Lie Ideals and Homoderivations in Semiprime Rings

Abstract

Let S be a 2-torsion free semiprime ring and U be a noncentral square-closed Lie ideal of $S.$ An additive mapping ℏ on S is defined as a homoderivation if $\hslash \left(ab\right)=\hslash \left(a\right)\hslash \left(b\right)+\hslash \left(a\right)b+a\hslash \left(a\right)$ for all $a,b\in S.$ In the present paper, we shall prove that ℏ is a commuting map on U if any one of the following holds: (i) $\hslash \left({\tilde{a}}_1{\tilde{a}}_2\right)+{\tilde{a}}_1{\tilde{a}}_2\in Z,$(ii) $\hslash \left({\tilde{a}}_1{\tilde{a}}_2\right)-{\tilde{a}}_1{\tilde{a}}_2\in Z,$(iii) $\hslash \left({\tilde{a}}_1\circ {\tilde{a}}_2\right)=0,$(iv) $\hslash \left({\tilde{a}}_1\circ {\tilde{a}}_2\right)=\left[{\tilde{a}}_1,{\tilde{a}}_2\right],$(v) $\hslash \left(\left[{\tilde{a}}_1,{\tilde{a}}_2\right]\right)=0,$(vi) $\hslash \left(\left[{\tilde{a}}_1,{\tilde{a}}_2\right]\right)=$ $({\tilde{a}}_1\circ {\tilde{a}}_2),$(vii) ${\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)\pm {\tilde{a}}_1{\tilde{a}}_2\in Z,$(viii) ${\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)\pm {\tilde{a}}_2{\tilde{a}}_1=0,$(ix) ${\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)\pm {\tilde{a}}_1\circ {\tilde{a}}_2=0,$(x) $[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_2]\pm {\tilde{a}}_1{\tilde{a}}_2=0,$(xi) $[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_2]\pm {\tilde{a}}_2{\tilde{a}}_1=0,$ for all ${\tilde{a}}_1,{\tilde{a}}_2\in U,$ where ℏ is a homoderivation on $S.$

1. Introduction

The symbol S used throughout this article denotes a ring with center Z. A ring S is called prime if $IJ\ne 0$ for any two nonzero ideals $I,J\subseteq S$, and semiprime if it contains no nonzero ideals whose square is zero. A proper ideal P of S is said to be prime if for any ${\tilde{a}}_1,{\tilde{a}}_2\in S,{\tilde{a}}_{1}S{\tilde{a}}_2\subseteq P$ implies that ${\tilde{a}}_1\in P$ or ${\tilde{a}}_2\in P.$ In other words, the ring S is prime ring if and only if $\left(0\right)$ is a prime ideal of $S,$ or equivalently, a ring S is prime if for ${\tilde{a}}_1,{\tilde{a}}_2\in S,$${\tilde{a}}_{1}S{\tilde{a}}_2=\left(0\right)$ implies either ${\tilde{a}}_1=0$ or ${\tilde{a}}_2=0.$ Recall that a proper ideal P of S is said to be semiprime if for any ${\tilde{a}}_1\in S,{\tilde{a}}_{1}S{\tilde{a}}_1\subseteq P$ implies that ${\tilde{a}}_1\in P$ and the ring S is a semiprime ring if $P=\left(0\right)$ is a semiprime ideal of $S.$ Every prime ideal is a semiprime ideal but the converse is not true in general. An additive subgroup U of S is called a Lie ideal of S if $[{\tilde{a}}_1,r]\in U$, for all ${\tilde{a}}_1\in U,$$r\in S$. U is called a square-closed Lie ideal of S if U is a Lie ideal and ${\tilde{a}}_1^2\in U$ for all ${\tilde{a}}_1\in U$. If U is a square-closed Lie ideal of $S,$ then we have ${({\tilde{a}}_1+{\tilde{a}}_2)}^2\in U$ and so ${({\tilde{a}}_1+{\tilde{a}}_2)}^2-{\tilde{a}}_1^2-{\tilde{a}}_2^2={\tilde{a}}_1{\tilde{a}}_2+{\tilde{a}}_2{\tilde{a}}_1\in U$ for all ${\tilde{a}}_1,{\tilde{a}}_2\in U$. Hence, we find $2{\tilde{a}}_1{\tilde{a}}_2\in U,$ for all ${\tilde{a}}_1,{\tilde{a}}_2\in U.$

An additive mapping $d:S\to S$ is called a derivation if $d\left({\tilde{a}}_1{\tilde{a}}_2\right)=d\left({\tilde{a}}_1\right){\tilde{a}}_2+{\tilde{a}}_{1}d\left({\tilde{a}}_2\right)$ holds for all ${\tilde{a}}_1,{\tilde{a}}_2\in S.$ Research on the subject of derivations in prime rings was initiated by E. C. Posner [1]. Regarding commuting mappings, Posner’s theorem, as discussed in this paper, is a crucial finding in the investigation of these mappings. It states that a prime ring possessing a nonzero commuting d mapping must be commutative. This theorem marks an important initial outcome in the study of commuting mappings. Afterwards, many researchers studied commutativity theorems for prime or semiprime rings admitting automorphisms or derivations on suitable subsets. A mapping $f:S\to S$ is called centralizing if $[f\left({\tilde{a}}_1\right),{\tilde{a}}_1]\in Z\left(S\right)$ holds for all ${\tilde{a}}_1\in S$; in the special case where $[f\left({\tilde{a}}_1\right),{\tilde{a}}_1]=0$ holds for all ${\tilde{a}}_1\in S$, the mapping f is said to be commuting. In 2000, El Soufi [2] defined a homoderivation on S as an additive mapping $\hslash :S\to S$ satisfying $\hslash \left({\tilde{a}}_1{\tilde{a}}_2\right)=\hslash \left({\tilde{a}}_1\right)\hslash \left({\tilde{a}}_2\right)+\hslash \left({\tilde{a}}_1\right){\tilde{a}}_2+{\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)$ for all ${\tilde{a}}_1,{\tilde{a}}_2\in S$. An example of such mapping is to let $\hslash \left({\tilde{a}}_1\right)=f\left({\tilde{a}}_1\right)-{\tilde{a}}_1,$ for all ${\tilde{a}}_1,{\tilde{a}}_2\in S$ where f is an endomorphism on S. It is clear that a homoderivation ℏ is also a derivation if $\hslash \left({\tilde{a}}_1\right)\hslash \left({\tilde{a}}_2\right)=0$ for all ${\tilde{a}}_1,{\tilde{a}}_2\in S$. In this case, $\hslash \left({\tilde{a}}_1\right)S\hslash \left({\tilde{a}}_2\right)=0$ for all ${\tilde{a}}_1,{\tilde{a}}_2\in S$. Therefore, if S is a prime ring, then the only additive mapping which is both a derivation and a homoderivation is the zero mapping. Another example of homoderivations is when the additive mapping $\hslash :S\to S$ defined by $\hslash \left(x\right)=-x$ is a homoderivation of S.

In Daif and Bell, Let S be semiprime ring, d is nonzero derivation of S and I is nonzero ideal of S. S contains a nonzero central ideal if one of the following conditions is provide; (i) $d\left([{\tilde{a}}_1,{\tilde{a}}_2]\right)=[{\tilde{a}}_1,{\tilde{a}}_2]$ (ii) $d\left([{\tilde{a}}_1,{\tilde{a}}_2]\right)=-[{\tilde{a}}_1,{\tilde{a}}_2]$ for all ${\tilde{a}}_1,{\tilde{a}}_2\in I$ has been proven in [3]. In [4], Quadri, Khan, and Rehman examined the following conditions for generalized derivation: (i) $f\left([{\tilde{a}}_1,{\tilde{a}}_2]\right)=[{\tilde{a}}_1,{\tilde{a}}_2],$ (ii) $f\left([{\tilde{a}}_1,{\tilde{a}}_2]\right)=-[{\tilde{a}}_1,{\tilde{a}}_2],$ (iii) $f\left({\tilde{a}}_{1}o{\tilde{a}}_2\right)={\tilde{a}}_{1}o{\tilde{a}}_2,$ (iv) $f\left({\tilde{a}}_{1}o{\tilde{a}}_2\right)=-\left({\tilde{a}}_{1}o{\tilde{a}}_2\right)$. The conditions discussed above by various authors have been studied by many authors in recent years for different structures and derivation. For more details, see the references [5,6,7,8,9].

In [2], El Soufi also proved the commutativity of prime rings, admitting a homoderivation ℏ that satisfies the condition $\hslash \left([{\tilde{a}}_1,{\tilde{a}}_2]\right)=\pm [{\tilde{a}}_1,{\tilde{a}}_2]$ for all ${\tilde{a}}_1,{\tilde{a}}_2\in I,$ with a nonzero two-sided ideal of S. Following this line of investigation, several authors studied homoderivations acting on appropriate subsets of the prime ring and semiprime rings. In [10], Asmaa Melaibari et al. studied the commutativity of rings admitting a homoderivation ℏ such that $\hslash \left([{\tilde{a}}_1,{\tilde{a}}_2]\right)=0$ for all ${\tilde{a}}_1,{\tilde{a}}_2\in U,$ where U is a nonzero ideal of S.

On the other hand, Ashraf and Rehman showed that if S is a prime ring with a nonzero ideal U of S and d is a derivation of S such that $d\left({\tilde{a}}_1{\tilde{a}}_2\right)\pm {\tilde{a}}_1{\tilde{a}}_2\in Z$, $d\left({\tilde{a}}_1{\tilde{a}}_2\right)\pm {\tilde{a}}_2{\tilde{a}}_1\in Z,$ for all ${\tilde{a}}_1,{\tilde{a}}_2\in U,$ then R is commutative in [11]. Alharfie and Muthana proved similar results regarding homoderivations in [12].

In light of all these results, our aim in this article is to explore a more general context of differential identities involving a Lie ideal of the semiprime ring with homoderivation. Some conditions about commutativity on Lie ideals of prime rings with homoderivation were proved in [13]. Futhermore, in Section 4 of the paper [14], it was shown that the study of differential identities on algebras plays a crucial role. This approach provides us with the opportunity to generalize the results obtained earlier.

2. Results

The symbol S throughout this article will denote a $2-$torsion free semiprime ring. In addition, U will symbolize a noncentral square-closed Lie ideal of S and ℏ will symbolize a homoderivation of S. In addition, for any ${\tilde{a}}_1,{\tilde{a}}_2\in S,$$[{\tilde{a}}_1,{\tilde{a}}_2]={\tilde{a}}_1{\tilde{a}}_2-{\tilde{a}}_2{\tilde{a}}_1$ and ${\tilde{a}}_{1}o{\tilde{a}}_2={\tilde{a}}_1{\tilde{a}}_2+{\tilde{a}}_2{\tilde{a}}_1$ will denote the well-known Lie and Jordan product, respectively.

We will make use of the following fundamental identities that apply to every ${\tilde{a}}_1,{\tilde{a}}_2,{\tilde{a}}_3\in S$ without explicitly mentioning them:

$$\begin{gathered} [{\tilde{a}}_1,{\tilde{a}}_2{\tilde{a}}_3]={\tilde{a}}_2[{\tilde{a}}_1,{\tilde{a}}_3]+[{\tilde{a}}_1,{\tilde{a}}_2]{\tilde{a}}_3 \\ [{\tilde{a}}_1{\tilde{a}}_2,{\tilde{a}}_3]=[{\tilde{a}}_1,{\tilde{a}}_3]{\tilde{a}}_2+{\tilde{a}}_1[{\tilde{a}}_2,{\tilde{a}}_3] \\ {\tilde{a}}_{1}o\left({\tilde{a}}_2{\tilde{a}}_3\right)=\left({\tilde{a}}_{1}o{\tilde{a}}_2\right){\tilde{a}}_3-{\tilde{a}}_2[{\tilde{a}}_1,{\tilde{a}}_3]={\tilde{a}}_2\left({\tilde{a}}_{1}o{\tilde{a}}_3\right)+[{\tilde{a}}_1,{\tilde{a}}_2]{\tilde{a}}_3 \\ \left({\tilde{a}}_1{\tilde{a}}_2\right)o{\tilde{a}}_3={\tilde{a}}_1\left({\tilde{a}}_{2}o{\tilde{a}}_3\right)-[{\tilde{a}}_1,{\tilde{a}}_3]{\tilde{a}}_2=\left({\tilde{a}}_{1}o{\tilde{a}}_3\right){\tilde{a}}_2+{\tilde{a}}_1[{\tilde{a}}_2,{\tilde{a}}_3]. \end{gathered}$$

Remark 1.

For all ${\tilde{a}}_1,{\tilde{a}}_2\in S,$ we obtain

$$\begin{array}{ccc}\hfill \hslash \left([{\tilde{a}}_1,{\tilde{a}}_2]\right)& =& \hslash ({\tilde{a}}_1{\tilde{a}}_2-{\tilde{a}}_2{\tilde{a}}_1)=\hslash \left({\tilde{a}}_1{\tilde{a}}_2\right)-\hslash \left({\tilde{a}}_2{\tilde{a}}_1\right)\hfill \\ & =& \hslash \left({\tilde{a}}_1\right)\hslash \left({\tilde{a}}_2\right)+\hslash \left({\tilde{a}}_1\right){\tilde{a}}_2+{\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)-\hslash \left({\tilde{a}}_2\right)\hslash \left({\tilde{a}}_1\right)-\hslash \left({\tilde{a}}_2\right){\tilde{a}}_1-{\tilde{a}}_2\hslash \left({\tilde{a}}_1\right)\hfill \\ & =& [\hslash \left({\tilde{a}}_1\right),\hslash \left({\tilde{a}}_2\right)]+[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_2]+[{\tilde{a}}_1,\hslash \left({\tilde{a}}_2\right)].\hfill \end{array}$$

Lemma 1

([15], Corollary 2.1). Let R be a $2-$torsion free semiprime ring, U a Lie ideal of R such that $U⊈Z\left(R\right)$ and $a,b\in U.$

(i)

If $aUa=0$, then $a=0$.

(ii)

If $aU=0$ ( or $Ua=0$), then $a=0.$

(iii)

If U is square-closed and $aUb=0$, then $ab=0$ and $ba=0.$

Theorem 1.

Let S be a $2-$torsion free semiprime ring, U a noncentral square-closed Lie ideal of S, and ℏ a homoderivation which is zero-power valued on U. If ℏ is the centralizing map on $U,$ then ℏ is the commuting map on $U.$

Proof. 

By the hypothesis, we have

$$[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]\in Z,\mathrm{for}\mathrm{all}{\tilde{a}}_1\in U.$$

(1)

Writing ${\tilde{a}}_1$ by $2{\tilde{a}}_1^2$ in the last equation, we have

$$\begin{array}{ccc}\hfill Z& \ni & 4[{\tilde{a}}_1^2,\hslash \left({\tilde{a}}_1^2\right)]=4[{\tilde{a}}_1^2,\hslash \left({\tilde{a}}_1\right)\hslash \left({\tilde{a}}_1\right)+{\tilde{a}}_1\hslash \left({\tilde{a}}_1\right)+\hslash \left({\tilde{a}}_1\right){\tilde{a}}_1]\hfill \\ & =& 4[{\tilde{a}}_1^2,\hslash \left({\tilde{a}}_1\right)\hslash \left({\tilde{a}}_1\right)+{\tilde{a}}_1\hslash \left({\tilde{a}}_1\right)+\hslash \left({\tilde{a}}_1\right){\tilde{a}}_1-{\tilde{a}}_1\hslash \left({\tilde{a}}_1\right)+{\tilde{a}}_1\hslash \left({\tilde{a}}_1\right)]\hfill \\ & =& 4[{\tilde{a}}_1^2,\hslash \left({\tilde{a}}_1\right)\hslash \left({\tilde{a}}_1\right)+2{\tilde{a}}_1\hslash \left({\tilde{a}}_1\right)-[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]]\hfill \end{array}$$

Since S is 2-torsion free, we obtain

$$[{\tilde{a}}_1^2,\hslash \left({\tilde{a}}_1\right)\hslash \left({\tilde{a}}_1\right)+2{\tilde{a}}_1\hslash \left({\tilde{a}}_1\right)-[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]]\in Z.$$

By the hypothesis, we have

$$[{\tilde{a}}_1^2,\hslash \left({\tilde{a}}_1^2\right)]\in Z.$$

Expanding this equation and using $[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]\in Z,$ we obtain

$$\begin{array}{ccc}\hfill [{\tilde{a}}_1^2,\hslash \left({\tilde{a}}_1^2\right)]& =& [{\tilde{a}}_1^2,\hslash \left({\tilde{a}}_1\right)\hslash \left({\tilde{a}}_1\right)+2{\tilde{a}}_1\hslash \left({\tilde{a}}_1\right)]={\tilde{a}}_1[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)\hslash \left({\tilde{a}}_1\right)]+[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)\hslash \left({\tilde{a}}_1\right)]{\tilde{a}}_1\hfill \\ & & +{\tilde{a}}_1[{\tilde{a}}_1,2{\tilde{a}}_1\hslash \left({\tilde{a}}_1\right)]+[{\tilde{a}}_1,2{\tilde{a}}_1\hslash \left({\tilde{a}}_1\right)]{\tilde{a}}_1\hfill \\ & =& {\tilde{a}}_1\hslash \left({\tilde{a}}_1\right)[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]+{\tilde{a}}_1[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]\hslash \left({\tilde{a}}_1\right)+\hslash \left({\tilde{a}}_1\right)[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]{\tilde{a}}_1\hfill \\ & & +[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]\hslash \left({\tilde{a}}_1\right){\tilde{a}}_1+2{\tilde{a}}_1^2[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]+2{\tilde{a}}_1[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]{\tilde{a}}_1\hfill \\ & =& 2{\tilde{a}}_1\hslash \left({\tilde{a}}_1\right)[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]+2\hslash \left({\tilde{a}}_1\right){\tilde{a}}_1[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]+4{\tilde{a}}_1^2[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]\hfill \\ & =& 2{\tilde{a}}_1({\tilde{a}}_1+\hslash \left({\tilde{a}}_1\right))[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]+2(\hslash \left({\tilde{a}}_1\right)+{\tilde{a}}_1){\tilde{a}}_1[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]\hfill \\ & =& (2{\tilde{a}}_1({\tilde{a}}_1+\hslash \left({\tilde{a}}_1\right))+2(\hslash \left({\tilde{a}}_1\right)+{\tilde{a}}_1){\tilde{a}}_1)[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]\hfill \\ & =& 2(2{\tilde{a}}_1^2+\hslash \left({\tilde{a}}_1\right){\tilde{a}}_1+{\tilde{a}}_1\hslash \left({\tilde{a}}_1\right))[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]\hfill \end{array}$$

Since S is $2-$torsion free, we have

$$(2{\tilde{a}}_1^2+\hslash \left({\tilde{a}}_1\right){\tilde{a}}_1+{\tilde{a}}_1\hslash \left({\tilde{a}}_1\right))[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]\in Z.$$

Commuting this term with ${\tilde{a}}_1,$ we find that

$$[(2{\tilde{a}}_1^2+\hslash \left({\tilde{a}}_1\right){\tilde{a}}_1+{\tilde{a}}_1\hslash \left({\tilde{a}}_1\right))[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)],{\tilde{a}}_1]=0.$$

Using the hypothesis, we obtain

$$[2{\tilde{a}}_1^2+\hslash \left({\tilde{a}}_1\right){\tilde{a}}_1+{\tilde{a}}_1\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_1][{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]=0$$

and so

$$([\hslash \left({\tilde{a}}_1\right){\tilde{a}}_1,{\tilde{a}}_1]+[{\tilde{a}}_1\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_1])=2{\tilde{a}}_1[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_1][{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]=0.$$

Since S is $2-$torsion free, we have

$${\tilde{a}}_1{[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_1]}^2=0.$$

(2)

Multiplying Equation (2) from the left side by $\hslash \left({\tilde{a}}_1\right)$, we find that

$$\hslash \left({\tilde{a}}_1\right){\tilde{a}}_1{[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_1]}^2=0.$$

(3)

Multiplying (2) from the left side by $\hslash \left({\tilde{a}}_1\right)$, we obtain

$${\tilde{a}}_1{[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_1]}^2\hslash \left({\tilde{a}}_1\right)=0.$$

Since ℏ is centralizing map on U, we have

$${\tilde{a}}_1\hslash \left({\tilde{a}}_1\right){[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_1]}^2=0.$$

(4)

Comparing (3) and (4), we find that

$${[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]}^3=0.$$

Since the center of a semiprime ring contains no nonzero nilpotent elements, we obtain $[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]=0$ for all ${\tilde{a}}_1\in U$. Hence, we conclude that ℏ is a commuting map on $U.$ □

Theorem 2.

Let S be a $2-$torsion free semiprime ring, U a noncentral square-closed Lie ideal of S, and ℏ a homoderivation which is zero-power valued on U. If any of the following holds for all ${\tilde{a}}_1,{\tilde{a}}_2\in U,$

(i)

$\hslash \left({\tilde{a}}_1{\tilde{a}}_2\right)+{\tilde{a}}_1{\tilde{a}}_2\in Z,$ or

(ii)

$\hslash \left({\tilde{a}}_1{\tilde{a}}_2\right)-{\tilde{a}}_1{\tilde{a}}_2\in Z,$

then ℏ is a commuting map on $U.$

Proof. 

$\left(i\right)$ By the hypothesis, we obtain

$$\hslash \left({\tilde{a}}_1{\tilde{a}}_2\right)+{\tilde{a}}_1{\tilde{a}}_2\in Z\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

That is,

$$\hslash \left({\tilde{a}}_1\right)\hslash \left({\tilde{a}}_2\right)+\hslash \left({\tilde{a}}_1\right){\tilde{a}}_2+{\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)+{\tilde{a}}_1{\tilde{a}}_2\in Z.$$

Substituting ${\tilde{a}}_2$ by $2{\tilde{a}}_2{\tilde{a}}_3,{\tilde{a}}_3\in U$ in the last equation, we obtain

$$2\hslash \left({\tilde{a}}_1\right)\hslash \left({\tilde{a}}_2\right)\hslash \left({\tilde{a}}_3\right)+2\hslash \left({\tilde{a}}_1\right)\hslash \left({\tilde{a}}_2\right){\tilde{a}}_3+2\hslash \left({\tilde{a}}_1\right){\tilde{a}}_2\hslash \left({\tilde{a}}_3\right)+2\hslash \left({\tilde{a}}_1\right){\tilde{a}}_2{\tilde{a}}_3$$

$$+2{\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)\hslash \left({\tilde{a}}_3\right)+2{\tilde{a}}_1\hslash \left({\tilde{a}}_2\right){\tilde{a}}_3+2{\tilde{a}}_1{\tilde{a}}_2\hslash \left({\tilde{a}}_3\right)+2{\tilde{a}}_1{\tilde{a}}_2{\tilde{a}}_3\in Z,$$

and so

$$2\left(\hslash \left({\tilde{a}}_1\right)\hslash \left({\tilde{a}}_2\right)+\hslash \left({\tilde{a}}_1\right){\tilde{a}}_2+{\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)+{\tilde{a}}_1{\tilde{a}}_2\right)\hslash \left({\tilde{a}}_3\right)$$

$$+2\left(\hslash \left({\tilde{a}}_1\right)\hslash \left({\tilde{a}}_2\right)+\hslash \left({\tilde{a}}_1\right){\tilde{a}}_2+{\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)+{\tilde{a}}_1{\tilde{a}}_2\right){\tilde{a}}_3\in Z.$$

Substituting this term with ${\tilde{a}}_3$, we obtain

$$2[\left(\hslash \left({\tilde{a}}_1\right)\hslash \left({\tilde{a}}_2\right)+\hslash \left({\tilde{a}}_1\right){\tilde{a}}_2+{\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)+{\tilde{a}}_1{\tilde{a}}_2\right)\hslash \left({\tilde{a}}_3\right),{\tilde{a}}_3]=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2,{\tilde{a}}_3\in U.$$

That is

$$[\left(\hslash \left(2{\tilde{a}}_1{\tilde{a}}_2\right)+2{\tilde{a}}_1{\tilde{a}}_2\right)\hslash \left({\tilde{a}}_3\right),{\tilde{a}}_3]=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2,{\tilde{a}}_3\in U.$$

Since ℏ is zero-power valued on U, there exists an integer $n>1$ such that ${\hslash }^n\left({\tilde{a}}_1\right)=0$ for all ${\tilde{a}}_1\in U$. Using ${\tilde{a}}_1,{\tilde{a}}_2\in U,$ we obtain that $2{\tilde{a}}_1{\tilde{a}}_2\in U.$ Replacing $2{\tilde{a}}_1{\tilde{a}}_2$ by $2{\tilde{a}}_1{\tilde{a}}_2-2\hslash \left({\tilde{a}}_1{\tilde{a}}_2\right)+2{\hslash }^2\left({\tilde{a}}_1{\tilde{a}}_2\right)+...+2{(-1)}^{n-1}{\hslash }^{n-1}\left({\tilde{a}}_1{\tilde{a}}_2\right)$ in this equation and using S is $2-$torsion free, we find that

$$[{\tilde{a}}_1{\tilde{a}}_2\hslash \left({\tilde{a}}_3\right),{\tilde{a}}_3]=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2,{\tilde{a}}_3\in U.$$

That is

$${\tilde{a}}_1{\tilde{a}}_2[\hslash \left({\tilde{a}}_3\right),{\tilde{a}}_3]+{\tilde{a}}_1[{\tilde{a}}_2,{\tilde{a}}_3]\hslash \left({\tilde{a}}_3\right)+[{\tilde{a}}_1,{\tilde{a}}_3]{\tilde{a}}_2\hslash \left({\tilde{a}}_3\right)=0.$$

(5)

Replacing ${\tilde{a}}_1$ by $2r{\tilde{a}}_1,r\in U$ in this equation, we obtain

$$2r{\tilde{a}}_1{\tilde{a}}_2[\hslash \left({\tilde{a}}_3\right),{\tilde{a}}_3]+2r{\tilde{a}}_1[{\tilde{a}}_2,{\tilde{a}}_3]\hslash \left({\tilde{a}}_3\right)+2r[{\tilde{a}}_1,{\tilde{a}}_3]{\tilde{a}}_2\hslash \left({\tilde{a}}_3\right)+2[r,{\tilde{a}}_3]{\tilde{a}}_1{\tilde{a}}_2\hslash \left({\tilde{a}}_3\right)=0.$$

Since S is $2-$torsion free, we see that

$$r{\tilde{a}}_1{\tilde{a}}_2[\hslash \left({\tilde{a}}_3\right),{\tilde{a}}_3]+r{\tilde{a}}_1[{\tilde{a}}_2,{\tilde{a}}_3]\hslash \left({\tilde{a}}_3\right)+r[{\tilde{a}}_1,{\tilde{a}}_3]{\tilde{a}}_2\hslash \left({\tilde{a}}_3\right)+[r,{\tilde{a}}_3]{\tilde{a}}_1{\tilde{a}}_2\hslash \left({\tilde{a}}_3\right)=0.$$

Using Equation (5), we have

$$[r,{\tilde{a}}_3]{\tilde{a}}_1{\tilde{a}}_2\hslash \left({\tilde{a}}_3\right)=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2,{\tilde{a}}_3,r\in U.$$

Replacing ${\tilde{a}}_2$ by $4{\tilde{a}}_2[r,{\tilde{a}}_3]{\tilde{a}}_1$ and using S is $2-$torsion free, we obtain

$$[r,{\tilde{a}}_3]{\tilde{a}}_1{\tilde{a}}_2[r,{\tilde{a}}_3]{\tilde{a}}_1\hslash \left({\tilde{a}}_3\right)=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2,{\tilde{a}}_3,r\in U.$$

Writing ${\tilde{a}}_1$ by $2{\tilde{a}}_1\hslash \left({\tilde{a}}_3\right)$ and again using S is $2-$torsion free, we have

$$[r,{\tilde{a}}_3]{\tilde{a}}_1\hslash \left({\tilde{a}}_3\right){\tilde{a}}_2[r,{\tilde{a}}_3]{\tilde{a}}_1\hslash \left({\tilde{a}}_3\right)=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2,{\tilde{a}}_3,r\in U.$$

By Lemma 1, we obtain that

$$[r,{\tilde{a}}_3]{\tilde{a}}_1\hslash \left({\tilde{a}}_3\right)=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_3,r\in U.$$

(6)

Now, writing r by $\hslash \left({\tilde{a}}_3\right)$ in (6) and multiplying this equation from the right by ${\tilde{a}}_3,$ we have

$$[\hslash \left({\tilde{a}}_3\right),{\tilde{a}}_3]{\tilde{a}}_1\hslash \left({\tilde{a}}_3\right){\tilde{a}}_3=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_3\in U.$$

(7)

Taking ${\tilde{a}}_1$ by $2{\tilde{a}}_1{\tilde{a}}_3$ in (7), we obtain that

$$2[\hslash \left({\tilde{a}}_3\right),{\tilde{a}}_3]{\tilde{a}}_1{\tilde{a}}_3\hslash \left({\tilde{a}}_3\right)=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2,{\tilde{a}}_3\in U.$$

Since S is $2-$torsion free, we obtain

$$[\hslash \left({\tilde{a}}_3\right),{\tilde{a}}_3]{\tilde{a}}_1{\tilde{a}}_3\hslash \left({\tilde{a}}_3\right)=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2,{\tilde{a}}_3\in U.$$

(8)

Combining (7) and (8), we find that

$$[\hslash \left({\tilde{a}}_3\right),{\tilde{a}}_3]{\tilde{a}}_1\left[\hslash \left({\tilde{a}}_3\right),{\tilde{a}}_3\right]=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2,{\tilde{a}}_3\in U.$$

(9)

Again by Lemma 1, we obtain that ℏ is a commuting map on U.

$\left(ii\right)$ If ℏ is a homoderivation satisfying $\hslash \left({\tilde{a}}_1{\tilde{a}}_2\right)-{\tilde{a}}_1{\tilde{a}}_2\in Z$, for all ${\tilde{a}}_1,{\tilde{a}}_2\in U$, then $(-\hslash )\left({\tilde{a}}_1{\tilde{a}}_2\right)-{\tilde{a}}_1{\tilde{a}}_2\in Z$. Since $(-\hslash )$ is a homoderivation on $S,$ we obtain that ℏ is a commuting map on U by Theorem 2$\left(i\right)$. □

Theorem 3.

Let S be a $2-$torsion free semiprime ring, U a noncentral square-closed Lie ideal of S, and ℏ a homoderivation on S. If any of the following holds for all ${\tilde{a}}_1,{\tilde{a}}_2\in U$

(i)

$\hslash \left([{\tilde{a}}_1,{\tilde{a}}_2]\right)={\tilde{a}}_1\circ {\tilde{a}}_2$, or

(ii)

$\hslash ({\tilde{a}}_1\circ {\tilde{a}}_2)=[{\tilde{a}}_1,{\tilde{a}}_2]$, or

(iii)

$\hslash ({\tilde{a}}_1\circ {\tilde{a}}_2)={\tilde{a}}_1\circ {\tilde{a}}_2$, or

(iv)

$\hslash \left([{\tilde{a}}_1,{\tilde{a}}_2]\right)=[{\tilde{a}}_1,{\tilde{a}}_2]$, or

then $U[S,S]=\left(0\right)$ and $[U,S]=\left(0\right).$ In particular, $[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_1]=0$ for all ${\tilde{a}}_1\in U$.

Proof. 

$\left(i\right)$ Assume that

$$\hslash \left([{\tilde{a}}_1,{\tilde{a}}_2]\right)={\tilde{a}}_1\circ {\tilde{a}}_2\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U$$

(10)

Putting ${\tilde{a}}_2={\tilde{a}}_1$ in (10), we have $0={\tilde{a}}_1\circ {\tilde{a}}_1\Rightarrow 2{\tilde{a}}_1^2=0$. Since S is 2-torsion free, we have ${\tilde{a}}_1^2=0$. By linearizing, we obtain

$${\tilde{a}}_1\circ {\tilde{a}}_2=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

(11)

Replacing ${\tilde{a}}_2$ by $2{\tilde{a}}_{2}r$, where $r\in S$ in this equation, we have $({\tilde{a}}_1\circ {\tilde{a}}_2)r-{\tilde{a}}_2[{\tilde{a}}_1,r]=0$, and by using (11), we obtain $[{\tilde{a}}_1,r]{\tilde{a}}_2[{\tilde{a}}_1,r]=0$ for all ${\tilde{a}}_1,{\tilde{a}}_2\in U$ and $r\in S$. Hence, $[{\tilde{a}}_1,r]U[{\tilde{a}}_1,r]=\left(0\right)$ for all ${\tilde{a}}_1,{\tilde{a}}_2\in U$ and $r\in S.$ Since $[{\tilde{a}}_1,r]\in U$ and by using Lemma 1, we obtain $[{\tilde{a}}_1,r]=0$ for all ${\tilde{a}}_1\in U$ and $r\in S$, that is, $[U,S]=\left(0\right)$, hence $U\subseteq Z\left(S\right)$, a contradiction with the fact that U is a noncentral. Therefore, we have to remove this condition.

Now, we will continue; we have $[{\tilde{a}}_1,r]=0$ for all ${\tilde{a}}_1\in U$ and $r\in S$. Replacing ${\tilde{a}}_1$ by $2{\tilde{a}}_{1}s$, where $s\in S$, we obtain ${\tilde{a}}_1[s,r]=0$ for all ${\tilde{a}}_1\in U$ and $r,s\in S$. That is, $U[S,S]=\left(0\right)$ and $[U,S]=\left(0\right)$.

$\left(ii\right)$ Assume that

$$\hslash ({\tilde{a}}_1\circ {\tilde{a}}_2)=[{\tilde{a}}_1,{\tilde{a}}_2]\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

(12)

Putting ${\tilde{a}}_2={\tilde{a}}_1$ in (12), we have $0=\hslash ({\tilde{a}}_1\circ {\tilde{a}}_1)\Rightarrow 2\hslash \left({\tilde{a}}_1^2\right)=0.$ Since S is 2-torsion free, we obtain $\hslash \left({\tilde{a}}_1^2\right)=0$. By linearizing, we have

$$\hslash ({\tilde{a}}_1\circ {\tilde{a}}_2)=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

(13)

Using (13) in (12), we obtain

$$[{\tilde{a}}_1,{\tilde{a}}_2]=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

(14)

Replacing ${\tilde{a}}_2$ by $2{\tilde{a}}_{2}r$, where $r\in S$, we obtain $2[{\tilde{a}}_1,{\tilde{a}}_2]r+2{\tilde{a}}_2[{\tilde{a}}_1,r]=0$, hence $[{\tilde{a}}_1,{\tilde{a}}_2]r+{\tilde{a}}_2[{\tilde{a}}_1,r]=0$, and by using (14), we obtain ${\tilde{a}}_2[{\tilde{a}}_1,r]=0$. Hence, $[{\tilde{a}}_1,r]{\tilde{a}}_2[{\tilde{a}}_1,r]=0$ for all ${\tilde{a}}_1,{\tilde{a}}_2\in U$ and $r\in S$. Thus,

$$[{\tilde{a}}_1,r]U[{\tilde{a}}_1,r]=\left(0\right)\mathrm{for}\mathrm{all}{\tilde{a}}_1\in U\mathrm{and}r\in S.$$

Since $[{\tilde{a}}_1,r]\in U$ and by using Lemma 1, we obtain $[{\tilde{a}}_1,r]=0$ for all ${\tilde{a}}_1\in U$ and $r\in S$; that is, $[U,S]=\left(0\right)$, hence $U\subseteq Z\left(S\right)$, a contradiction with the fact that U is a noncentral. Therefore, we have to remove this condition. Now, we have $[{\tilde{a}}_1,r]=0$ for all ${\tilde{a}}_1\in U$ and $r\in S$. Replacing ${\tilde{a}}_1$ by $2{\tilde{a}}_{1}s$, where $s\in S$, we obtain ${\tilde{a}}_1[s,r]=0$ for all ${\tilde{a}}_1\in U$ and $r,s\in S$. That is, $U[S,S]=\left(0\right)$ and $[U,S]=\left(0\right)$.

$\left(iii\right)$ Assume that

$$\hslash ({\tilde{a}}_1\circ {\tilde{a}}_2)={\tilde{a}}_1\circ {\tilde{a}}_2\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U$$

(15)

Replacing ${\tilde{a}}_2$ by $2{\tilde{a}}_2{\tilde{a}}_1$ in (15), we have

$$\hslash \left(({\tilde{a}}_1\circ {\tilde{a}}_2){\tilde{a}}_1\right)=({\tilde{a}}_1\circ {\tilde{a}}_2){\tilde{a}}_1$$

and so

$$\hslash ({\tilde{a}}_1\circ {\tilde{a}}_2)\hslash \left({\tilde{a}}_1\right)+\hslash ({\tilde{a}}_1\circ {\tilde{a}}_2){\tilde{a}}_1+({\tilde{a}}_1\circ {\tilde{a}}_2)\hslash \left({\tilde{a}}_1\right)=({\tilde{a}}_1\circ {\tilde{a}}_2){\tilde{a}}_1.$$

Using (15),

$$({\tilde{a}}_1\circ {\tilde{a}}_2)\hslash \left({\tilde{a}}_1\right)+({\tilde{a}}_1\circ {\tilde{a}}_2){\tilde{a}}_1+({\tilde{a}}_1\circ {\tilde{a}}_2)\hslash \left({\tilde{a}}_1\right)=({\tilde{a}}_1\circ {\tilde{a}}_2){\tilde{a}}_1$$

and so

$$\begin{array}{ccc}\hfill ({\tilde{a}}_1\circ {\tilde{a}}_2)\hslash \left({\tilde{a}}_1\right)+({\tilde{a}}_1\circ {\tilde{a}}_2)\hslash \left({\tilde{a}}_1\right)& =& 0\hfill \\ \hfill 2({\tilde{a}}_1\circ {\tilde{a}}_2)\hslash \left({\tilde{a}}_1\right)& =& 0\hfill \\ \hfill ({\tilde{a}}_1\circ {\tilde{a}}_2)\hslash \left({\tilde{a}}_1\right)& =& 0.\hfill \end{array}$$

Replacing ${\tilde{a}}_1$ by ${\tilde{a}}_1^2$ in the last relation, we obtain $({\tilde{a}}_1^2\circ {\tilde{a}}_2)\hslash \left({\tilde{a}}_1^2\right)=0$. Putting ${\tilde{a}}_2={\tilde{a}}_1$ in (15) and using this in last equation we obtain

$$({\tilde{a}}_1^2\circ {\tilde{a}}_2){\tilde{a}}_1^2=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

(16)

Replacing ${\tilde{a}}_2$ by $2r{\tilde{a}}_2$, where $r\in S,2r({\tilde{a}}_1^2\circ {\tilde{a}}_2){\tilde{a}}_1^2+2[{\tilde{a}}_1^2,r]{\tilde{a}}_2{\tilde{a}}_1^2=0$. Using (16) in this relation, we obtain $2[{\tilde{a}}_1^2,r]{\tilde{a}}_2{\tilde{a}}_1^2=0,$ and so $[{\tilde{a}}_1^2,r]{\tilde{a}}_2{\tilde{a}}_1^2=0\Rightarrow [{\tilde{a}}_1^2,r]{\tilde{a}}_2[{\tilde{a}}_1^2,r]=0$ for all ${\tilde{a}}_1\in U$ and $r\in S$. By Lemma 1, we obtain $[{\tilde{a}}_1^2,r]=0$ for all ${\tilde{a}}_1\in U$ and $r\in S$. By linearizing, we obtain

$$[{\tilde{a}}_1\circ {\tilde{a}}_2,r]=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U,r\in S.$$

(17)

Replacing ${\tilde{a}}_2$ by $2{\tilde{a}}_{2}r$, we obtain $[({\tilde{a}}_1\circ {\tilde{a}}_2)r-{\tilde{a}}_2[{\tilde{a}}_1,r],r]=0$ for all ${\tilde{a}}_1\in U$ and $r\in S$. Using (17), we obtain

$${\tilde{a}}_2[[{\tilde{a}}_1,r],r]-[{\tilde{a}}_2,r][{\tilde{a}}_1,r]=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U,r\in S$$

(18)

Replacing ${\tilde{a}}_2$ by $2{\tilde{a}}_1{\tilde{a}}_2$, in (18), we obtain

$${\tilde{a}}_1{\tilde{a}}_2[[{\tilde{a}}_1,r],r]-{\tilde{a}}_1[{\tilde{a}}_2,r][{\tilde{a}}_1,r]-[{\tilde{a}}_1,r]{\tilde{a}}_2[{\tilde{a}}_1,r]=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U,r\in S$$

and so

$${\tilde{a}}_1({\tilde{a}}_2[[{\tilde{a}}_1,r],r]-[{\tilde{a}}_2,r][{\tilde{a}}_1,r])-[{\tilde{a}}_1,r]{\tilde{a}}_2[{\tilde{a}}_1,r]=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U,r\in S.$$

Using (18) in the last relation, we obtain $[{\tilde{a}}_1,r]{\tilde{a}}_2[{\tilde{a}}_1,r]=0$ for all ${\tilde{a}}_1,{\tilde{a}}_2\in U$ and $r\in S$. By Lemma 1, we obtain $[{\tilde{a}}_1,r]=0$ for all ${\tilde{a}}_1\in U$ and $r\in S$. That is, $[U,S]=\left(0\right)$, hence $U\subseteq Z\left(S\right)$, a contradiction with the fact that U is a noncentral. Therefore, we have to remove this condition.

Now, we will continue; we have $[{\tilde{a}}_1,r]=0$ for all ${\tilde{a}}_1\in U$ and $r\in S$. Replacing ${\tilde{a}}_1$ by $2{\tilde{a}}_{1}s$, where $s\in S$, we obtain ${\tilde{a}}_1[s,r]=0$ for all ${\tilde{a}}_1\in U$ and $r,s\in S$. That is, $U[S,S]=\left(0\right)$ and $[U,S]=\left(0\right)$.

$\left(iv\right)$ Assume that

$$\hslash \left([{\tilde{a}}_1,{\tilde{a}}_2]\right)=[{\tilde{a}}_1,{\tilde{a}}_2]\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

(19)

As in (iii), we obtain $[{\tilde{a}}_1,{\tilde{a}}_2]\hslash \left({\tilde{a}}_1\right)=0$ for all ${\tilde{a}}_1,{\tilde{a}}_2\in U$. Replacing ${\tilde{a}}_1$ by $[{\tilde{a}}_1,r]$, where $z\in U$, we obtain $[[{\tilde{a}}_1,z],{\tilde{a}}_2]\hslash \left([{\tilde{a}}_1,z]\right)=0$ for all ${\tilde{a}}_1,{\tilde{a}}_2,z\in U$. Using (19), we obtain $[[{\tilde{a}}_1,z],{\tilde{a}}_2][{\tilde{a}}_1,z]=0$ for all ${\tilde{a}}_1,{\tilde{a}}_2,z\in$$U.$ Replacing ${\tilde{a}}_2$ by $2r{\tilde{a}}_2$, where $r\in S$, we obtain $[[{\tilde{a}}_1,z],r]{\tilde{a}}_2[{\tilde{a}}_1,z]=0$ for all ${\tilde{a}}_1,{\tilde{a}}_2,z\in U$ and $r\in S$. That is, $[[{\tilde{a}}_1,z],r]{\tilde{a}}_2[[{\tilde{a}}_1,z],r]=0$ for all ${\tilde{a}}_1,{\tilde{a}}_2,z\in U$ and $r\in S$. By Lemma 1, we obtain $[[{\tilde{a}}_1,z],r]=0$ for all ${\tilde{a}}_1,z\in U$ and $r\in S$. Replacing z by $z{\tilde{a}}_1$, we obtain $[{\tilde{a}}_1,z][{\tilde{a}}_1,r]=0$ for all ${\tilde{a}}_1,z\in U$ and $r\in S$. Replacing z by $rz$, we obtain $[{\tilde{a}}_1,r]z[{\tilde{a}}_1,r]=0$ for all ${\tilde{a}}_1,z\in U$ and $r\in S$. By Lemma 1, we obtain $[{\tilde{a}}_1,r]=0$ for all ${\tilde{a}}_1\in U$ and $r\in S$. That is, $[U,S]=\left(0\right)$, hence $U\subseteq Z\left(S\right)$, a contradiction with fact that U is a noncentral. So we have to remove this condition. Now we have $[{\tilde{a}}_1,r]=0$ for all ${\tilde{a}}_1\in$U and $r\in S$. Replacing ${\tilde{a}}_1$ by $2{\tilde{a}}_{1}s$, where $s\in S$, we obtain ${\tilde{a}}_1[s,r]=0$ for all ${\tilde{a}}_1\in U$ and $r,s\in S$. That is, $U[S,S]=\left(0\right)$ and $[U,S]=\left(0\right)$. □

Example 1.

Let $S=\left\{\left(\begin{array}{cccc}0& a& b& c\\ 0& 0& 0& b\\ 0& 0& 0& -a\\ 0& 0& 0& 0\end{array}\right):a,b,c\in \mathbb{Z}\right\}$ be a 2-torsion free ring, and is not a semiprime ring $U=\left\{\left(\begin{array}{cccc}0& a& a& c\\ 0& 0& 0& a\\ 0& 0& 0& -a\\ 0& 0& 0& 0\end{array}\right):a,c\in \mathbb{Z}\right\}$ be a noncentral square-closed Lie ideal of S. $\hslash :S\to S$ $\hslash \left(\left(\begin{array}{cccc}0& a& b& c\\ 0& 0& 0& b\\ 0& 0& 0& -a\\ 0& 0& 0& 0\end{array}\right)\right)=\left(\begin{array}{cccc}0& 0& -b& c\\ 0& 0& 0& -b\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$. Then, it easy to check that ℏ is a homoderivation of S. The conditions $\left(i\right)-\left(iv\right)$ given in Theorem 3 are satisfied on U. However, S is not semiprime and h is not commuting in U. Thus, the conditions’ semiprimeness is essential.

Theorem 4.

Let S be a $2-$torsion semiprime ring, U a noncentral square-closed Lie ideal of S, and ℏ a zero-power valued on $U.$ Suppose that S admits a homoderivation ℏ such that for all ${\tilde{a}}_1,{\tilde{a}}_2\in U$

(i)

${\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)\pm {\tilde{a}}_1{\tilde{a}}_2\in Z,$ or

(ii)

${\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)\pm {\tilde{a}}_2{\tilde{a}}_1=0,$ or

(iii)

${\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)\pm {\tilde{a}}_1\circ {\tilde{a}}_2=0,$ or

(iv)

$[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_2]\pm {\tilde{a}}_1{\tilde{a}}_2=0,$ or

(v)

$[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_2]\pm {\tilde{a}}_2{\tilde{a}}_1=0$.

Then ℏ is a commuting map on U.

Proof. 

$\left(i\right)$ By the hypothesis, we obtain

$${\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)+{\tilde{a}}_1{\tilde{a}}_2\in Z,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

That is

$${\tilde{a}}_1(\hslash \left({\tilde{a}}_2\right)+{\tilde{a}}_2)\in Z,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

Since ℏ is zero-power valued on $U,$ there exists an integer $n>1$ such that ${\hslash }^n\left({\tilde{a}}_1\right)=0$ for all ${\tilde{a}}_1\in U$. Replacing ${\tilde{a}}_2$ by ${\tilde{a}}_2-\hslash \left({\tilde{a}}_2\right)+{\hslash }^2\left({\tilde{a}}_2\right)+...+{(-1)}^{n-1}{\hslash }^{n-1}\left({\tilde{a}}_2\right)$ in this equation, we obtain that

$${\tilde{a}}_1{\tilde{a}}_2\in Z,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

Commuting this term with $r\in S$, we obtain

$$0=[{\tilde{a}}_1{\tilde{a}}_2,r]=[{\tilde{a}}_1,r]{\tilde{a}}_2+{\tilde{a}}_1[{\tilde{a}}_2,r].$$

Replacing ${\tilde{a}}_1$ by $2{\tilde{a}}_3{\tilde{a}}_1,$${\tilde{a}}_3\in U$ in this equation and using this equation, we obtain

$$2[{\tilde{a}}_3,r]{\tilde{a}}_1{\tilde{a}}_2=0.$$

Since S is $2-$torsion free ring, we see that

$$[{\tilde{a}}_3,r]{\tilde{a}}_1{\tilde{a}}_2=0.$$

Taking ${\tilde{a}}_2$ by $[{\tilde{a}}_3,r]$ in this equation, we have

$$[{\tilde{a}}_3,r]{\tilde{a}}_1[{\tilde{a}}_3,r]=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_3\in U,r\in S.$$

Replacig r by $\hslash \left({\tilde{a}}_3\right)$ in this equation, we obtain

$$[{\tilde{a}}_3,\hslash \left({\tilde{a}}_3\right)]{\tilde{a}}_1[{\tilde{a}}_3,\hslash \left({\tilde{a}}_3\right)]=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_3\in U.$$

By Lemma 1, we obtain that ℏ is a commuting map on U.

${\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)-{\tilde{a}}_1{\tilde{a}}_2\in Z$ is proved similarly. We complete the proof.

$\left(ii\right)$ We obtain

$${\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)+{\tilde{a}}_2{\tilde{a}}_1=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

Replacing ${\tilde{a}}_2$ by ${\tilde{a}}_1{\tilde{a}}_2,$${\tilde{a}}_2\in U$ in this equation and using this, we have

$${\tilde{a}}_1\hslash \left({\tilde{a}}_1\right)(\hslash \left({\tilde{a}}_2\right)+{\tilde{a}}_2)=0.$$

Since ℏ is zero-power valued on U, there exists an integer $n>1$ such that ${\hslash }^n\left({\tilde{a}}_1\right)=0$ for all ${\tilde{a}}_1\in U$. Replacing ${\tilde{a}}_2$ by ${\tilde{a}}_2-\hslash \left({\tilde{a}}_2\right)+{\hslash }^2\left({\tilde{a}}_2\right)+...+{(-1)}^{n-1}{\hslash }^{n-1}\left({\tilde{a}}_2\right)$ in this equation, we obtain that

$${\tilde{a}}_1\hslash \left({\tilde{a}}_1\right){\tilde{a}}_2=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

Writing ${\tilde{a}}_2$ by $4{\tilde{a}}_2{\tilde{a}}_1\hslash \left({\tilde{a}}_1\right)$ in the last equation, we have

$$4{\tilde{a}}_1\hslash \left({\tilde{a}}_1\right){\tilde{a}}_2{\tilde{a}}_1\hslash \left({\tilde{a}}_1\right)=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

Since S is $2-$torsion free ring, we obtain that

$${\tilde{a}}_1\hslash \left({\tilde{a}}_1\right){\tilde{a}}_2{\tilde{a}}_1\hslash \left({\tilde{a}}_1\right)=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

By Lemma 1, we obtain

$${\tilde{a}}_1\hslash \left({\tilde{a}}_1\right)=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

(20)

By the hypothesis, we obtain

$${\tilde{a}}_1\hslash \left({\tilde{a}}_1\right)+{\tilde{a}}_1^2=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1\in U.$$

Using Equation (20), we obtain that

$${\tilde{a}}_1^2=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1\in U.$$

(21)

Replacing ${\tilde{a}}_1$ by ${\tilde{a}}_1+{\tilde{a}}_2$ in this equation, we have

$${\tilde{a}}_1\circ {\tilde{a}}_2=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

Replacing ${\tilde{a}}_2$ by $2{\tilde{a}}_2{\tilde{a}}_3,$${\tilde{a}}_3\in U$ in the above expression and using this, we obtain

$$2[{\tilde{a}}_1,{\tilde{a}}_2]{\tilde{a}}_3=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2,{\tilde{a}}_3\in U.$$

Since S is $2-$torsion free ring, we have

$$[{\tilde{a}}_1,{\tilde{a}}_2]{\tilde{a}}_3=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2,{\tilde{a}}_3\in U.$$

Replacing ${\tilde{a}}_3$ by $2{\tilde{a}}_3[{\tilde{a}}_1,{\tilde{a}}_2]$ in this equation and using S is $2-$torsion free ring, we have

$$[{\tilde{a}}_1,{\tilde{a}}_2]U[{\tilde{a}}_1,{\tilde{a}}_2]=\left(0\right),\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

By Lemma 1, we find that

$$[{\tilde{a}}_1,{\tilde{a}}_2]=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

Replacing ${\tilde{a}}_2$ by $\hslash \left({\tilde{a}}_1\right)$ in this equation, we obtain $[{\tilde{a}}_1,\hslash \left({\tilde{a}}_1\right)]=0$. Hence, we conclude that ℏ is a commuting map on U. This completes the proof.

${\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)-{\tilde{a}}_2{\tilde{a}}_1=0$ is proved similarly.

$\left(iii\right)$ By the hypothesis, we obtain

$${\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)\pm {\tilde{a}}_1\circ {\tilde{a}}_2=0.$$

Replacing ${\tilde{a}}_2$ by $2{\tilde{a}}_2{\tilde{a}}_1$ in this equation and using S is $2-$torsion free ring, we get

$${\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)\hslash \left({\tilde{a}}_1\right)+{\tilde{a}}_1\hslash \left({\tilde{a}}_2\right){\tilde{a}}_1+{\tilde{a}}_1{\tilde{a}}_2\hslash \left({\tilde{a}}_1\right)\pm ({\tilde{a}}_1\circ {\tilde{a}}_2){\tilde{a}}_1=0.$$

Using the hypothesis, we get

$${\tilde{a}}_1\hslash \left({\tilde{a}}_2\right)\hslash \left({\tilde{a}}_1\right)+{\tilde{a}}_1{\tilde{a}}_2\hslash \left({\tilde{a}}_1\right)=0.$$

That is

$${\tilde{a}}_1(\hslash \left({\tilde{a}}_2\right)+{\tilde{a}}_2)\hslash \left({\tilde{a}}_1\right)=0.$$

Since ℏ is zero-power valued on $U,$ there exists an integer $n>1$ such that ${\hslash }^n\left({\tilde{a}}_1\right)=0$ for all ${\tilde{a}}_1\in U$. Replacing ${\tilde{a}}_2$ by ${\tilde{a}}_2-\hslash \left({\tilde{a}}_2\right)+{\hslash }^2\left({\tilde{a}}_2\right)+\cdots +{(-1)}^{n-1}{\hslash }^{n-1}\left({\tilde{a}}_2\right)$ in this equation, we obtain that

$${\tilde{a}}_1{\tilde{a}}_2\hslash \left({\tilde{a}}_1\right)=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

Taking ${\tilde{a}}_2$ by 4$\hslash \left({\tilde{a}}_1\right){\tilde{a}}_2{\tilde{a}}_1$ in the above equation and using S is $2-$torsion free ring, we see that

$${\tilde{a}}_1\hslash \left({\tilde{a}}_1\right){\tilde{a}}_2{\tilde{a}}_1\hslash \left({\tilde{a}}_1\right)=0.$$

By Lemma 1, we have

$${\tilde{a}}_1\hslash \left({\tilde{a}}_1\right)=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

By our hypothesis, we obtain

$${\tilde{a}}_1\hslash \left({\tilde{a}}_1\right)\pm {\tilde{a}}_1^2=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1\in U$$

and so

$${\tilde{a}}_1^2=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1\in U.$$

The rest of the proof is the same as Equation (21).

$\left(iv\right)$ We obtain

$$[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_2]\pm {\tilde{a}}_1{\tilde{a}}_2=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

Taking ${\tilde{a}}_2$ by 2${\tilde{a}}_2{\tilde{a}}_1$ in this equation and using S is $2-$torsion free ring, we obtain

$$[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_2]{\tilde{a}}_1+{\tilde{a}}_2[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_1]\pm {\tilde{a}}_1{\tilde{a}}_2{\tilde{a}}_1=0.$$

By the hypothesis, we find that

$${\tilde{a}}_2[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_1]=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

(22)

Replacing ${\tilde{a}}_2$ by $2[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_1]{\tilde{a}}_2$ in this equation and using S is $2-$torsion free ring, we obtain

$$[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_1]r[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_1]=0.$$

By Lemma 1, we have

$$[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_1]=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1\in U.$$

This completes the proof.

$\left(v\right)$ By the hypothesis, we obtain

$$[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_2]\pm {\tilde{a}}_2{\tilde{a}}_1=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

Replacing ${\tilde{a}}_2$ by $2{\tilde{a}}_1{\tilde{a}}_2$ in this equation, we have

$$2{\tilde{a}}_1[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_2]+2[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_1]{\tilde{a}}_2\pm 2{\tilde{a}}_1{\tilde{a}}_2{\tilde{a}}_1=0.$$

Since S is $2-$torsion free ring, we can write

$${\tilde{a}}_1[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_2]+[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_1]{\tilde{a}}_2\pm {\tilde{a}}_1{\tilde{a}}_2{\tilde{a}}_1=0.$$

By our hypothesis, we obtain

$$[\hslash \left({\tilde{a}}_1\right),{\tilde{a}}_1]{\tilde{a}}_2=0,\mathrm{for}\mathrm{all}{\tilde{a}}_1,{\tilde{a}}_2\in U.$$

The rest of the proof is the same as Equation (22). Hence, we complete the proof. □

3. Open Problems

Our assumptions focus on a noncentral square-closed Lie ideal within a 2-torsion free semiprime ring. As a direction for future research, similar investigations can be conducted without imposing the square-closed condition on Lie ideals of rings. Additionally, this study extends to various algebraic structures, including alternative rings, algebras, and near-rings.

Ruth N. Ferreira and Bruno L. M. Ferreira established the following theorem in ([16], Theorem 1.1):

Theorem 5.

Let S be a $3-$torsion free alternative ring. S is a prime ring if and only if $aS$·$b=0$ (or a·$Sb=0$) implies $a=0$ or $b=0$ for $a,b\in S.$

It is well known that the 3-torsion free condition is unnecessary in the case of associative rings. Therefore, the findings in this paper can also be explored within the framework of alternative rings. Several studies in the literature address related topics (see [16,17,18], etc.).

4. Conclusions

In this work, we examine the commutativity of semiprime rings influenced by the action of homoderivations. Our study generalizes prior results by identifying conditions under which a noncentral square-closed Lie ideal of a 2-torsion free semiprime ring admits a homoderivation. To demonstrate the necessity of these conditions, we present an illustrative example within the framework of our theorem. The findings contribute to the broader landscape of commutativity theorems, offering new insights into the structural properties of rings with derivations. Moreover, this research paves the way for further exploration of derivations in algebraic systems, with potential applications in operator algebras, noncommutative geometry, and other areas where ring theory plays a fundamental role.