Action of $n$-derivations and $n$-multipliers on ideals of (semi)-prime rings

1.   Introduction

Throughout, $\mathcal{R}$ will be an associative ring with $\mathcal{Z}(\mathcal{R})$ as its center. A ring $\mathcal{R}$ is said to be prime if $\vartheta \mathcal{R}\ell = \{0\}$ implies that either $\vartheta = 0$ or $\ell = 0$ and semiprime if $\vartheta \mathcal{R}\vartheta = \{0\}$ implies that $\vartheta = 0$, where $\vartheta, \ell \in \mathcal{R}$. The symbols $[\vartheta, \ell ]$ and $\vartheta \circ \ell$ denote the commutator $\vartheta \ell -\ell \vartheta$ and the anti-commutator $\vartheta \ell + \ell \vartheta$, respectively, for any $\vartheta, \ell \in \mathcal{R}$. A ring $\mathcal{R}$ is said to be $n$-torsion free if $n\vartheta = 0$ implies that $\vartheta = 0 \; \forall \; \vartheta \in \mathcal{R}$. If $\mathcal{R}$ is $n!$-torsion free, then it is $m$-torsion free for every divisor $m$ of $n!$. An additive mapping $\mathcal{D}:\mathcal{R} \rightarrow \mathcal{R}$ is called a derivation if $\mathcal{D}(\vartheta \ell) = \mathcal{D}(\vartheta)\ell +\vartheta \mathcal{D}(\ell)$ holds $\forall \; \vartheta, \ell \in \mathcal{R}$. In order to broaden the scope of derivation, Maksa ^[12] introduced the notion of symmetric bi-derivations on rings, which Vukman examined in greater detail in ^[17,18]. A bi-additive map $\mathfrak{D}:\mathcal{R} \times \mathcal{R} \rightarrow \mathcal{R}$ is said to be a bi-derivation if

hold for any $\vartheta, \vartheta ', \ell, \ell ' \in \mathcal{R}$. The foregoing conditions are identical if $\mathfrak{D}$ is also a symmetric map, that is, if $\mathfrak{D}(\vartheta, \ell) = \mathfrak{D}(\ell, \vartheta)$ for every $\vartheta, \ell \in \mathcal{R}$. In this case, $\mathfrak{D}$ is referred to as a symmetric bi-derivation on $\mathcal{R}$. Several authors have studied symmetric bi-derivations on rings (see ^[3,11,16] and references therein) and produced highly helpful outcomes.

The study of tri-derivation was initiated in ^[13], by $\ddot{\text{O}}$zt$\ddot{\text{u}}$rk, in which he proved various results. Several results have been obtained by various authors in this direction (see ^[13,19] and references therein). In light of the concepts of bi-derivation and tri-derivation, Park ^[14] introduced the concept of permuting $n$-derivation as follows:

Definition 1.1. Let $n \geq 2$ be a fixed integer, and $\mathcal{R}^n = \underbrace{\mathcal{R} \times \mathcal{R} \times \cdots \times \mathcal{R}}_{n-\text{times}}$. A map $\mathfrak{D}:\mathcal{R}^n \rightarrow \mathcal{R}$ is said to be symmetric (permuting) if

for all permutations $\pi(t) \in S_n$ and $\vartheta _t \in R$, where $t = 1, 2, \ldots, n$.

Definition 1.2. Let $n \geq 2$ be a fixed integer. An $n$-additive mapping (i.e., additive in each argument) $\mathfrak{D} : \mathcal{R}^n \rightarrow \mathcal{R}$ is called an $n$-derivation on $\mathcal{R}$ if the relations

hold for all $\vartheta _t, \vartheta '_t \in \mathcal{R}, \; t = 1, 2, \ldots, n{ }$.

If, in addition, $\mathfrak{D}$ is a permuting map, then all the above conditions are equivalent, and in that case $\mathfrak{D}$ is called a permuting $n$-derivation on $\mathcal{R}$.

Of course, $1$-derivation is a derivation, a $2$-derivation is a symmetric bi-derivation, and for $n = 3$, $\mathfrak{D}$ is referred to as a permuting 3-derivation (or tri-derivation) on rings (see ^[17,19] for details).

A map $\mathcal{d}:\mathcal{R} \rightarrow \mathcal{R}$ defined by $\mathcal{d}(\vartheta) = \mathfrak{D}(\vartheta, \vartheta, \ldots, \vartheta)$ is called the trace of $\mathfrak{D}$. If $\mathfrak{D} : \mathcal{R}^n \rightarrow \mathcal{R}$ is permuting and $n$-additive, then the trace $\mathcal{d}$ of $\mathfrak{D}$ satisfies the relation

$\forall \; \vartheta, \ell \in \mathcal{R}$, where ${}^{n}C_{k} = \binom{n}{k}$ and

Let $S$ be a nonempty subset of $\mathcal{R}$. A mapping $\mathcal{d} :\mathcal{R} \rightarrow \mathcal{R}$ is said to be commuting (respectively, centralizing) on $\mathcal{R}$ if $[\mathcal{d}(\vartheta), \vartheta] = 0$ (respectively, $[\mathcal{d}(\vartheta), \vartheta] \in \mathcal{Z}(\mathcal{R}))$ for all $\vartheta \in \mathcal{R}$. The study of commuting and centralizing mappings on a prime ring was initiated by Posner ^[15], who proved that if a prime ring $\mathcal{R}$ admits a nonzero centralizing derivation, then $\mathcal{R}$ is commutative. Being inspired by this result, Bre$\breve{s}$ar [9, Theorem 4.1] proved this for left ideals. In fact, he proved that if $\mathcal{R}$ is a prime ring, $\mathcal{W}$ is a nonzero left ideal of $\mathcal{R}$, and $\mathcal{d}$ and $\mathcal{g}$ are nonzero derivations of $\mathcal{R}$ satisfying $\mathcal{d}(\vartheta)\vartheta -\vartheta \mathcal{g}(\vartheta) \in \mathcal{Z}(\mathcal{R}) \; \forall \; \vartheta \in \mathcal{W}$, then $\mathcal{R}$ is commutative. In ^[2], Argaç gave a partial extension of Bre$\breve{s}$ar's result in the setting of semiprime rings. Motivated by the classical result due to Posner ^[15], Vukman obtained some results concerning the trace of symmetric bi-derivations in prime rings (see ^[17,18] for more details). In ^[3], Ashraf established similar results for semiprime rings. Further, Ashraf et al. ^[6,8] obtained commutativity of rings admitting $n$-derivations whose traces satisfy certain polynomial conditions. Recently, Ashraf et al. ^[4] introduced the concepts of permuting $n$-multipliers and proved that for a fixed integer $n \geq 2$, if $\mathcal{R}$ is a non-commutative $n!$-torsion free prime ring admitting a permuting generalized $n$-derivation $\mathcal{G}$ with associated $n$-derivation $\mathcal{D}$ such that the trace $\omega$ of $\mathcal{G}$ is commuting on $\mathcal{R}$, then $\mathcal{G}$ is a left $n$-multiplier on $\mathcal{R}$. Many authors have studied various identities involving traces of bi-derivations and $n$-derivations and have obtained several interesting results (viz., ^{[3,4,6,11,16,17,18]} and references therein).

The primary aim of this paper is to prove analogous results related to permuting $n$-derivations in the setting of prime and semiprime rings. In fact, we investigate the structure of (semi)prime rings and describe the forms of maps (traces of $n$-derivations) satisfying certain functional identities. More precisely, we prove that: let $n \geq 2$ be a fixed integer, $\mathcal{R}$ be an $n!$-torsion free semiprime ring and $\mathcal{W}$ be a nonzero ideal of $\mathcal{R}.$ If $\mathcal{R}$ admits two nonzero symmetric $n$-derivations $\mathfrak{D}:\mathcal{R}^n \rightarrow \mathcal{R}$ with trace $\mathcal{d} : \mathcal{R} \rightarrow \mathcal{R}$ and $\mathcal{G}:\mathcal{R}^n \rightarrow \mathcal{R}$ with trace $\mathcal{g} : \mathcal{R} \rightarrow \mathcal{R}$ satisfying $\mathcal{d}(\vartheta)\ell \pm \vartheta\mathcal{g}(\ell) \in \mathcal{Z}(\mathcal{R}) \; \forall \; \vartheta, \ell \in \mathcal{W}$, then $\mathcal{R}$ has a nonzero central ideal (Theorem 2.5). Further, in the last section, we establish that if $\mathcal{R}$ is an $n!$-torsion free prime ring admitting two symmetric $n$-derivations $\mathfrak{D} : \mathcal{R}^n \rightarrow \mathcal{R}$ and $\mathcal{G} : \mathcal{R}^n \rightarrow \mathcal{R}$ with traces $\mathcal{f}$ and $\mathcal{g}$, respectively, satisfying $\mathcal{f}(\vartheta)\vartheta +\vartheta \mathcal{g}(\vartheta) = 0 \; \forall \; \vartheta \in \mathcal{W}$, a left ideal of $\mathcal{R}$, then either $\mathcal{R}$ is commutative or $\mathcal{G}$ acts as a left $n$-multiplier on $\mathcal{W}$ (Theorem 3.2). Moreover, we also characterize the traces of $q$-iterations of $n$-derivations in prime rings and prove that for a fixed integer $n \geq 2$, if $\mathcal{R}$ is an $n!$-torsion free prime ring and $q \geq 1$, a fixed integer admitting $q$-iterations of $n$-derivations $\mathfrak{D}_1, \mathfrak{D}_2, \ldots, \mathfrak{D}_q : \mathcal{R}^n \rightarrow \mathcal{R}$ such that the product of the traces of $\mathfrak{D}_1, \mathfrak{D}_2, \ldots, \mathfrak{D}_q$, respectively, is zero on a nonzero ideal of $\mathcal{R}$, then either $\mathfrak{D}_1 = 0$ or the rest of $\mathfrak{D}_i's$ act as $n$-multipliers on $\mathcal{R}$ (Theorem 3.6).

2.   Results

In the present section, we state and prove the main results of this article. In order to establish the proofs of our main theorems, we first state a number of well-known results.

Lemma 2.1. ^[14] Let $n$ be a fixed positive integer and $\mathcal{R}$ an $n!$-torsion free ring. Suppose that $a_1, a_2, \ldots, a_n \in \mathcal{R}$ satisfy $\lambda a_1 +\lambda^2 a_2+ \cdots +\lambda^n a_n = 0\; (or \in \mathcal{Z}(\mathcal{R}))$ for $\lambda = 1, 2, \ldots, n$. Then, $a_t = 0\; (or \in \mathcal{Z}(\mathcal{R}))$ for $t = 1, 2, \ldots, n.$

Lemma 2.2. ^[10] If $\mathcal{R}$ is a semiprime ring, then the center of a nonzero ideal of $\mathcal{R}$ is contained in the center of $\mathcal{R}$.

Lemma 2.3. ^[16] Let $\mathcal{R}$ be a 2-torsion free semiprime ring and $\mathcal{W}$ be a nonzero ideal of $\mathcal{R}$. If $[\mathcal{W}, \mathcal{W} ] \subseteq \mathcal{Z}(\mathcal{R})$, then $\mathcal{R}$ contains a nonzero central ideal.

Lemma 2.4. ^[16] Let $\mathcal{R}$ be a 2-torsion free semiprime ring and $\mathcal{W}$ be a nonzero ideal of $\mathcal{R}$. If $\mathcal{W} \circ \mathcal{W} \subseteq \mathcal{Z}(\mathcal{R})$, then $\mathcal{R}$ contains a nonzero central ideal.

The first main result of this paper is the following theorem:

Theorem 2.5. Let $n \geq 2$ be a fixed integer, $\mathcal{R}$ be an $n!$-torsion free semiprime ring and $\mathcal{W}$ be a nonzero ideal of $\mathcal{R}.$ If $\mathcal{R}$ admits two nonzero symmetric $n$-derivations $\mathfrak{D}:\mathcal{R}^n \rightarrow \mathcal{R}$ with trace $\mathcal{d} : \mathcal{R} \rightarrow \mathcal{R}$ and $\mathcal{G}:\mathcal{R}^n \rightarrow \mathcal{R}$ with trace $\mathcal{g} : \mathcal{R} \rightarrow \mathcal{R}$ satisfying $\mathcal{d}(\vartheta)\ell \pm \vartheta\mathcal{g}(\ell) \in \mathcal{Z}(\mathcal{R}) \; \forall \; \vartheta, \ell \in \mathcal{W}$, then $\mathcal{R}$ has a nonzero central ideal.

Proof. It is given that

Replacing $\ell$ by $\ell +m \mathcal{k}$ for $\mathcal{k} \in \mathcal{W}$ and $1 \leq m \leq n-1$, we obtain

Solving further, we get

$\forall \; \vartheta, \ell, \mathcal{k} \in \mathcal{W}$. Taking account of the given condition, we find that

which implies that

where $h_t(\ell; \mathcal{k})$ represents the term in which $\mathcal{k}$ appears $t$- times.

The application of Lemma 2.1 yields

Since $\mathcal{R}$ is $n!$-torsion free, we get

Replacing $\mathcal{k}$ by $\ell$, we find that

Hence, by the hypothesis, we see that

Now, on commuting with $r$ where $r\in \mathcal{R}$, we get

Replacing $\ell$ by $\ell \mathcal{k}$ where $\mathcal{k} \in \mathcal{W}$ in (2.2) and using (2.2), we get

Now, replacing $r$ by $\mathcal{d}(\vartheta)$ in the above equation, we obtain

Multiplying by $\mathcal{k}$ from left, we get

Taking $\mathcal{k} \ell$ in place of $\ell$ in (2.3), we see that

Subtracting (2.5) from (2.4), we get

i.e.,

i.e.,

Since $\mathcal{R}$ is a semiprime ring, the last expression gives

Replacing $\ell$ by $r[\mathcal{k}, \mathcal{d}(\vartheta)]$, we get

From the semiprimeness of $\mathcal{R}$, we see that

Invoking Lemma 2.2, we have

Now, again replacing $\vartheta$ by $\vartheta+mw_1$ for $w_1 \in \mathcal{W}$ and $1 \leq m \leq n-1$ in (2.7) and using (2.7), we obtain

which implies that

$\forall \; \vartheta, w_1 \in \mathcal{W}$. Invoking Lemma 2.1 and using the fact that $\mathcal{R}$ is $n!$-torsion free, we get

Replace $\vartheta$ by $\vartheta+mw_2$ for $w_2 \in \mathcal{W}$ and $1 \leq m \leq n-1$ in the above equation to get

and on further solving and using torsion restriction, we get

Continuing in the same manner, we get

On commuting with $r$, we get

After replacing $w_1$ by $w_1w_1'$ in the above equation and using torsion restriction of $\mathcal{R}$, we arrive at

Now taking $r$ to be $rr'$ where $r' \in \mathcal{R}$, we get

i.e.,

Since $\mathcal{R}$ is a semiprime ring, it must contain a family of prime ideals of $\mathcal{R}$ whose intersection is zero. Let $\mathfrak{P} = \{P_j \; |\; j \in \Lambda \}$ be the family of all prime ideals such that $\bigcap P_j = \{0\}$. Let $P$ be a typical member of $\mathfrak{P}.$ From (2.10), we conclude that for a fixed $w_1 \in \mathcal{W}$,

Let us set $\mathcal{U} = \{w_1 \in \mathcal{W} \; \; |\; [w_1, \mathcal{R}] \subseteq P\}$ and $\mathcal{V} = \{w_1 \in \mathcal{W} \; |\; \mathfrak{D}(w_1, w_2, \ldots, w_n) \in P\; \forall\; w_2, \ldots, w_n \in \mathcal{W}\}$. Both $\mathcal{U}$ and $\mathcal{V}$ are additive subgroups of $\mathcal{W}$ such that $\mathcal{W} = \mathcal{U} \cup \mathcal{V}$, but a group cannot be the union of two of its proper subgroups. Hence, either $\mathcal{W} = \mathcal{U}$ or $\mathcal{W} = \mathcal{V}$. Let us suppose that $\mathcal{W} \neq \mathcal{U}$. Then, we have $\mathcal{W} = \mathcal{V}$, i.e., $\mathfrak{D}(w_1, w_2, \ldots, w_n) \in P \; \forall \; w_1, w_2, \ldots, w_n \in \mathcal{W}$. Replace $w_1$ by $w_1r_1$, i.e., $\mathfrak{D}(w_1r_1, w_2, \ldots, w_n) \in P$ for any $r_1 \in \mathcal{R}$. On solving, we get $w_1\mathfrak{D}(r_1, w_2, \ldots, w_n) \in P$. Using primeness of $P$, we get either $w_1 \in P$ or $\mathfrak{D}(r_1, w_2, \ldots, w_n) \in P$ for all $w_1, w_2, \ldots, w_n \in \mathcal{W}, \; r_1 \in \mathcal{R}$. However, $w_1 \in P$ implies that $[w_1, \mathcal{R}] \subseteq P$, which leads to a contradiction. Thus, we have $\mathfrak{D}(r_1, w_2, \ldots, w_n) \in P$ for all $w_2, \ldots, w_n \in \mathcal{W}, \; r_1 \in \mathcal{R}$. Again replace $w_2$ by $w_2r_2$, and using the same procedure, we get $\mathfrak{D}(r_1, r_2, w_3, \ldots, w_n) \in P$ for all $w_3, \ldots, w_n \in \mathcal{W}, \; r_1, r_2 \in \mathcal{R}$. Continuing in a similar manner, we arrive at

Since $P$ was an arbitrary element of $\mathfrak{P}$,

which implies that $\mathfrak{D}(\mathcal{R}, \mathcal{R}, \ldots, \mathcal{R}) = \{0\}$. Hence, we arrive at a contradiction. Therefore, $\mathcal{W} = \mathcal{U}$, i.e., $[w_1, \mathcal{R}] \subseteq P$ for all $w_1 \in \mathcal{W}$ or $[\mathcal{W}, \mathcal{R}] \subseteq \bigcap P_j = \{0\}$. That is, $[\mathcal{W}, \mathcal{R}] = \{0\}$. Therefore, $\mathcal{W}$ is a nonzero central ideal of $\mathcal{R}$. Hence, $\mathcal{R}$ has a nonzero central ideal. □

Theorem 2.6. For a fixed integer $n \geq 2$, let $\mathcal{R}$ be an $n!$-torsion free semiprime ring and $\mathcal{W}$ be a nonzero ideal of $\mathcal{R}$. Suppose that $\mathcal{R}$ admits two nonzero symmetric $n$-derivations $\mathfrak{D}:\mathcal{R}^n \rightarrow \mathcal{R}$ with trace $\mathcal{d}:\mathcal{R} \rightarrow \mathcal{R}$ and $\mathcal{G}:\mathcal{R}^n \rightarrow \mathcal{R}$ with trace $\mathcal{g}:\mathcal{R} \rightarrow \mathcal{R}$ satisfying any one of the following conditions:

(1) $[\mathcal{d}(\vartheta), \ell] = \pm\; \vartheta \circ \mathcal{g}(\ell) \; \forall \; \vartheta, \ell \in \mathcal{W},$

(2) $\mathcal{d}([\vartheta, \ell]) = [\mathcal{d}(\vartheta), \ell]+[\mathcal{d}(\ell), \vartheta] \; \forall \; \vartheta, \ell \in \mathcal{W},$

(3) $\mathcal{d}(\vartheta) \circ \ell = \pm\; \vartheta \circ \mathcal{g}(\ell) \; \forall \; \vartheta, \ell \in \mathcal{W}.$

Then, $\mathcal{R}$ contains a nonzero central ideal.

Proof. ${(i)}$ It is given that

Now, replace $\ell$ by $\ell+m\mathcal{k}$ for $\mathcal{k} \in \mathcal{W}$ and $1 \leq m \leq n-1$, and we get

$\forall \; \vartheta, \ell, \mathcal{k} \in \mathcal{W}$. By using the hypothesis, we get

which implies that

where

denotes the sum of terms in which $\mathcal{k}$ appears $t$-times. The application of Lemma 2.1 yields that

Using the torsion free restriction in $\mathcal{R}$, we find that

After replacing $\mathcal{k}$ by $\ell$, we get

Again using the hypothesis, we get

which is the same as (2.6). Hence, proceeding in the same pattern as we have done so far, we conclude that $\mathcal{R}$ contains a nonzero central ideal.

${(ii)}$ It is given that

On replacing $\ell$ by $\ell+m\mathcal{k}$ for $\mathcal{k} \in \mathcal{W}$ and $1 \leq m \leq n-1$, we get

$\forall \; \vartheta, \ell, \mathcal{k} \in \mathcal{W}.$ On simplifying, we get

$\; \forall\; \vartheta, \ell, \mathcal{k} \in \mathcal{W}$. Using the hypothesis, we get

$\forall\; \vartheta, \ell, \mathcal{k} \in \mathcal{W}$. This leads us to the following:

where

denotes the sum of terms in which $\mathcal{k}$ appears $t$- times.

Taking account of Lemma 2.1 and the torsion free restriction in $\mathcal{R}$, we get

Replacing $\mathcal{k}$ by $\ell$, we get

Using the hypothesis once again, we obtain

which is the same as (2.6). Hence, the result follows by using the same argument as discussed in Theorem 2.5.

${(iii)}$ It is given that

On replacing $\ell$ by $\ell+m\mathcal{k}$ for $\mathcal{k} \in \mathcal{W}$ and $1 \leq m \leq n-1$, we get

On simplifying, we get

$\forall\; \vartheta, \ell, \mathcal{k} \in \mathcal{W}$. On using the given condition, we find that

Application of Lemma 2.1 gives

Since $\mathcal{R}$ is $n!$-torsion free, we have

On replacing $\mathcal{k}$ by $\ell$, we get

Using the hypothesis one more time, we see that

Replacing $\ell$ by $\ell \mathcal{k}$ where $\mathcal{k} \in \mathcal{W}$, we find that

Replacing $\ell$ by $[\mathcal{k}, \mathcal{d}(\vartheta)]r$ in the above equation, we have

Since $\mathcal{R}$ is a semiprime ring, we get

which is the same as (2.6). Hence, proceeding in the same way, we conclude that $\mathcal{R}$ contains a nonzero central ideal. □

In ^[7], Ashraf et al. proved that $\mathcal{R}$ is commutative if it satisfies any one of the following conditions: (i) $F(\vartheta \ell) \pm \vartheta \ell \in \mathcal{Z}(\mathcal{R})$, (ii) $F(\vartheta \ell) \pm \ell \vartheta \in \mathcal{Z}(\mathcal{R})$, and (iii) $F(\vartheta)F(\ell)-\vartheta \ell \in \mathcal{Z}(\mathcal{R}) \; \forall \; \vartheta, \ell \in \mathcal{R}$, where $F$ is a generalized derivation on $\mathcal{R}$. In our next result, we extend Theorems 2.1 and 2.3 of ^[7] for the traces of permuting $n$-derivations on semiprime rings.

Theorem 2.7. Let $n \geq 2$ be a fixed integer and let $\mathcal{R}$ be an $n!$-torsion free semiprime ring and $\mathcal{W}$ be an nonzero ideal of $\mathcal{R}$. Suppose that $\mathcal{R}$ admits a symmetric $n$-derivation $\mathfrak{D} : \mathcal{R}^n \rightarrow \mathcal{R}$ with trace $\mathcal{d}: \mathcal{R} \rightarrow \mathcal{R}$ such that any one of the following conditions hold:

(1) $\mathcal{d}(\vartheta \ell) \pm \vartheta \ell \in \mathcal{Z}(\mathcal{R}) \; \forall\; \vartheta, \ell \in \mathcal{W},$

(2) $\mathcal{d}(\vartheta \ell) \pm \ell \vartheta \in \mathcal{Z}(\mathcal{R}) \; \forall\; \vartheta, \ell \in \mathcal{W},$

(3) $\mathcal{d}(\vartheta \ell) \pm [\vartheta, \ell ] \in \mathcal{Z}(\mathcal{R}) \; \forall\; \vartheta, \ell \in \mathcal{W},$

(4) $\mathcal{d}(\vartheta \ell) \pm \vartheta \circ \ell \in \mathcal{Z}(\mathcal{R}) \; \forall\; \vartheta, \ell \in \mathcal{W}.$

Then, $\mathcal{R}$ contains a nonzero central ideal.

Proof. ${(i)}$ It is given that

Replace $\ell$ by $\ell +m\mathcal{k}$ for $\mathcal{k} \in \mathcal{W}$ and $1 \leq m \leq n-1$, and we get

That is,

$\forall\; \vartheta, \ell, \mathcal{k} \in \mathcal{W}$. On using the given condition, we see that

Now, use Lemma 2.1 and the fact that $\mathcal{R}$ is $n!$- torsion free to get

Replace $\mathcal{k}$ by $\ell$ to get

Again using the hypothesis, we get

Commuting with $r\in \mathcal{R}$, we obtain

and so

Replacing $\ell$ by $\ell \mathcal{k}$ in (2.12) and using (2.12), we see that

On replacing $\vartheta$ by $[\mathcal{k}, r]$, we get

That is,

Since $\mathcal{R}$ is a semiprime ring, we have

Taking $\ell$ to be $t[\mathcal{k}, r]$, $t \in \mathcal{R}$, we see that

By the semiprimeness of $\mathcal{R}$, we get $\mathcal{W} \subseteq \mathcal{Z}(\mathcal{R})$. Thus, $\mathcal{R}$ contains a nonzero central ideal.

${(ii)}$ Use similar arguments as used in $\it{(i)}$ to get the required result.

${(iii)}$ It is given that

Replace $\ell$ by $\ell +m\mathcal{k}$ for $\mathcal{k} \in \mathcal{W}$ and $1 \leq m \leq n-1$, and we get

That is,

$\forall\; \vartheta, \ell, \mathcal{k} \in \mathcal{W}$. Using the hypothesis, we see that

Invoking Lemma 2.1 and using the torsion free restriction of $\mathcal{R}$, we get

Replace $\mathcal{k}$ by $\ell$, and we obtain

On using the hypothesis, we see that

That is, $[\mathcal{W}, \mathcal{W}] \subset \mathcal{Z}(\mathcal{R})$. Hence, by Lemma 2.3, $\mathcal{R}$ contains a nonzero central ideal.

${(iv)}$ It is given that

Taking $\ell +m\mathcal{k}$ in the place of $\ell$ for $\mathcal{k} \in \mathcal{W}$ and $1 \leq m \leq n-1$, we get

That is,

for all $\vartheta, \ell, \mathcal{k} \in \mathcal{W}$. With the help of the given condition, we see that

Now, using Lemma 2.1 and the fact that $\mathcal{R}$ is $n!$-torsion free, we obtain

Replace $\mathcal{k}$ by $\ell$, and we get

Making use of the hypothesis, we see that

That is, $\mathcal{W} \circ \mathcal{W} \in \mathcal{Z}(\mathcal{R})$. Hence, by using Lemma 2.4, $\mathcal{R}$ contains a nonzero central ideal. The proof is complete. □

Based on the preceding findings, we obtain the following known result:

Corollary 2.8. ^[8] For any fixed integer $n \geq 2$, let $\mathcal{R}$ be an $n!$-torsion free semiprime ring. If $\mathcal{R}$ admits a nonzero permuting $n$-derivation $\Delta : \mathcal{R}^n \rightarrow \mathcal{R}$ with trace $\mathcal{d} : \mathcal{R} \rightarrow \mathcal{R}$ satisfying any one of the conditions

(1) $\mathcal{d}(\vartheta \ell) \pm \vartheta \ell \in \mathcal{Z}(\mathcal{R}) \; \forall\; \vartheta, \ell \in \mathcal{R},$

(2) $\mathcal{d}(\vartheta \ell) \pm \ell \vartheta \in \mathcal{Z}(\mathcal{R}) \; \forall\; \vartheta, \ell \in \mathcal{R},$

then $\mathcal{R}$ is commutative.

Theorem 2.9. For a fixed integer $n \geq 2$, let $\mathcal{R}$ be an $n!$-torsion free semiprime ring and $\mathcal{W}$ be a nonzero ideal of $\mathcal{R}$. Suppose that $\mathcal{R}$ admits two nonzero symmetric $n$-derivations $\mathfrak{D} : \mathcal{R}^n \rightarrow \mathcal{R}$ and $\mathcal{G} : \mathcal{R}^n \rightarrow \mathcal{R}$ with $\mathcal{d}:\mathcal{R} \rightarrow \mathcal{R}$ and $\mathcal{g}: \mathcal{R} \rightarrow \mathcal{R}$ as traces of $\mathfrak{D}$ and $\mathcal{G}$ satisfying any one of the following conditions:

(1) $\mathcal{g}(\vartheta \ell)+\mathcal{d}(\vartheta)\mathcal{d}(\ell) \pm \vartheta \ell \in \mathcal{Z}(\mathcal{R}) \; \forall\; \vartheta, \ell \in \mathcal{W},$

(2) $\mathcal{g}(\vartheta \ell)+\mathcal{d}(\vartheta)\mathcal{d}(\ell) \pm \ell \vartheta \in \mathcal{Z}(\mathcal{R}) \; \forall\; \vartheta, \ell \in \mathcal{W},$

(3) $\mathcal{g}([\vartheta, \ell ])+[\mathcal{d}(\vartheta), \mathcal{d}(\ell)] \pm [\vartheta, \ell ] \in \mathcal{Z}(\mathcal{R}) \; \forall\; \vartheta, \ell \in \mathcal{W},$

(4) $\mathcal{g}(\vartheta \circ \ell)+\mathcal{d}(\vartheta) \circ \mathcal{d}(\ell) \pm \vartheta \circ \ell \in \mathcal{Z}(\mathcal{R}) \; \forall\; \vartheta, \ell \in \mathcal{W}.$

Then, $\mathcal{R}$ contains a nonzero central ideal.

Proof. ${(i)}$ It is given that

Replacing $\ell$ by $\ell +m\mathcal{k}$ for $\mathcal{k} \in \mathcal{W}$ and $1 \leq m \leq n-1$, we arrive at

Using the given condition, we get

$\forall \; \vartheta, \ell, \mathcal{k} \in \mathcal{W}$. Using Lemma 2.1, we see that

Since $\mathcal{R}$ is $n!$-torsion free, we get

Writing $\ell$ in place of $\mathcal{k}$, we get

Using the hypothesis, we obtain that

On using the same arguments as after (2.11), we get the required result.

${(ii)}$ Following the same steps as in $(i)$, we discover that $\mathcal{R}$ contains a nonzero central ideal.

${(iii)}$ It is given that

Replacing $\ell$ by $\ell +m\mathcal{k}$ for $\mathcal{k} \in \mathcal{W}$ and $1 \leq m \leq n-1$, we conclude that

On using the hypothesis, we get

$\forall \; \vartheta, \ell, \mathcal{k} \in \mathcal{W}$. Using Lemma 2.1 and the fact that $\mathcal{R}$ is $n!-$torsion free, we have

Writing $\ell$ in place of $\mathcal{k}$, we obtain

Using the hypothesis, we obtain that

By Lemma 2.3, we conclude that $\mathcal{R}$ contains a nonzero central ideal.

${(iv)}$ It is given that

Replacing $\ell$ by $\ell +m\mathcal{k}$ for $\mathcal{k} \in \mathcal{W}$ and $1 \leq m \leq n-1$, we arrive at

On using the hypothesis, we get

$\forall \; \vartheta, \ell, \mathcal{k} \in \mathcal{W}$. Using Lemma 2.1 and using the fact that $\mathcal{R}$ is $n!-$torsion free, we get

Write $\ell$ in place of $\mathcal{k}$ to get

Using the hypothesis, we obtain that

We conclude by Lemma 2.4 that $\mathcal{R}$ contains a nonzero central ideal. The proof is complete. □

3.   Permuting $n$-multipliers

This section deals with the study of permuting $n$-multipliers. The idea of a permuting $n$-multiplier was initially suggested by Ashraf et al. in ^[4], and they proved some interesting results. In the present section, we examine the action of symmetric $n$-derivations satisfying the functional identity $\mathcal{f}(i)i +i g(i) = 0 \; \forall \; i \in \mathcal{W}$, a nonzero left ideal of $\mathcal{R}$ where $\mathcal{f}$ and $g$ are the traces of symmetric $n$-derivations $\mathfrak{D}$ and $\mathcal{G}$, respectively. We begin with the following:

Definition 3.1. A permuting $n$-additive map $\Lambda: \mathcal{R}^n \rightarrow \mathcal{R}$ is called a permuting left $n$-multiplier (resp., permuting right $n$-multiplier) if

holds $\forall \; i_t, i'_t \in \mathcal{R}, \; t = 1, 2, \ldots, n$. If $\Lambda$ is both a permuting left $n$-multiplier and a permuting right $n$-multiplier, it is referred to as a permuting $n$-multiplier. For related results, see ^[4,5].

According to Bre$\breve{s}$ar's proof in [9, Theorem 4.1], if $\mathcal{R}$ is a prime ring, $\mathcal{W}$ is a nonzero left ideal of $\mathcal{R}$, and $\mathcal{d}$ and $\mathcal{g}$ are nonzero derivations of $\mathcal{R}$ satisfying $\mathcal{d}(i)i -i \mathcal{g}(i) \in \mathcal{Z}(\mathcal{R}) \; \forall \; i \in \mathcal{W}$, then $\mathcal{R}$ is commutative. We expand the previous result by demonstrating the following theorem for the trace of $n$-derivation of $\mathcal{R}$.

Theorem 3.2. Let $\mathcal{R}$ be an $n!$-torsion free prime ring and $\mathcal{W}$ be a nonzero left ideal of $\mathcal{R}$. Suppose that $\mathcal{R}$ admits two symmetric $n$-derivations $\mathfrak{D} : \mathcal{R}^n \rightarrow \mathcal{R}$ and $\mathcal{G} : \mathcal{R}^n \rightarrow \mathcal{R}$ with $\mathcal{f}$ and $\mathcal{g}$ as traces of $\mathfrak{D}$ and $\mathcal{G}$, respectively. If $\mathcal{f}(i)i +i \mathcal{g}(i) = 0 \; \forall \; i \in \mathcal{W}$, then either $\mathcal{R}$ is commutative or $\mathcal{G}$ acts as a left n-multiplier on $\mathcal{W}$. Furthermore, in the last case, either $\mathfrak{D} = 0$ or $\mathcal{W}[\mathcal{W}, \mathcal{W}] = 0$.

Proof. By hypothesis, we have

Replacing $i$ by $i +m\ell$ for $\ell \in \mathcal{W}$ and $1 \leq m \leq n-1$, we get

On using the definition of $\mathcal{f}$ and $\mathcal{g}$, we see that

On using the given condition, we get

$\forall \; i, \ell \in \mathcal{W}$. On using Lemma 2.1, we get

$\forall \; i, \ell \in \mathcal{W}$. Replace $i$ by $i \mathcal{k}$ to obtain

On comparing (3.1) and (3.2), we get

This implies that

$\forall \; i, \ell, \mathcal{k} \in \mathcal{W}$. Substitute $ri$ for $i$ in (3.3) to get

$\forall \; i, \ell, \mathcal{k} \in \mathcal{W}, r \in \mathcal{R}$. Compare (3.4) and (3.3).

Since $\mathcal{R}$ is $n!-$torsion free, we obtain

$\forall \; i, \ell, \mathcal{k} \in \mathcal{W}, r \in \mathcal{R}$. Replacing $\ell$ by $\mathcal{k}$ in (3.5), we see that

Substituting $ri$ for $i$, we get

Since $\mathcal{R}$ is a prime ring, it yields that either $[\mathcal{k}, r] = 0$ or $i \mathcal{g}(\mathcal{k}) = 0$. If $[\mathcal{k}, r] = 0 \; \forall\; \mathcal{k} \in \mathcal{W} \; \text{and}\; r \in \mathcal{R},$ then replacing $\mathcal{k}$ by $s\mathcal{k}$, we get $[s, r]\mathcal{k} = 0\; \forall \; \mathcal{k} \in \mathcal{W}, r, s \in \mathcal{R}$. Again, replace $\mathcal{k}$ by $r\mathcal{k}$ such that $[s, r]r\mathcal{k} = 0 \; \forall \; \mathcal{k} \in \mathcal{W}, r, s \in \mathcal{R}$. Since $\mathcal{R}$ is a prime ring, we conclude that $\mathcal{R}$ is commutative. Next, if $i \mathcal{g}(\mathcal{k}) = 0 \; \forall\; i, \mathcal{k} \in \mathcal{W}$, then replacing $\mathcal{k}$ by $\mathcal{k} +m\ell$, we get

That is,

By using Lemma 2.1 and the fact that $\mathcal{R}$ is $n!-$torsion free, we get

This implies that

Hence, $\mathcal{G}$ acts as a left $n$-multiplier. Since $i \mathcal{G}(\mathcal{k}, \ell \ldots, \ell) = 0 \; \forall\; i, \ell, \mathcal{k} \in \mathcal{W},$ using (3.5), we arrive at

Replace $r$ by $sr$ to get

Primeness of $\mathcal{R}$ yields that either $\mathfrak{D}(s, \ell, \ldots, \ell) = 0$ or $i [\mathcal{k}, \ell ] = 0 \; \forall\; i, \ell, \mathcal{k} \in \mathcal{W}, s \in \mathcal{R}$. If $\mathfrak{D} \neq 0$, the latter results in $\mathcal{W}[\mathcal{W}, \mathcal{W}] = 0$. □

Following the same vein, we can also demonstrate the following:

Theorem 3.3. Let $\mathcal{R}$ be an $n!-$ torsion free prime ring and $\mathcal{W}$ be a nonzero right ideal of $\mathcal{R}$. Assume that $\mathfrak{D}$ and $\mathcal{G}$ are two symmetric $n$-derivations of $\mathcal{R}$ with trace $\mathcal{f}$ and $\mathcal{g}$, respectively. If $\mathcal{f}(i)i +i \mathcal{g}(i) = 0 \; \forall \; i \in \mathcal{W}$, then either $\mathcal{R}$ is commutative or $\mathfrak{D}$ acts as a left n-multiplier on $\mathcal{W}$. Furthermore, in the last case, either $\mathcal{G} = 0$ or $\mathcal{W}[\mathcal{W}, \mathcal{W}] = 0$.

In view of the above result, we obtain the following known results:

Corollary 3.4. ^[1] Let $\mathcal{R}$ be a prime ring of characteristic not two, $\mathcal{W}$ be a nonzero left ideal of $\mathcal{R}$ and $\Delta_1$, $\Delta_2$ be symmetric bi-derivations of $\mathcal{R}$ with traces $\mathcal{d}_1$ and $\mathcal{d}_2$, respectively. If $\Delta_1(i, i)i +i \Delta_2(i, i) = 0 \; \forall \; i \in \mathcal{W}$, then either $\mathcal{R}$ is commutative or $\Delta_2$ acts as a left bi-multiplier on $\mathcal{W}$. Moreover, in the last case either $\Delta_1 = 0$ or $\mathcal{W}[\mathcal{W}, \mathcal{W}] = 0$.

Corollary 3.5. ^[1] Let $\mathcal{R}$ be a prime ring of characteristic not two, $\mathcal{W}$ be a nonzero right ideal of $\mathcal{R}$ and $\Delta_1$, $\Delta_2$ be symmetric bi-derivations of $\mathcal{R}$ with traces $\mathcal{d}_1$ and $\mathcal{d}_2$, respectively. If $\Delta_1(i, i)i +i \Delta_2(i, i) = 0 \; \forall \; i \in \mathcal{W}$, then either $\mathcal{R}$ is commutative or $\Delta_1$ acts as a left bi-multiplier on $\mathcal{W}$. Moreover, in the last case either $\Delta_2 = 0$ or $\mathcal{W}[\mathcal{W}, \mathcal{W}] = 0$.

The next result is the generalization of Vukman's result ^[18]. Indeed, Vukman showed that if $\mathcal{R}$ is a prime ring of characteristic different from two and three, and there exist symmetric bi-derivations $\mathfrak{D}_1:\mathcal{R} \times \mathcal{R} \rightarrow \mathcal{R}$ and $\mathfrak{D}_2:\mathcal{R} \times \mathcal{R} \rightarrow \mathcal{R}$, such that $\mathcal{\mathcal{f}_1}(a)\mathcal{\mathcal{f}_2}(a) = 0, \; \forall \; a \in \mathcal{R}$ holds, where $\mathcal{\mathcal{f}_1}$ and $\mathcal{\mathcal{f}_2}$ are the traces of $\mathfrak{D}_1$ and $\mathfrak{D}_2$ respectively, then either $\mathfrak{D}_1 = 0$ or $\mathfrak{D}_2 = 0$. We extend this theorem for $q$-iterations of $n$-derivations.

Theorem 3.6. Let $\mathcal{R}$ be an $n$!-torsion free prime ring, $\mathcal{W}$ be a nonzero ideal of $\mathcal{R}$ and $q \geq 1$, be a fixed integer. Consider $\mathfrak{D}_1, \mathfrak{D}_2, \ldots, \mathfrak{D}_q : \mathcal{R}^n \rightarrow \mathcal{R}$ to be $n$-derivations on $\mathcal{R}$ such that $\mathcal{d}_1(i _1)\mathcal{d}_2(i _2)\cdots \mathcal{d}_q(i_q) = 0 \; \forall \; i_1, i_2, \ldots, i_q \in \mathcal{W}$ where $\mathcal{d}_i's$, are traces of $\mathfrak{D}_i's$, respectively. Then, one of the following holds:

(1) $\mathcal{d}_1(i _1) = 0 \; \forall \; i_1 \in \mathcal{W}$;

(2) All $\mathfrak{D}_p$ act as left $n$-multipliers on $\mathcal{R}$ for $p = 2, 3, \ldots, q$.

Proof. We will prove it through induction. If we put $q = 1$ in our hypothesis, then it is obvious that $\mathcal{d}_1(i_1) = 0 \; \forall \; i_1 \in \mathcal{W}$. Now, consider $q = 2$, and by the hypothesis, we have

Replacing $i_2$ by $i_2 +m\ell_2$ for $\ell_2 \in \mathcal{W}$ and $1 \leq m \leq n-1$, we get

On simplifying, we get

$\forall \; i_1, i_2, \ell_2 \in \mathcal{W}$. Compare (3.6) and (3.7) and use Lemma 2.1 to get

Since $\mathcal{R}$ is $n!$-torsion free, we obtain

Replacing $\ell_2$ by $\ell_2r$ in (3.8), we obtain

i.e.,

Since $\mathcal{R}$ is a prime ring, we can find either $\mathcal{d}_1(i_1)\ell_2 = 0$ or $\mathfrak{D}_2(i _2, \ldots, i _2, r) = 0$. Consider the first case, $\mathcal{d}_1(i _1)\ell_2 = 0$. Again, $\mathcal{R}$ is a prime ring, and we get $\mathcal{d}_1(i _1) = 0$. Now, consider the latter case, $\mathfrak{D}_2(i _2, \ldots, i _2, r) = 0 \; \forall \; i _2 \in \mathcal{W}$, $r \in \mathcal{R}$. A straightforward modification shows that $\mathfrak{D}_2(i _2, \ldots, i _2, w_1r) = \mathfrak{D}_2(i _2, \ldots, i _2, w_1)r \; \forall \; w_1 \in \mathcal{W}, r \in \mathcal{R}$. Hence, $\mathfrak{D}_2$ acts as a left $n$-multiplier as desired.

Next, suppose that it is true for $n = q-1$, and we shall prove it for $n = q$. Let us assume the hypothesis:

Replacing $i _q$ by $i _q+m\ell _q$ for $\ell_q \in \mathcal{W}$ and $1 \leq m \leq n-1$ in (3.9) and taking account of Lemma 2.1, we get

$\forall \; i _1, i _2, \ldots, i _q, \ell _q\in \mathcal{W}$. Since $\mathcal{R}$ is $n!$-torsion free, we see that

Substituting $\ell _qu$ for $\ell _q$ in (3.10) and using (3.10), we arrive at

i.e.,

$\forall \; i _1, i _2, \ldots, i _q, \ell _q\in \mathcal{W}, u\in \mathcal{R}$. Primeness of $\mathcal{R}$ gives that either $\mathcal{d}_1(i _1)\mathcal{d}_2(i _2)\cdots \mathcal{d}_{q-1}(i _{q-1}) = 0$ or $\mathfrak{D}_q(i _q, \ldots, i _q, u) = 0 \; \forall \; i _1, i _2, \ldots, i _q \in \mathcal{W}, \; u\in \mathcal{R}$. If $\mathcal{d}_1(i _1)\mathcal{d}_2(i _2)\cdots \mathcal{d}_{q-1}(i _{q-1}) = 0$, then we are done by the former case. If $\mathfrak{D}_q(i _q, \ldots, i _q, u) = 0 \; \forall \; i _q \in \mathcal{W}, \; u \in \mathcal{R}$, then we can easily compute that $\mathfrak{D}_q(i _q, \ldots, i _q, w_{q-1}u) = \mathfrak{D}_q(i _q, \ldots, i _q, w_{q-1})u \; \forall \; i _q, w_{q-1} \in \mathcal{W}, \; u \in \mathcal{R}$. Hence, $\mathfrak{D}_q$ acts as a left $n$-multiplier on $\mathcal{R}$ as desired. The theorem's proof is completed with this conclusion. □

4.   Conclusions and further remarks

In this article, we discussed some results concerning the containment of a nonzero central ideal in a ring $\mathcal{R}$ satisfying certain functional identities involving the traces $\mathcal{d}$ and $\mathcal{g}$ of symmetric $n$-derivations $\mathcal{D}$ and $\mathcal{G}$, respectively. Besides proving some results concerning the traces of permuting $n$-derivations, some results related to permuting $n$-multipliers are also discussed in the last section. In fact, we characterized symmetric $n$-derivations of prime rings in terms of left $n$-multipliers. In future, it would be interesting to study these functional identities in the setting of generalized permuting $n$-derivations and its related maps in rings with involution.

Acknowledgments

The authors are very thankful to the anonymous referees for their valuable comments and suggestions which have improved the manuscript immensely. Moreover, the authors extend their appreciation to King Saud University, Deanship of Scientific Research, College of Science Research Center for funding this research.

Conflict of interest

The authors declare that they have no conflicts of interest.