}{0ex}}+[D(({r}_{1},{r}_{2},...,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{2},{x}_{3},...,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})),D(({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))({r}_{1},{r}_{2},....,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})]\phantom{\rule{0ex}{0ex}}+[D(({r}_{1},{r}_{2},...,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{2},{x}_{3},...,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})),[({r}_{1},{r}_{2},....,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n}),({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})]]=0.$$

We add the term

$[D(({r}_{1},{r}_{2},...,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{2},{x}_{3},...,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})),D({r}_{1},{r}_{2},...,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n},{x}_{2},{x}_{3},...,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})){x}_{1}]$ to both sided and using

D is symmetric mapping, we conclude that

$$2[D(({r}_{1},{r}_{2},...,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{2},{x}_{3},...,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})),D(({r}_{1},{r}_{2},...,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{1},{x}_{2},{x}_{3},...,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))]\phantom{\rule{0ex}{0ex}}+[({r}_{1},{r}_{2},....,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n}),({x}_{1},{x}_{2},....,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})]\phantom{\rule{0ex}{0ex}}=D(({r}_{1},{r}_{2},...,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{2},{x}_{3},...,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))[D(({r}_{1},{r}_{2},...,\stackrel{`}{{r}_{i}},{r}_{i}...,{r}_{n})({x}_{2},{x}_{3},...,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})),{x}_{1}].$$

Replacing

${x}_{2}$ by

${x}_{1}{x}_{2}$ and using

${x}_{i}^{2}={x}_{1}$ with putting

${x}_{i}$ for

${r}_{i}$ of this relation, we find that

Left-multiplying by ${[D({x}_{1},{x}_{2},{x}_{3},...,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))}^{2},{x}_{1}]R$ and right-multiplying by $RD({x}_{1},{x}_{2},{x}_{3},...,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}){)}^{2}$. Also, using Lemma 2 and $aRb\subset Z(R),a,b\in R$ with applying the same previous technique which used of the above part of our proof, we satisfy two options either $D({x}_{1},{x}_{2},{x}_{3},...,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}){)}^{2}\in U$ or ${[D({x}_{1},{x}_{2},{x}_{3},...,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))}^{2},{x}_{1}]\in U$.

Definitely, the first case and using the condition $D\ne 0$ give us $D({x}_{1},{x}_{2},{x}_{3},...,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))\in M-set$.

Whereas, the second case implies to ${[D({x}_{1},{x}_{2},{x}_{3},...,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n}))}^{2},{x}_{1}]=0.$ Writing x instead of ${x}_{1}={x}_{2}={x}_{3}=...=\stackrel{`}{{x}_{i}}={x}_{i}...={x}_{n})$. Hence, we conclude that $[D{(x)}^{2},x]=0.$ Linearization of this equation and using it. Applying that R is 2-torsion free, we conclude that $D({R}^{2})$ is commuting on R.

(ii) Obviously, if $D=0$ then the main relation becomes $[R,U]\subseteq Z(R)$. It is clear that R contains a non-zero central ideal. This completes the proof. □

**Remark** **1.** The condition ${x}_{i}^{2}={x}_{i}$ which appeared in Branch(i) of the previous theorem is not superfluous. Indeed, the evidence of this fact can be obtained from Example 1.

**Theorem** **7.** Let R be a 2-torsion free semiprime ring, U be a non-zero ideal of R and D be a permuting n-generalized semiderivation with a trace μ. Suppose D satisfies the identity $D([R,U])\mp \mu (R\circ U)\subseteq Z(R)$. If

- $(i)$
$D(R)\ne 0$ then $D({U}^{2})\subseteq Z(R)$.

- $(ii)$
$D(R)=0$ then $\mu ({U}^{2})\subseteq Z(R)$.

**Proof.** (i) Given that $D([R,U])\mp \mu (R\circ U)\subseteq Z(R)$. Evidently, replacing R by U reduces this relation into $\mp \psi ({U}^{2})\subseteq Z(R)$.

Putting

$x+y$ for all

$x,y\in U$ instead of

U of above relation, for any arbitrary element of

R say

r, regard that

$x=({x}_{1},{x}_{2},{x}_{3},...,\stackrel{`}{{x}_{i}},{x}_{i}...,{x}_{n})$ and

$y=({y}_{1},{y}_{2},{y}_{3},...,\stackrel{`}{{y}_{i}},{y}_{i}...,{y}_{n})$, we conclude that

$[\mu ({x}^{2}),r]+[\mu (xy+yx),r]+[\mu ({y}^{2}),r]=0$ for all

$x,y\in U$,

$r\in R.$ According to the identity

$\mu ({U}^{2})\subseteq Z(R)$, this relation modifies to

It is clear to be seen that

$D:{R}^{n}\u27f6R$ is permuting and

n-additive mapping, then the trace

$\mu $ of

D satisfies the following relation

where

x appears

$n-i$-times and

y appears

i-times.

Let

$\mu \ne 0$. Since

$\mu (xy+yx)\in Z(R)$ for all

$x,y\in U$, replacing

y by

$y+kz$ for all

$x,y,z\in U$,

$1\le k\le n-1$, in the this relation, we find that

for all

$x,y\in U.$Moreover, we observe that

for all

$x,y\in U$,

$r\in R.$ Using Relation (23), we notice that

for all

$x,y\in U$,

$r\in R.$Applying Lemma 3 gives that $n[D(x\circ z,x\circ y,x\circ y,x\circ y...,x\circ y),r]=0.$

Due to the fact

R is 2-torsion free, we receive that

Particularly, we achieve that $D(x\circ y,x\circ y,x\circ y,x\circ y...,x\circ y)\in Z(R)$ for all $x,y\in U$. Putting x instead of y of this relation. This yields $D({x}_{1}^{2},{x}_{2}^{2},...,{\stackrel{`}{{x}_{i}}}^{2},{x}_{i}^{2},...,{x}_{n}^{2})\in Z(R)$ for all $\stackrel{`}{{x}_{i}},{x}_{i}\in U$.

Writing ${x}^{2}$ instead of ${x}_{1}^{2}={x}_{2}^{2}=...={\stackrel{`}{{x}_{i}}}^{2}={x}_{i}^{2}=...={x}_{n}^{2}$ in this identity, we conclude that $[D({x}^{2}),r]=0$ for all $r\in R$ yields $D({x}^{2})\in Z(R)$.

(ii) Certainly, if $D(R)=0$ then the main relation reduces to $\mu (R\circ U)\subseteq Z(R)$, which means $2\mu ({U}^{2})\subseteq Z(R)$. By reason of the fact R is 2-torsion free, we find that $\mu ({U}^{2})\subseteq Z(R)$. This finishes the proof. □

**Theorem** **8.** Let R be a 2-torsion free semiprime ring and D be a non-zero permuting n-generalized semiderivation associated with function ψ such that $\psi (R)=a-bR$, $a,b\in R$. If D satisfies the relation $[\Delta (R),D(R)]\subseteq Z(R)$ then R has a weak zero divisor.

**Proof.** For the convenience, let us rewrite the main condition as $[\Delta (R),D(({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})({y}_{1},{y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n}))]\in Z(R)$ for all $\stackrel{`}{{x}_{i}},{x}_{i},\stackrel{`}{{y}_{i}},{y}_{i}\in R$, $i=1,2,...n.$

Suppose there is an arbitrary element of

R with using the main relation such that

$$D(({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})({y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n}))[[\Delta (R),{y}_{1}],r]\phantom{\rule{0ex}{0ex}}+[D({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})({y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n}),r][\Delta (R),{y}_{1}]\phantom{\rule{0ex}{0ex}}+[\Delta (R),D({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})({y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n})][{y}_{1},r]\phantom{\rule{0ex}{0ex}}+[\Delta (R),\psi ({x}_{n})][\Delta ({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n-1})({y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n}),r]\phantom{\rule{0ex}{0ex}}+[[\Delta (R),\psi ({x}_{n})],r]\Delta (({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n-1})({y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n}))=0$$

for all

$\stackrel{`}{{x}_{i}},{x}_{i},\stackrel{`}{{y}_{i}},{y}_{i},r\in R$.

Without loss of generality we replace

r by

$\Delta (R)$. In this case this relation becomes

$$D(({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})({y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n}))[[\Delta (R),{y}_{1}],\Delta (R)]\phantom{\rule{0ex}{0ex}}+2[D({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})({y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n}),\Delta (R)][\Delta (R),{y}_{1}]\phantom{\rule{0ex}{0ex}}+[\Delta (R),\psi ({x}_{n})][\Delta ({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n-1})({y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n}),\Delta (R)]\phantom{\rule{0ex}{0ex}}+[[\Delta (R),\psi ({x}_{n})],\Delta (R)]\Delta (({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n-1})({y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n}))=0.$$

Replacing

R by

${y}_{1}$, we find that

Compatibility with the fact that the associated function

$\psi $ acts as

$\psi (R)=a-bR$, where

a and

b are fixed element of

R, we conclude that

In this relation, putting

$-{x}_{n}$ for the place of

${x}_{n}$ and combining the result with above relation, we see that

Applying the fact that

R is 2-torsion free, we see that

Replacing

${y}_{2}$ by

${y}_{1}{y}_{2}$ of this identity, we arrive to

Arguing in a similar technique as we have done in the proof of Theorem 6, we separate the proof in two cases:

Either $[[\Delta ({y}_{1}),a],\Delta ({y}_{1})]\in U$ or $\Delta (R)\in U$. Now if we continue with the process inductively then from the second case, we arrive to $\Delta (R)=0$.

In agreement with $\Delta \ne 0$, the latter result leads to a contradiction.

If second case holds, that is, $[[\Delta ({y}_{1}),a],\Delta ({y}_{1})]=0$ for all ${y}_{1}\in R.$

Consequently, multiplying this relation by R. This case implies to $R[[\Delta ({y}_{1}),a],\Delta ({y}_{1})]=0$ for all ${y}_{1}\in R.$ In other words, this result shows that for each $x\in R\backslash \left\{0\right\}$, we satisfy this result. Basically, R has not a zero divisors. Hence, we assume that $R\Delta ({y}_{1})\ne 0.$ Therefore, R has a weak zero divisor. This completes the proof. □

**Theorem** **9.** Let R be a 2-torsion free semiprime ring, U be an ideal and D be a non-zero permuting n-generalized semiderivation associated with function ψ such that $\psi (R)=Ra-b{R}^{2}$, $a,b\in R$ and $\lambda (R)=-{R}^{2}$ an associated function of Δ. Suppose R satisfies the identity $D([[{R}_{1},{R}_{2}],({R}_{1}\circ {R}_{2})])\subseteq Z(R)$. Then either ${R}^{2}\subseteq M-set$ or ${b}^{4}\in M-set$ or $b\circ R=0$ or ${b}^{3}\in Z(R)$.

**Proof.** Directly, from the main relation we find that

$[D([[{R}_{1}{R}_{2},{R}_{2}{R}_{1}])-D([{R}_{2}{R}_{1},{R}_{1}{R}_{2}])),t]=0$ for all

$t\in R.$ Simplify this relation, we arrive to

for all

$t\in R.$Applying Definition 2 on the last term and putting

$({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})$ instead of

${R}_{1}$ with

${y}_{1},{y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n}$ taking the place of

${R}_{2}$ yields

$$[D([{R}_{1},{R}_{2}]{R}_{1}{R}_{2}),t]+[D({R}_{1}{R}_{2}[{R}_{2},{R}_{1}]),t]-[D([{R}_{2},{R}_{1}]{R}_{2}{R}_{1}),t]\phantom{\rule{0ex}{0ex}}-[D(({y}_{1},{y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n})({x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})[({y}_{1},{y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n}),({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})]){x}_{1}\phantom{\rule{0ex}{0ex}}+\psi ({y}_{n})\Delta (({y}_{1},{y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n-1})({x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})[({y}_{1},{y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n}),({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})]),t]\phantom{\rule{0ex}{0ex}}=0.$$

Without loss of generality, this relation can be rewritten as the following

$$[D([{R}_{1},{R}_{2}]{R}_{1}{R}_{2}),t]+[D({R}_{1}{R}_{2}[{R}_{2},{R}_{1}]),t]-[D([{R}_{2},{R}_{1}]{R}_{2}{R}_{1}),t]\phantom{\rule{0ex}{0ex}}-[D(({y}_{1},{y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n})({x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})[({y}_{1},{y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n}),({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})]){x}_{1},t]\phantom{\rule{0ex}{0ex}}-[\psi ({y}_{n})\Delta (({y}_{1},{y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n-1})({x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})[({y}_{1},{y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n})({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})]),t]\phantom{\rule{0ex}{0ex}}=0.$$

Now in view of our hypothesis we see

$\psi $ acts as

$\psi (R)=a-b{R}^{2}$,

a and

b are fixed element of

R. Hence, this relation becomes

$$[D([{R}_{1},{R}_{2}]{R}_{1}{R}_{2}),t]+[D({R}_{1}{R}_{2}[{R}_{2},{R}_{1}]),t]-[D([{R}_{2},{R}_{1}]{R}_{2}{R}_{1}),t]\phantom{\rule{0ex}{0ex}}-[D(({y}_{1},{y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n})({x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n}))[({y}_{1},{y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n}),({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})]){x}_{1},t]\phantom{\rule{0ex}{0ex}}-[{y}_{n}a\Delta (({y}_{1},{y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n-1})({x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})[({y}_{1},{y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n}),({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})]),t]\phantom{\rule{0ex}{0ex}}+[b{y}_{n}^{2}\Delta (({y}_{1},{y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n-1})({x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})[({y}_{1},{y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n}),({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})]),t]\phantom{\rule{0ex}{0ex}}=0.$$

In particular for

${y}_{n}=-{y}_{n}$ of this relation. Combining this result with the relations, we arrive to

$$[D([{R}_{1},{R}_{2}]{R}_{1}{R}_{2}),t]+[D({R}_{1}{R}_{2}[{R}_{2},{R}_{1}]),t]-[D([{R}_{2},{R}_{1}]{R}_{2}{R}_{1}),t]\phantom{\rule{0ex}{0ex}}-[D(({y}_{1},{y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n})({x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})[({y}_{1},{y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n}),({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})]){x}_{1},t]\phantom{\rule{0ex}{0ex}}-[a\Delta (({y}_{1},{y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n-1})({x}_{2},{x}_{3},...,\stackrel{`}{{x}_{i}},{x}_{i},...,{x}_{n})[({y}_{1},{y}_{2},...,\stackrel{`}{{y}_{i}},{y}_{i},...,{y}_{n}),({x}_{1},{x}_{2},...,\stackrel{`}{{x}_{i}}{x}_{i},...,{x}_{n})]),t]\phantom{\rule{0ex}{0ex}}=0.$$

Obviously, substituting this relation in the above relation, we conclude that