Commutativity of a 3-Prime near Ring Satisfying Certain Differential Identities on Jordan Ideals

Abstract

The purpose of this study is to obtain the commutativity of a 3-prime near ring satisfying some differential identities on Jordan ideals involving derivations and multiplicative derivations. Further, herein we discuss some examples to show the necessity of the hypothesis to our results.

1. Introduction

A left near ring $\mathcal{N}$ is a triplet $(\mathcal{N},+,·)$, where + and · are two binary operations such that (i) $(\mathcal{N},+)$ is a group, (ii) $(\mathcal{N},·)$ is a semigroup, and (iii) $u·(v+w)=u·v+u·w$ for every $u,v,w\in \mathcal{N}$. Analogously, if instead of (iii), $\mathcal{N}$ satisfies the right distributive law, then $\mathcal{N}$ is said to be a right near ring. Therefore, near rings are generalized rings, need not be commutative, and most importantly, only one distributive law is postulated (e.g., Example 1.4, Pilz [1]). A near ring $\mathcal{N}$ is known as zero-symmetric if $0u=0$ for every $u\in \mathcal{N}$ (left distributive law gives that $u0=0$). Throughout the manuscript, $\mathcal{N}$ represents a zero-symmetric left near ring with $\mathcal{Z}\left(\mathcal{N}\right)$ as its multiplicative center. For $v,w\in \mathcal{N}$, the symbols $[v,w]$ and $v\circ w$ denote the commutator $vw-wv$ and the anticommutator $vw+wv$, respectively. A near ring $\mathcal{N}$ is known as 2-torsion free if $2u=0$⇒$u=0$ for every $u\in \mathcal{N}$. A near ring $\mathcal{N}$ is known as 3-prime if for $v,w\in \mathcal{N}$, $v\mathcal{N}w=\left\{0\right\}$⇒$v=0$ or $w=0$. Bell and Mason [2] initiated the study of derivation in near rings. An additive mapping $d:\mathcal{N}\to \mathcal{N}$ is known as a derivation on a near ring $\mathcal{N}$ if $d\left(vw\right)=d\left(v\right)w+vd\left(w\right)$ or equivalently as in [3], $d\left(vw\right)=vd\left(w\right)+d\left(v\right)w$ for all $v,w\in \mathcal{N}$. The commutative property of prime (semiprime) rings with some suitable constraints on derivations has been established by various authors (see [4,5,6,7,8,9,10,11]). Some comparable results on near rings have also been obtained, (c.f. [2,3,12,13,14,15,16,17]). An additive map $f:\mathcal{N}\to \mathcal{N}$ is known as commuting on a non empty subset $\mathcal{S}$ of a near ring $\mathcal{N}$ if $\left[f\right(u),u]=0$ for all $u\in \mathcal{S}$. An additive subgroup $\mathcal{J}$ of a near ring $\mathcal{N}$ is known as a Jordan ideal of $\mathcal{N}$ if $k\circ u\in \mathcal{J}$ and $u\circ k\in \mathcal{J}$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$. Daif and Bell [7] established the following result: Let $\mathcal{I}$ be a nonzero ideal of a prime ring $\mathcal{R}$. If d is a derivation on $\mathcal{R}$ satisfying $d\left(\right[v,w\left]\right)=\pm [v,w]$ for all $v,w\in \mathcal{I}$, then $\mathcal{R}$ is commutative. In [18], Boua and Oukhtite proved that if a 3-prime near ring $\mathcal{N}$ with a nonzero derivation d satisfying either $d\left(\right[v,w\left]\right)=\pm [v,w]$ or $d(v\circ w)=\pm (v\circ w)$ for every $v,w\in \mathcal{N}$, then $\mathcal{N}$ is a commutative ring. Further, Boua [19] proved that if $\mathcal{U}$ is a semigroup ideal of a 3-prime near ring $\mathcal{N}$ and d is a derivation on $\mathcal{N}$ satisfying any one of the following conditions: (i) $d\left(\right[v,w\left]\right)=[v,w]$, (ii) $d\left(\right[v,w\left]\right)=\left[d\right(v),w]$, (iii) $\left[d\right(v),w]=[v,w]$, (iv) $d(v\circ w)=d\left(v\right)\circ w$, or (v) $d\left(v\right)\circ w=(v\circ w)$ for all $v,w\in \mathcal{U}$, then $\mathcal{N}$ is a commutative ring.

A mapping $d:\mathcal{R}\to \mathcal{R}$ is known as a multiplicative derivation on a ring $\mathcal{R}$ if $d\left(vw\right)=d\left(v\right)w+vd\left(w\right)$ for every $v,w\in \mathcal{R}$. In [20], the concept of multiplicative derivation in rings was introduced by Daif and it was inspired by Martindale [21]. In [22], Goldmann and Šemrl studied these mappings and provided the full description of such mappings (for more details, we refer to [20] and [22]). Let $\mathcal{R}=C[0,1]$ be a ring of all real valued continuous functions. Define a map $d:\mathcal{R}\to \mathcal{R}$ by

$$\begin{array}{c}\hfill d\left(g\right)\left(u\right)=\left\{\begin{array}{c}g\left(u\right)log\mid g\left(u\right)\mid ,\mathrm{where}g\left(u\right)\ne 0\\ 0,\mathrm{otherwise}.\end{array}\right.\end{array}$$

Then it is easy to verify that $d\left(gh\right)=d\left(g\right)h+gd\left(h\right)$ for all $g,h\in C[0,1]$ but $d(g+h)\ne d\left(g\right)+d\left(h\right)$. A mapping (not necessarily additive) $d:\mathcal{N}\to \mathcal{N}$ is known as a multiplicative derivation on a near ring $\mathcal{N}$ if $d\left(vw\right)=d\left(v\right)w+vd\left(w\right)$ for every $v,w\in \mathcal{N}$.

In this manuscript, we show the commutativity condition for a 3-prime near ring $\mathcal{N}$ if any one of the following holds: (i) $[d_1\left(u\right),d_2\left(k\right)]=[u,k]$, (ii) $d\left(\right[k,u\left]\right)=\left[d\right(k),u]$, (iii) $\left[d\right(u),k]=[u,k]$, (iv) $d\left(\right[k,u\left]\right)=d\left(k\right)\circ u$, (v) $\left[d\right(k),d(u\left)\right]=0$ for all $u\in \mathcal{N}$ and $k\in \mathcal{J}$, a Jordan ideal of $\mathcal{N}$, where d, $d_1$, $d_2$ are derivations on $\mathcal{N}$.

2. Preliminaries

In this section, we state some basic lemmas to establish our main results.

Lemma 1.

([2], Lemma 3) Let d be a nonzero derivation on a 3-prime near ring $\mathcal{N}$.

(i)

If $\mathcal{N}$ is 2-torsion free, then $d^2\ne 0$.

(ii)

If $u\in \mathcal{N}$ and $d\left(\mathcal{N}\right)u=\left\{0\right\}$, then $u=0$.

Lemma 2.

([23], Lemma 3) Let $\mathcal{J}$ be a nonzero Jordan ideal of a 2-torsion free 3-prime near ring $\mathcal{N}$. If $\mathcal{J}\subseteq \mathcal{Z}\left(\mathcal{N}\right)$, then $\mathcal{N}$ is a commutative ring.

Lemma 3.

([24], Corollary 3) Let $\mathcal{J}$ be a nonzero Jordan ideal of a 2-torsion free 3-prime near ring $\mathcal{N}$. If d is a derivation on $\mathcal{N}$ such that $d\left(\mathcal{J}\right)=\left\{0\right\}$, then either $d=0$ or the elements of $\mathcal{J}$ commute under the multiplication of $\mathcal{N}$.

Lemma 4.

Let d be a multiplicative derivation on a near ring $\mathcal{N}$. Then

$$\begin{array}{c}\hfill \left(d\left(u\right)v+ud\left(v\right)\right)w=d\left(u\right)vw+ud\left(v\right)w,\forall u,v,w\in \mathcal{N}.\end{array}$$

Proof. 

By solving $d\left(uvw\right)$ in two different ways, we have

$$\begin{array}{cc}\hfill d\left(uvw\right)& =d\left(uv\right)w+uvd\left(w\right)\hfill \\ \hfill & =\left(d\left(u\right)v+ud\left(v\right)\right)w+uvd\left(w\right)\hfill \end{array}$$

(1)

and

$$\begin{array}{cc}\hfill d\left(uvw\right)& =d\left(u\right)vw+ud\left(vw\right)\hfill \\ \hfill & =d\left(u\right)vw+ud\left(v\right)w+uvd\left(w\right).\hfill \end{array}$$

(2)

Comparing (1) and (2), we obtain

$$\left(d\left(u\right)v+ud\left(v\right)\right)w=d\left(u\right)vw+ud\left(v\right)w,\forall u,v,w\in \mathcal{N}.$$

□

3. Main Results

3.1. Commutativity Conditions Involving Derivations

Bell and Daif [6] showed the following result: If $\mathcal{R}$ is a 2-torsion free prime ring admitting a strong commutativity preserving (in short, SCP) derivation d, i.e., d satisfies $\left[d\right(v),d(w\left)\right]=[v,w]$ for every $v,w\in \mathcal{R}$, then $\mathcal{R}$ is commutative. In this section, we extend this result for a 3-prime near ring in two directions. First of all, we consider two derivations instead of one derivation, and secondly, we prove the commutativity of a 3-prime near ring $\mathcal{N}$ in place of a ring $\mathcal{R}$ in case of a Jordan ideal of $\mathcal{N}$.

Theorem 1.

Let $\mathcal{J}$ be a nonzero Jordan ideal of a 2-torsion free 3-prime near ring $\mathcal{N}$. If $d_1,d_2$ are two nonzero derivations on $\mathcal{N}$ such that $d_2$ is commuting on $\mathcal{J}$ and $[d_1\left(u\right),d_2\left(k\right)]=[u,k]$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$, then either $d_1=0$ on $\mathcal{J}$ or $\mathcal{N}$ is a commutative ring.

Proof. 

By hypothesis,

$$[d_1\left(u\right),d_2\left(k\right)]=[u,k],\forall k\in \mathcal{J},u\in \mathcal{N}.$$

(3)

Replacing u by $ku$ in (3) and using $[ku,k]=k[u,k]$, we get

$$\begin{array}{cc}\hfill [d_1\left(ku\right),d_2\left(k\right)]& =[ku,k]=k[u,k]\hfill \\ \hfill d_1\left(ku\right)d_2\left(k\right)-d_2\left(k\right)d_1\left(ku\right)& =k[d_1\left(u\right),d_2\left(k\right)],\forall k\in \mathcal{J},u\in \mathcal{N}.\hfill \end{array}$$

Applying the definition of $d_1$ and using Lemma 4, we arrive at

$$k{d}_1\left(u\right)d_2\left(k\right)+d_1\left(k\right)u{d}_2\left(k\right)-d_2\left(k\right)d_1\left(k\right)u-d_2\left(k\right)k{d}_1\left(u\right)=k{d}_1\left(u\right)d_2\left(k\right)-k{d}_2\left(k\right)d_1\left(u\right).$$

Since $d_2$ is commuting on $\mathcal{J}$, so the last expression yields that

$$d_1\left(k\right)u{d}_2\left(k\right)=d_2\left(k\right)d_1\left(k\right)u,\forall k\in \mathcal{J},u\in \mathcal{N}.$$

(4)

Substituting $uw$ in place of u in (4) and using (4), we find that

$$d_1\left(k\right)\mathcal{N}[d_2\left(k\right),w]=\left\{0\right\},\forall k\in \mathcal{J},w\in \mathcal{N}.$$

By 3-primeness of $\mathcal{N}$, we obtain

$$d_1\left(k\right)=0\mathrm{or}{d}_2\left(k\right)\in \mathcal{Z}\left(\mathcal{N}\right),\forall k\in \mathcal{J}.$$

(5)

If $d_2\left(k\right)\in \mathcal{Z}\left(\mathcal{N}\right)$ for all $k\in \mathcal{J}$, then our hypothesis becomes $[u,k]=0$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$ which means that $k\in \mathcal{Z}\left(\mathcal{N}\right)$ for all $k\in \mathcal{J}$. Therefore, (5) becomes

$$d_1\left(k\right)=0\mathrm{or}k\in \mathcal{Z}\left(\mathcal{N}\right),\forall k\in \mathcal{J}.$$

Hence, by Lemma 2, we obtain the result. □

The example given below illustrates that we cannot omit the 3-prime condition in Theorem 1.

Example 1.

Suppose that $\mathcal{S}$ is a zero-symmetric non abelian left near ring and let

$$\begin{array}{c}\hfill \mathcal{N}=\left\{\left(\begin{array}{ccc}0& u& v\\ 0& 0& 0\\ 0& 0& w\end{array}\right)\mid u,v,w\in \mathcal{S}\right\}.\end{array}$$

Then $\mathcal{N}$ is zero-symmetric non abelian left near ring w.r.t. addition and multiplication of matrices. Define mappings $d_1,d_2:\mathcal{N}\to \mathcal{N}$ by

$$d_1\left(\begin{array}{ccc}0& u& v\\ 0& 0& 0\\ 0& 0& w\end{array}\right)=\left(\begin{array}{ccc}0& 0& v\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)and{d}_2\left(\begin{array}{ccc}0& u& v\\ 0& 0& 0\\ 0& 0& w\end{array}\right)=\left(\begin{array}{ccc}0& u& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right).$$

Now consider

$$\mathcal{J}=\left\{\left(\begin{array}{ccc}0& p& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)\mid p\in \mathcal{S}\right\}.$$

It is easy to verify that $d_1$, $d_2$ are nonzero derivations on a non 3-prime near ring $\mathcal{N}$, $\mathcal{J}$ is a nonzero Jordan ideal of $\mathcal{N}$ satisfying $[d_1\left(u\right),d_2\left(k\right)]=[u,k]$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$. However, $\mathcal{N}$ is not commutative.

Theorem 2.

Let $\mathcal{J}$ be a nonzero Jordan ideal of a 2-torsion free 3-prime near ring $\mathcal{N}$. If d is a nonzero derivation on $\mathcal{N}$ satisfying $d\left(\right[k,u\left]\right)=\left[d\right(k),u]$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$, then $\mathcal{N}$ is a commutative ring.

Proof. 

By hypothesis,

$$d\left(\right[k,u\left]\right)=\left[d\right(k),u],\forall k\in \mathcal{J},u\in \mathcal{N}.$$

(6)

Replacing u by $ku$ in (6), we get

$$d\left(k\right)[k,u]+kd\left(\right[k,u\left]\right)=\left[d\right(k),ku].$$

Using the hypothesis, we have

$$d\left(k\right)[k,u]+k\left[d\right(k),u]=\left[d\right(k),ku],$$

which reduces to

$$kd\left(k\right)u=d\left(k\right)uk,\forall k\in \mathcal{J},u\in \mathcal{N}.$$

(7)

Substituting $uw$ for u in (7) and using it again, we find that

$$d\left(k\right)ukw=d\left(k\right)uwk,\forall k\in \mathcal{J},u,w\in \mathcal{N}$$

$$d\left(k\right)u[k,w]=0,\forall k\in \mathcal{J},u,w\in \mathcal{N};$$

i.e., $d\left(k\right)\mathcal{N}[k,w]=\left\{0\right\}$ for all $k\in \mathcal{J}$ and $w\in \mathcal{N}$. As $\mathcal{N}$ is 3-prime, we obtain

$$d\left(k\right)=0\mathrm{or}k\in \mathcal{Z}\left(\mathcal{N}\right),\forall k\in \mathcal{J}.$$

(8)

Assume that there exists $k_0\in \mathcal{J}$ such that $d\left(k_0\right)=0$, then by hypothesis

$$d\left(k_{0}u\right)=d\left(u{k}_0\right),\forall u\in \mathcal{N},$$

which gives

$$k_{0}d\left(u\right)=d\left(u\right)k_0,\forall u\in \mathcal{N}.$$

(9)

Replacing u by $d\left(u\right)w$ in (9) and applying the definition of d, we have

$$k_0\{d^2\left(u\right)w+d\left(u\right)d\left(w\right)\}=\{d^2\left(u\right)w+d\left(u\right)d\left(w\right)\}{k}_0.$$

Applying Lemma 4 and using (9) in above expression, we get

$$k_0{d}^2\left(u\right)w=d^2\left(u\right)w{k}_0,\forall u,w\in \mathcal{N}.$$

(10)

Replacing w by $wx$ in (10) and again using (10), we find that

$$d^2\left(u\right)w[k_0,x]=0,\forall u,w,x\in \mathcal{N},$$

which means that $d^2\left(u\right)\mathcal{N}[k_0,x]=\left\{0\right\}$ for all $u,x\in \mathcal{N}$. By 3-primeness of $\mathcal{N}$ and Lemma 1(i), we conclude that $k_0\in \mathcal{Z}\left(\mathcal{N}\right)$. Therefore in all cases, (8) gives $\mathcal{J}\subseteq \mathcal{Z}\left(\mathcal{N}\right)$, and hence $\mathcal{N}$ is a commutative ring by Lemma 2. □

Theorem 3.

Let $\mathcal{J}$ be a nonzero Jordan ideal of a 2-torsion free 3-prime near ring $\mathcal{N}$. If d is a nonzero derivation on $\mathcal{N}$ satisfying $\left[d\right(u),k]=[u,k]$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$, then either the elements of $\mathcal{J}$ commute under the multiplication of $\mathcal{N}$ or $\mathcal{N}$ is a commutative ring.

Proof. 

By hypothesis,

$$\left[d\right(u),k]=[u,k],\forall k\in \mathcal{J},u\in \mathcal{N}.$$

(11)

Replacing u by $ku$ in (11) and using the hypothesis, we get

$$d\left(ku\right)k-kd\left(ku\right)=k\left[d\right(u),k],\forall k\in \mathcal{J},u\in \mathcal{N},$$

which implies that

$$kd\left(u\right)k+d\left(k\right)uk-kd\left(k\right)u-k^{2}d\left(u\right)=kd\left(u\right)k-k^{2}d\left(u\right).$$

The last expression yields that

$$d\left(k\right)uk=kd\left(k\right)u,\forall k\in \mathcal{J},u\in \mathcal{N}.$$

(12)

Substituting $uw$ for u in (12) and using (12), we find that

$$d\left(k\right)\mathcal{N}[k,w]=\left\{0\right\},\forall k\in \mathcal{J},w\in \mathcal{N}.$$

Since $\mathcal{N}$ is 3-prime, we obtain

$$d\left(k\right)=0\mathrm{or}k\in \mathcal{Z}\left(\mathcal{N}\right),\forall k\in \mathcal{J}.$$

(13)

If $d\left(k\right)=0$ for all $k\in \mathcal{J}$, then replacing k by $(k\circ v)$, we have

$$kd\left(v\right)+d\left(v\right)k=0,\forall k\in \mathcal{J},v\in \mathcal{N}.$$

Putting $v{k}^{\prime }$ instead of v in the previous expression and using it again, we see that

$$d\left(v\right)[k,k^{\prime }]=0,\forall k,k^{\prime }\in \mathcal{J},v\in \mathcal{N}.$$

Applying Lemma 1(ii), we obtain $[k,k^{\prime }]=0$ for all $k,k^{\prime }\in \mathcal{J}$. Therefore, (13) together with Lemma 2 yield that either the elements of $\mathcal{J}$ commute under multiplication of $\mathcal{N}$ or $\mathcal{N}$ is commutative. □

Theorem 4.

Let $\mathcal{J}$ be a nonzero Jordan ideal of a 2-torsion free 3-prime near ring $\mathcal{N}$. If d is a derivation on $\mathcal{N}$ satisfying $d\left(\right[k,u\left]\right)=d\left(k\right)\circ u$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$, then either $d=0$ or the elements of $\mathcal{J}$ commute under the multiplication of $\mathcal{N}$.

Proof. 

By hypothesis,

$$d\left(\right[k,u\left]\right)=d\left(k\right)\circ u,\forall k\in \mathcal{J},u\in \mathcal{N}.$$

(14)

Replacing u by $ku$ in (14), we obtain

$$d\left(k\right)[k,u]+kd\left(\right[k,u\left]\right)=d\left(k\right)\circ ku,$$

which gives

$$d\left(k\right)[k,u]+k\left(d\right(k)\circ u)=d\left(k\right)\circ ku.$$

After solving this expression, we find that

$$kd\left(k\right)u=d\left(k\right)uk,\forall k\in \mathcal{J},u\in \mathcal{N}.$$

(15)

Substituting $uw$ for u in (15), we get

$$kd\left(k\right)uw=d\left(k\right)uwk,\forall k\in \mathcal{J},u,w\in \mathcal{N},$$

which gives $d\left(k\right)\mathcal{N}[k,w]=\left\{0\right\}$ for all $k\in \mathcal{J}$ and $w\in \mathcal{N}$. Since $\mathcal{N}$ is 3-prime, we obtain

$$d\left(k\right)=0\mathrm{or}k\in \mathcal{Z}\left(\mathcal{N}\right),\forall k\in \mathcal{J}.$$

Invoking Lemma 2, the last expression yields that $d\left(k\right)=0$ for all $k\in \mathcal{J}$ or $\mathcal{N}$ is a commutative ring.

If $\mathcal{N}$ is commutative, then our hypothesis becomes $2d\left(k\right)u=0$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$. By 2-torsion freeness of $\mathcal{N}$, we have $d\left(k\right)u=0$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$. Replacing u by $ud\left(k\right)$ and using the fact that $\mathcal{N}$ is 3-prime, we find that $d\left(k\right)=0$ for all $k\in \mathcal{J}$. Therefore in both cases, we arrive at $d\left(k\right)=0$ for all $k\in \mathcal{J}$. Hence, by Lemma 3, we conclude the result. □

In [9], Herstein established that if $\mathcal{R}$ is a prime ring of $char\left(\mathcal{R}\right)\ne 2$ and d is a derivation on $\mathcal{R}$ such that $\left[d\right(u),d(v\left)\right]=0$ for all $u,v\in \mathcal{R}$, then $\mathcal{R}$ is commutative. Motivated by Herstein’s result, Bell and Mason [2] extended the result for near rings as follows: A 2-torsion free 3-prime near ring $\mathcal{N}$ is a commutative ring if it admits a nonzero derivation d satisfying $\left[d\right(v),d(w\left)\right]=0$ for every $v,w\in \mathcal{N}$. Now we prove the following result:

Theorem 5.

Let $\mathcal{J}$ be a nonzero Jordan ideal of a 3-prime near ring $\mathcal{N}$. If d is a nonzero derivation on $\mathcal{N}$ satisfying $\left[d\right(k),d(u\left)\right]=0$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$, then $d\left(\mathcal{J}\right)\subseteq \mathcal{Z}\left(\mathcal{N}\right)$.

Proof. 

By hypothesis,

$$\left[d\right(k),d(u\left)\right]=0,\forall k\in \mathcal{J},u\in \mathcal{N}.$$

(16)

Replacing u by $uw$ in (16), we obtain

$$d\left(k\right)\left\{d\right(u)w+ud(w\left)\right\}=\left\{d\right(u)w+ud(w\left)\right\}d\left(k\right),\forall k\in \mathcal{J},u,w\in \mathcal{N}.$$

Using Lemma 4, we find that

$$d\left(k\right)d\left(u\right)w+d\left(k\right)ud\left(w\right)=d\left(u\right)wd\left(k\right)+ud\left(w\right)d\left(k\right),\forall k\in \mathcal{J},u,w\in \mathcal{N}.$$

Substituting $d\left(u\right)$ in place of u in above expression, we get

$$d\left(k\right)d^2\left(u\right)w+d\left(k\right)d\left(u\right)d\left(w\right)=d^2\left(u\right)wd\left(k\right)+d\left(u\right)d\left(w\right)d\left(k\right),$$

which implies that

$$d\left(k\right)d^2\left(u\right)w=d^2\left(u\right)wd\left(k\right),\forall k\in \mathcal{J},u,w\in \mathcal{N}.$$

(17)

Replacing w by $wx$ in (17) and using it again, we obtain $d^2\left(u\right)w[d\left(k\right),x]=0$ for all $k\in \mathcal{J}$ and $u,w,x\in \mathcal{N}$. In view of 3-primeness of $\mathcal{N}$ together with Lemma 1(i), we obtain $d\left(\mathcal{J}\right)\subseteq \mathcal{Z}\left(\mathcal{N}\right)$, which completes the proof. □

The following example shows the necessity of $\mathcal{N}$ to be 3-prime in the hypothesis in Theorems 2–5.

Example 2.

Suppose that $\mathcal{S}$ is a zero-symmetric non abelian left near ring. Let

$$\begin{array}{c}\hfill \mathcal{N}=\left\{\left(\begin{array}{ccc}0& u& v\\ 0& 0& w\\ 0& 0& 0\end{array}\right)\mid u,v,w\in \mathcal{S}\right\}and\mathcal{J}=\left\{\left(\begin{array}{ccc}0& 0& q\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)\mid q\in \mathcal{S}\right\}.\end{array}$$

It is easy to verify that $\mathcal{N}$ is a non 3-prime, zero-symmetric, non abelian, left near ring w.r.t. addition and multiplication of matrices, and $\mathcal{J}$ is a nonzero Jordan ideal of $\mathcal{N}$. Define $d:\mathcal{N}\to \mathcal{N}$ by $d\left(\begin{array}{ccc}0& u& v\\ 0& 0& w\\ 0& 0& 0\end{array}\right)=\left(\begin{array}{ccc}0& u& v\\ 0& 0& 0\\ 0& 0& 0\end{array}\right).$ It can be easily seen that d is a nonzero derivation on $\mathcal{N}$ satisfying

(i)

$d\left(\right[k,u\left]\right)=\left[d\right(k),u]$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$;

(ii)

$\left[d\right(u),k]=[u,k]$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$;

(iii)

$\left[d\right(k),d(u\left)\right]=0$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$;

(iv)

$d\left(\right[k,u\left]\right)=d\left(k\right)\circ u$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$.

However, $\mathcal{N}$ is not commutative.

3.2. Commutativity Conditions Involving Multiplicative Derivation

Recently, Bedir and Gölbaşi [25] proved that a 3-prime near ring $\mathcal{N}$ with multiplicative derivation d is commutative if one of the following holds: (i) $d\left(vw\right)=d\left(v\right)d\left(w\right)$, (ii) $d\left(vw\right)=d\left(w\right)d\left(v\right)$, (iii) $d\left(\right[v,w\left]\right)=\left[d\right(v),w]$, or (iv) $\left[d\right(v),w]=\left[d\right(v),d(w\left)\right]$, (v) $\left[d\right(v),w]\in \mathcal{Z}\left(\mathcal{N}\right)$ for every $v,w\in \mathcal{N}$. More recently, Mamouni et al. [26] proved that a 2-torsion free prime ring $\mathcal{R}$ equipped with a generalized derivation $\mathcal{F}$ is commutative if any one of the following holds: (i) $\mathcal{F}(v\circ w)\in \mathcal{Z}\left(\mathcal{R}\right)$, (ii) $\mathcal{F}(v\circ w)-(v\circ w)\in \mathcal{Z}\left(\mathcal{R}\right)$, or (iii) $\mathcal{F}(v\circ w)+(v\circ w)\in \mathcal{Z}\left(\mathcal{R}\right)$ for every $v,w\in \mathcal{J}$, a Jordan ideal of $\mathcal{R}$. Further, they obtained the same results for $\mathcal{R}$ to be a *-prime ring and $\mathcal{J}$ to be a *-Jordan ideal.

In this line of investigation, we obtain the commutativity of a 3-prime near ring $\mathcal{N}$ with a multiplicative derivation d in case of a Jordan ideal of $\mathcal{N}$ satisfying one of the following: (i) $\left[d\right(u),k]=0$, (ii) $d\left(\right[u,k\left]\right)=[u,k]$, (iii) $d\left(\right[u,k\left]\right)=0$, or (iv) $d\left(\right[u,k\left]\right)=(u\circ k)$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$, where $\mathcal{J}$ is a Jordan ideal of $\mathcal{N}$.

Theorem 6.

Let $\mathcal{J}$ be a nonzero Jordan ideal of a 2-torsion free 3-prime near ring $\mathcal{N}$. Then $\mathcal{N}$ admits no multiplicative derivation d satisfying $d\left(\right[u,k\left]\right)=(u\circ k)$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$.

Proof. 

By hypothesis,

$$d\left(\right[u,k\left]\right)=u\circ k,\forall k\in \mathcal{J},u\in \mathcal{N}.$$

(18)

Replacing u by $ku$ in (18), we get

$$d\left(k\right)[u,k]+kd\left(\right[u,k\left]\right)=k(u\circ k),\forall k\in \mathcal{J},u\in \mathcal{N}$$

$$d\left(k\right)uk=d\left(k\right)ku,\forall k\in \mathcal{J},u\in \mathcal{N}.$$

(19)

Substituting $uw$ for u in (19) and using (19) again, we find that $d\left(k\right)\mathcal{N}[k,w]=\left\{0\right\}$ for all $k\in \mathcal{N}$ and $w\in \mathcal{N}$. Since $\mathcal{N}$ is 3-prime, we have

$$d\left(k\right)=0\mathrm{or}k\in \mathcal{Z}\left(\mathcal{N}\right),\forall k\in \mathcal{J}.$$

(20)

Assume that $d\left(k_0\right)=0$ for all $k_0\in \mathcal{J}$ and since $d\left([k_0,k_0]\right)=k_0\circ k_0$ together with 2-torsion freeness gives $k_0^2=0$ for all $k_0\in \mathcal{J}$. Now replacing u by $k_{0}u$ in our hypothesis, we have

$$\begin{array}{cc}\hfill d\left([k_{0}u,k_0]\right)& =k_{0}u\circ k_0\hfill \\ \hfill d(k_{0}u{k}_0-k_0^{2}u)& =k_{0}u{k}_0+k_0^{2}u,\hfill \end{array}$$

which reduces to

$$k_{0}d\left(u\right)k_0=k_{0}u{k}_0,\forall u\in \mathcal{N}.$$

(21)

Replacing u by $u{k}_{0}v$ in (21), we arrive at

$$\begin{array}{cc}\hfill k_0\{d\left(u\right)k_{0}v+ud\left(k_{0}v\right)\}{k}_0& =k_{0}u{k}_{0}v{k}_0\hfill \\ \hfill k_{0}d\left(u\right)k_{0}v{k}_0+k_{0}u\{d\left(k_0\right)v+k_{0}d\left(v\right)\}{k}_0& =k_{0}u{k}_{0}v{k}_0\hfill \\ \hfill k_{0}d\left(u\right)k_{0}v{k}_0+k_{0}u{k}_{0}d\left(v\right)k_0& =k_{0}u{k}_{0}v{k}_0.\hfill \end{array}$$

Using (21), we find that $k_{0}u{k}_{0}v{k}_0=0$ for all $k_0\in \mathcal{J}$ and $u,v\in \mathcal{N}$. By 3-primeness of $\mathcal{N}$, we get $k_0=0$ for all $k_0\in \mathcal{J}$, a contradiction. Therefore, (20) yields that $k\in \mathcal{Z}\left(\mathcal{N}\right)$ for all $k\in \mathcal{J}$ and by an application of Lemma 2, we get that $\mathcal{N}$ is commutative. Hence, our hypothesis becomes $2uk=0$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$. Since $\mathcal{N}$ is 2-torsion free, we obtain $uk=0$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$. Putting $ku$ in place of u and since $\mathcal{N}$ is 3-prime, we find that $k=0$ for all $k\in \mathcal{J}$, a contradiction. □

Theorem 7.

Let $\mathcal{J}$ be a nonzero Jordan ideal of a 2-torsion free 3-prime near ring $\mathcal{N}$. If d is a multiplicative derivation on $\mathcal{N}$ satisfying one of the following conditions:

(i)

$\left[d\right(u),k]=0$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$;

(ii)

$d\left(\right[u,k\left]\right)=[u,k]$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$;

(iii)

$d\left(\right[u,k\left]\right)=0$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$,

then either $d=0$ on $\mathcal{J}$ or $\mathcal{N}$ is a commutative ring.

Proof. 

$\left(i\right)$ Suppose that

$$\left[d\right(u),k]=0,\forall k\in \mathcal{J},u\in \mathcal{N}.$$

(22)

Replacing by u by $ku$ in (22), we get

$$\left(d\right(k)u+kd(u\left)\right)k=k\left(d\right(k)u+kd(u\left)\right),\forall k\in \mathcal{J},u\in \mathcal{N}.$$

Applying Lemma 4 and (22), we obtain

$$d\left(k\right)uk=kd\left(k\right)u,\forall k\in \mathcal{J},u\in \mathcal{N}.$$

(23)

Putting $uw$ instead of u in (23) and using (23) again, we get $d\left(k\right)\mathcal{N}[k,w]=\left\{0\right\}$ for all $k\in \mathcal{J}$ and $w\in \mathcal{N}$. As $\mathcal{N}$ is 3-prime, we have either $d\left(k\right)=0$ or $k\in \mathcal{Z}\left(\mathcal{N}\right)$ for all $k\in \mathcal{J}$. Hence, by Lemma 2, the last expression gives either $d\left(k\right)=0$ for all $k\in \mathcal{J}$ or $\mathcal{N}$ is a commutative ring.

$\left(ii\right)$ By hypothesis,

$$d\left(\right[u,k\left]\right)=[u,k],\forall k\in \mathcal{J},u\in \mathcal{N}.$$

(24)

Replacing u by $ku$ in (24), we obtain

$$d\left(k\right)[u,k]+kd\left(\right[u,k\left]\right)=k[u,k],\forall k\in \mathcal{J},u\in \mathcal{N}.$$

By our hypothesis, we have

$$d\left(k\right)uk=d\left(k\right)ku,\forall k\in \mathcal{J},u\in \mathcal{N}.$$

(25)

Now substituting $uw$ in place of u in (25) and using (25) again, we get $d\left(k\right)\mathcal{N}[k,w]=\left\{0\right\}$ for all $k\in \mathcal{J}$ and $w\in \mathcal{N}$. Invoking Lemma 2 together with the 3-primeness of $\mathcal{N}$, we conclude the result.□

$\left(iii\right)$ By hypothesis,

$$d\left(\right[u,k\left]\right)=0,\forall k\in \mathcal{J},u\in \mathcal{N}.$$

(26)

Replacing u by $ku$ in (26), we find that

$$d\left(k\right)[u,k]+kd\left(\right[u,k\left]\right)=0,\forall k\in \mathcal{J},u\in \mathcal{N},$$

which reduces to

$$d\left(k\right)uk=d\left(k\right)ku,\forall k\in \mathcal{J},u\in \mathcal{N}.$$

(27)

Since (27) is same as (25), then arguing in the similar manner as above, we obtain the result. □

Theorem 8.

Let $\mathcal{J}$ be a nonzero Jordan ideal of a 3-prime near ring $\mathcal{N}$. If d is a multiplicative derivation on $\mathcal{N}$ satisfying $d\left(u\right)d\left(k\right)=(k\circ u)$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$, then $\mathcal{J}\subseteq \mathcal{Z}\left(\mathcal{N}\right)$.

Proof. 

By hypothesis,

$$d\left(u\right)d\left(k\right)=(k\circ u),\forall k\in \mathcal{J},u\in \mathcal{N}.$$

(28)

Replacing u by $ku$ in (28) and using $k\circ ku=k(k\circ u)$, we get

$$\left(d\right(k)u+kd(u\left)\right)d\left(k\right)=k(k\circ u),\forall k\in \mathcal{J},u\in \mathcal{N}.$$

Applying Lemma 4 and (28), we obtain

$$d\left(k\right)ud\left(k\right)=0,\forall k\in \mathcal{J},u\in \mathcal{N}.$$

Since $\mathcal{N}$ is 3-prime, we find that $d\left(k\right)=0$ for all $k\in \mathcal{J}$. Therefore, our hypothesis gives $ku=-uk$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$. Putting $uv$ instead of u, we have $u[k,v]=0$ for all $k\in \mathcal{J}$ and $u,v\in \mathcal{N}$. Replacing u by $[k,v]u$ in the previous expression and using the 3-primeness of $\mathcal{N}$, we obtain our result. □

Now we discuss an example which demonstrates that the 3-prime condition in Theorems 6–8 is essential.

Example 3.

Suppose that $\mathcal{S}$ is a zero-symmetric non abelian left near ring. Consider

$$\begin{array}{c}\hfill \mathcal{N}=\left\{\left(\begin{array}{ccc}0& u& v\\ 0& 0& 0\\ 0& w& 0\end{array}\right)\mid u,v,w\in \mathcal{S}\right\}and\mathcal{J}=\left\{\left(\begin{array}{ccc}0& p& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)\mid p\in \mathcal{S}\right\}.\end{array}$$

It is easy to check that $\mathcal{N}$ is a non 3-prime, zero-symmetric, non abelian, left near ring w.r.t. addition and multiplication of matrices and $\mathcal{J}\ne \left\{0\right\}$, a Jordan ideal of $\mathcal{N}$. Define $d:\mathcal{N}\to \mathcal{N}$ by $d\left(\begin{array}{ccc}0& u& v\\ 0& 0& 0\\ 0& w& 0\end{array}\right)=\left(\begin{array}{ccc}0& uw& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right).$ It is easy to see that d is a nonzero multiplicative derivation on $\mathcal{N}$ satisfying

(i)

$d\left(\right[u,k\left]\right)=u\circ k$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$;

(ii)

$\left[d\right(u),k]=0$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$;

(iii)

$d\left(\right[u,k\left]\right)=[u,k]$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$;

(iv)

$d\left(\right[u,k\left]\right)=0$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$;

(v)

$d\left(u\right)d\left(k\right)=k\circ u$ for all $k\in \mathcal{J}$ and $u\in \mathcal{N}$.

However, $\mathcal{N}$ is not commutative.

For a near ring $\mathcal{N}$—a graph in which vertices are the elements of $\mathcal{N}$, and for any two vertices $u,v$, we have $u\mathcal{N}v=\left\{0\right\}$ or $v\mathcal{N}u=\left\{0\right\}$ if u and v are adjacent—it is known as the prime graph ([27]) of $\mathcal{N}$ and is represented by $\mathcal{G}\left(\mathcal{N}\right)$. Easily, we observe that $\mathcal{G}\left(\mathcal{N}\right)$ is a star graph if $\mathcal{N}$ is prime. For a commutative ring $\mathcal{R}$, a graph in which the vertices are the set of nonzero zero-divisors of $\mathcal{R}$ and for any two vertices $u,v$, we have $u\ne v$, and $uv=0$ if u and v are joined by an edge, is known as the zero-divisor graph of $\mathcal{R}$. We have the following corollary:

Corollary 1.

Suppose that $\mathcal{N}$ is a 3-prime near ring and its prime graph $\mathcal{G}\left(\mathcal{N}\right)$ is star. If d is a derivation on $\mathcal{N}$ satisfying any one of the following: (i) $d\left(\right[u,v\left]\right)=\left[d\right(u),v]$, (ii) $\left[d\right(u),v]=[u,v]$, (iii) $\left[d\right(u),v]=0$, (iv) $d\left(\right[u,v\left]\right)=[u,v]$, or (v) $d\left(\right[u,v\left]\right)=0$ for all $u,v\in \mathcal{N}$, then the zero-divisor graph of $\mathcal{N}$ is a subgraph of $\mathcal{G}\left(\mathcal{N}\right)$.

4. Discussion

Near rings are generalized rings, since addition is not commutative and the most important fact is only one distributive law is needed. Upon comparing with the standard class of rings, endomorphism rings of abelian groups, we can see that ring theory describes a "linear theory of group mappings," while near rings deal the general "nonlinear theory." A great number of linear results have been transferred to the general nonlinear case with some suitable changes. In the present manuscript, we have generalized the results which have been established for "abelian group mappings" to "non-abelian group mappings." The results of near rings can be used in various fields inside and outside of pure mathematics. We can construct efficient codes and block designs with the help of finite near rings. Inside mathematics, there are several applications of near ring theory in functional analysis, algebraic topology, and category theory, and outside mathematics, there are applications in digital computing, automata theory, sequential mechanics, and combinatorics (see [28] and the references therein).

5. Conclusions

In future research, one can discuss the following issues: (i) Theorems 1–8 can be proven by replacing derivation d by a generalized derivation (or multiplicative generalized derivation), keeping more constraints on derivations. (ii) The commutativity of semiprime near rings is another interesting work for the future.