Notes on Semiprime Ideals with Symmetric Bi-Derivation

Abstract

In this paper, we prove many algebraic identities that include symmetric bi-derivation in rings which contain a semiprime ideal. We intend to generalize previous results obtained for semiprime rings with symmetric derivation using semiprime ideals in rings.

1. Introduction

Several studies in the literature were made about additive mappings of prime and semiprime rings. Many authors investigated the structure of rings in these papers. An ideal $\Pi$ in a ring $\mathsf{ϝ}$ is said to be prime if $\Pi \ne \mathsf{ϝ}$ and for any ideals ${\Im }_1,{\Im }_2$ in $\mathsf{ϝ},$${\Im }_1{\Im }_2\subseteq \Pi$ then ${\Im }_1\subseteq \Pi$ or ${\Im }_2\subseteq \Pi .$ A ring $\mathsf{ϝ}$ is said to be a prime ring if the zero ideal is a prime ideal (that is, if ${\Im }_1,{\Im }_2$ are ideals such that ${\Im }_1{\Im }_2=0,$ then ${\Im }_1=0$ or ${\Im }_2=0$). A ring $\mathsf{ϝ}$ is semiprime if and only if $\mathsf{ϝ}$ has no nonzero nilpotent ideals. Every prime ring is semiprime since 0 is a prime ideal. Also, we know that $\mathsf{ϝ}/\Pi$ is a prime ring where $\Pi$ is a prime ideal of $\mathsf{ϝ}.$ The relationships among prime ideals, prime rings and semiprime rings are analogous to the relationships between mappings and commutativity.

An additive mapping $\delta :\mathsf{ϝ}\to \mathsf{ϝ}$ is called a derivation if $\delta \left(\kappa {\kappa }^{\prime }\right)=\delta \left(\kappa \right){\kappa }^{\prime }+\kappa \delta \left({\kappa }^{\prime }\right)$ holds for all $\kappa ,{\kappa }^{\prime }\in \mathsf{ϝ}.$ This definition was originated by Posner in [1]. In 1957, E. C. Posner [1] established a fundamental result stating that if the product of two derivations on a prime ring of characteristic not two is itself a derivation, then at least one of the derivations must be trivial. This result has since inspired extensive research, particularly in the broader setting of prime and semiprime rings under various structural constraints. Subsequent work by Vukman [2,3] extended Posner’s theorem to the framework of bi-derivations, leading to significant advancements in the study of symmetric bi-derivations and their applications in ring theory. Further developments have explored fuzzification in rings, yielding important theoretical insights (see [4,5,6,7,8,9], and references therein). The study of derivations in ring theory remains essential not only as a structural tool but also due to its deep connections with analysis and applied mathematics. Investigating analogous properties of derivations within algebraic frameworks continues to be a key area of exploration with implications for both pure and applied mathematical disciplines.

More than a few authors proved the commutativity of prime or semiprime rings with certain algebraic identities using derivation. The definition of derivation was extended different ways such as semiderivation, generalized derivation, Jordan derivation, etc. One of these generalizations is the definition of symmetric derivation given by Maksa in [10].

Definition 1.

A mapping $\Delta (.,.):\mathsf{ϝ}\times \mathsf{ϝ}\to \mathsf{ϝ}$ is said to be symmetric if $\Delta (\kappa ,{\kappa }^{\prime })=\Delta ({\kappa }^{\prime },\kappa )$ for all $\kappa ,{\kappa }^{\prime }\in \mathsf{ϝ}.$ A mapping $\delta :\mathsf{ϝ}\to \mathsf{ϝ}$ is called the trace of $\Delta (.,.)$ if $\delta \left(\kappa \right)=\Delta (\kappa ,\kappa )$ for all $\kappa \in \mathsf{ϝ}.$ It is obvious that if $\Delta (.,.)$ is bi-additive (i.e., additive in both arguments), then the trace δ of $\Delta (.,.).$

The following are some basic properties of a symmetric bi-additive mapping $\Delta (.,.):\mathsf{ϝ}\times \mathsf{ϝ}\to \mathsf{ϝ}$ with the trace $\delta$ of $\Delta$. The proofs of these properties are straightforward and hence omitted.

(1) $\delta (\kappa +{\kappa }^{\prime })=\delta \left(\kappa \right)+\delta \left({\kappa }^{\prime }\right)+2\Delta (\kappa ,{\kappa }^{\prime })$ for all $\kappa$, ${\kappa }^{\prime }\in \mathsf{ϝ}$,

(2) $\Delta (\kappa ,0)=\Delta (0,\kappa )=0$ for all $\kappa \in \mathsf{ϝ}$,

(3) $\Delta (-\kappa ,{\kappa }^{\prime })=-\Delta (\kappa ,{\kappa }^{\prime })=\Delta (\kappa ,-{\kappa }^{\prime })$ for all $\kappa$, ${\kappa }^{\prime }\in \mathsf{ϝ}$,

(4) $\delta (-\kappa )=\delta \left(\kappa \right)$ for all $\kappa \in \mathsf{ϝ}$,

(5) $\delta (\kappa +{\kappa }^{\prime })+\delta (\kappa -{\kappa }^{\prime })=2\delta \left(\kappa \right)+2\delta \left({\kappa }^{\prime }\right)$ for all $\kappa$, ${\kappa }^{\prime }\in \mathsf{ϝ}$.

Definition 2.

Let $\Delta (.,.):\mathsf{ϝ}\times \mathsf{ϝ}\to \mathsf{ϝ}$ be a symmetric bi-additive mapping. We call Δ symmetric bi-derivation on ϝ if it satisfies the following condition

$$\Delta (\kappa {\kappa }^{\prime },{\kappa }_3)=\Delta (\kappa ,{\kappa }_3){\kappa }^{\prime }+\kappa \Delta ({\kappa }^{\prime },{\kappa }_3)$$

and

$$\Delta (\kappa ,{\kappa }^{\prime }{\kappa }_3)=\Delta (\kappa ,{\kappa }^{\prime }){\kappa }_3+{\kappa }^{\prime }\Delta (\kappa ,{\kappa }_3),$$

for all $\kappa ,{\kappa }^{\prime },{\kappa }_3\in \mathsf{ϝ}.$

We can give the following example for symmetric bi-derivation:

Example 1.

Suppose the ring $\mathsf{ϝ}=\left\{\left.\left(\begin{array}{cc}a& 0\\ b& 0\end{array}\right)\right|a,b\in \mathbb{R}\right\}.$ Define maps $\Delta :\mathsf{ϝ}\times \mathsf{ϝ}\to \mathsf{ϝ}$ as follows:

$$\Delta \left(\left(\begin{array}{cc}a& 0\\ b& 0\end{array}\right),\left(\begin{array}{cc}x& 0\\ y& 0\end{array}\right)\right)=\left(\begin{array}{cc}0& 0\\ ax& 0\end{array}\right).$$

Then, it is easy to verify that Δ is a symmetric bi-derivation of ϝ.

The notion of bi-derivations was introduced by Maksa [10] and later examined in the context of functional equations in [11], where symmetric bi-derivations were shown to be linked to general solutions of certain functional identities. Investigations into symmetric bi-derivations in prime rings have been further explored in [2,3].

Brešar [12] established that every bi-derivation $\Delta$ on a noncommutative prime ring $\mathsf{ϝ}$ takes the form $\Delta (\kappa ,{\kappa }^{\prime })=\lambda [\kappa ,{\kappa }^{\prime }]$ for some $\lambda$ in the extended centroid of $\mathsf{ϝ}$. This result was later generalized to semiprime rings using functional identities [13]. Extensive research has since been conducted on bi-derivations and related mappings in prime and semiprime rings as well as in various algebraic structures. Further generalizations have introduced the concepts of 33 derivations and, more broadly, n derivations, extending results initially established by Posner and Vukman to these higher-order derivations in prime and semiprime settings. The study of such mappings continues to play a crucial role in the structural analysis of rings and algebras.

Throughout this paper, $\mathsf{ϝ}$ will represent an associative ring. For any $\kappa ,{\kappa }^{\prime }\in \mathsf{ϝ},$ the symbol $[\kappa ,{\kappa }^{\prime }]$ stands for the commutator $\kappa {\kappa }^{\prime }-{\kappa }^{\prime }\kappa$, and the symbol $\kappa \circ {\kappa }^{\prime }$ stands for the anti-commutator $\kappa {\kappa }^{\prime }+{\kappa }^{\prime }\kappa$. The following fundamental identities will be utilized throughout the discussion without explicit reference:

(i) $[\kappa ,{\kappa }^{\prime }{\kappa }_3]={\kappa }^{\prime }[\kappa ,{\kappa }_3]+[\kappa ,{\kappa }^{\prime }]{\kappa }_3$

(ii) $[\kappa {\kappa }^{\prime },{\kappa }_3]=[\kappa ,{\kappa }_3]{\kappa }^{\prime }+\kappa [{\kappa }^{\prime },{\kappa }_3]$

(iii) $\kappa {\kappa }^{\prime }\circ {\kappa }_3=(\kappa \circ {\kappa }_3){\kappa }^{\prime }+\kappa [{\kappa }^{\prime },{\kappa }_3]=\kappa ({\kappa }^{\prime }\circ {\kappa }_3)-[\kappa ,{\kappa }_3]{\kappa }^{\prime }$

(iv) $\kappa \circ {\kappa }^{\prime }{\kappa }_3={\kappa }^{\prime }(\kappa \circ {\kappa }_3)+[\kappa ,{\kappa }^{\prime }]{\kappa }_3=(\kappa \circ {\kappa }^{\prime }){\kappa }_3+{\kappa }^{\prime }[{\kappa }_3,\kappa ].$

Daif and Bell [14] established that if a semiprime ring $\mathsf{ϝ}$ admits a derivation $\delta$ satisfying a specific condition, then ℑ must be a central ideal, there exists a nonzero ideal ℑ of $\mathsf{ϝ}$ such that either $\delta \left([\kappa ,{\kappa }^{\prime }]\right)=[\kappa ,{\kappa }^{\prime }]$ for all $\kappa ,{\kappa }^{\prime }\in \Im$ or $\delta \left([\kappa ,{\kappa }^{\prime }]\right)=-[\kappa ,{\kappa }^{\prime }]$ for all $\kappa ,{\kappa }^{\prime }\in \Im$. This result was extended for semiprime rings in [15].

Let S be a nonempty subset of $\mathsf{ϝ}$. A mapping from $\mathsf{ϝ}$ to $\mathsf{ϝ}$ is called commuting on S if $[\Delta (\kappa ),\kappa ]=0,$ for all $\kappa \in S$. Every additive commuting mapping $\Delta :\mathsf{ϝ}\to \mathsf{ϝ}$ gives rise to a bi-derivation on $\mathsf{ϝ}$. Namely, linearizing $[\Delta (\kappa ),\kappa ]=0$, we obtain $[\Delta \left(\kappa \right),{\kappa }^{\prime }]=[\kappa ,\Delta \left({\kappa }^{\prime }\right)]$, and we note that the map $(\kappa ,{\kappa }^{\prime })⟼[\Delta \left(\kappa \right),{\kappa }^{\prime }]$ is a bi-derivation.

$\Delta$ is called strong commutativity preserving (simply, SCP) on $\mathsf{ϝ}$ if $[\kappa ,{\kappa }^{\prime }]=[\Delta \left(\kappa \right),\Delta \left({\kappa }^{\prime }\right)]$ for all $\kappa ,{\kappa }^{\prime }\in \mathsf{ϝ}.$ Derivations and SCP mappings have been extensively examined in the study of operator algebras, prime rings, and semiprime rings. For further insights into SCP mappings, see [16,17,18] and the references therein. In [19], Ashraf and Rehman established that a prime ring $\mathsf{ϝ}$ with a nonzero ideal ℑ must be commutative if it admits a derivation $\delta$ satisfying either of the conditions: $\delta \left(\kappa {\kappa }^{\prime }\right)-\kappa {\kappa }^{\prime }\in Z\left(\mathsf{ϝ}\right)$ or $\delta \left(\kappa {\kappa }^{\prime }\right)-{\kappa }^{\prime }\kappa \in Z\left(\mathsf{ϝ}\right),$ the center of $\mathsf{ϝ}$ and all $\kappa ,{\kappa }^{\prime }\in \mathsf{ϝ}.$

Recently, some of the results obtained for prime or semiprime rings with derivations were obtained for any ring that is not prime or semiprime but contains a prime or semiprime ideal (see [20,21,22,23] and references therein).

In this article, we prove many algebraic identities that include the symmetric bi-derivation of any ring which contains a semiprime ideal. This work generalizes the results obtained for prime or semiprime rings with derivation and symmetric derivation in the literature.

2. Results

Theorem 1.

Let ϝ be a ring, Π a semiprime ideal of $\mathsf{ϝ} and\Im$ a nonzero ideal of ϝ which contains Π and $char(\mathsf{ϝ}/\Pi )\ne 2.$ Suppose that ϝ admits $\Delta :\mathsf{ϝ}\times \mathsf{ϝ}\to \mathsf{ϝ}$ a symmetric bi-derivation with a trace of δ such that for all $\kappa ,{\kappa }^{\prime }\in \Im ,$

(i) $\delta \left([\kappa ,{\kappa }^{\prime }]\right)\pm [\kappa ,{\kappa }^{\prime }]\in \Pi ,$

(ii) $\left[\delta \left(\kappa \right),\delta \left({\kappa }^{\prime }\right)\right]\pm \left[\kappa ,{\kappa }^{\prime }\right]\in \Pi ,$

then $\Im /\Pi$ is commutative.

Proof. 

(i) By the hypothesis, we obtain

$$\delta \left([\kappa ,{\kappa }^{\prime }]\right)\pm [\kappa ,{\kappa }^{\prime }]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Replacing ${\kappa }^{\prime }$ by ${\kappa }^{\prime }+{\kappa }_3,{\kappa }_3\in \Im$ in above expression, we have

$$\begin{array}{cc}\hfill \Pi & \ni \delta \left([\kappa ,{\kappa }^{\prime }+{\kappa }_3]\right)\pm [\kappa ,{\kappa }^{\prime }+{\kappa }_3]\hfill \\ \hfill & =\delta ([\kappa ,{\kappa }^{\prime }]+[\kappa ,{\kappa }_3])\pm [\kappa ,{\kappa }^{\prime }]\pm [\kappa ,{\kappa }_3]\hfill \\ \hfill & =\delta \left([\kappa ,{\kappa }^{\prime }]\right)+\delta \left([\kappa ,{\kappa }_3]\right)+2\Delta ([\kappa ,{\kappa }^{\prime }],[\kappa ,{\kappa }_3])\pm [\kappa ,{\kappa }^{\prime }]\pm [\kappa ,{\kappa }_3].\hfill \end{array}$$

Using the hypothesis, we obtain that

$$2\Delta ([\kappa ,{\kappa }^{\prime }],[\kappa ,{\kappa }_3])\in \Pi .$$

Since $char(\mathsf{ϝ}/\Pi )\ne 2,$ we have

$$\Delta ([\kappa ,{\kappa }^{\prime }],[\kappa ,{\kappa }_3])\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im .$$

Replacing ${\kappa }_3$ by ${\kappa }^{\prime }$ in last expression, we obtain

$$\Delta ([\kappa ,{\kappa }^{\prime }],[\kappa ,{\kappa }^{\prime }])\in \Pi$$

and so

$$\delta \left([\kappa ,{\kappa }^{\prime }]\right)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Using this in our hypothesis, we find that

$$[\kappa ,{\kappa }^{\prime }]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

(1)

Hence, we obtain

$$\kappa {\kappa }^{\prime }-{\kappa }^{\prime }\kappa \in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

(2)

and so

$$\kappa {\kappa }^{\prime }\equiv {\kappa }^{\prime }\kappa (mod\Pi ),\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Thus, we conclude that $\Im /\Pi$ is commutative. This completes the proof.

(ii) Let assume that

$$\left[\delta \left(\kappa \right),\delta \left({\kappa }^{\prime }\right)\right]\pm \left[\kappa ,{\kappa }^{\prime }\right]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

The linearization of this expression, we obtain

$$\left[\delta \left(\kappa \right),\delta \left({\kappa }^{\prime }\right)]+2[\delta \left(\kappa \right),\Delta ({\kappa }^{\prime },{\kappa }_3)]+[\delta \left(\kappa \right),\delta \left({\kappa }_3\right)\right]\pm \left[\kappa ,{\kappa }^{\prime }]\pm [\kappa ,{\kappa }_3\right]\in \Pi .$$

Using the hypothesis, we have

$$2[\delta \left(\kappa \right),\Delta ({\kappa }^{\prime },{\kappa }_3)]\in \Pi$$

Since $char(\mathsf{ϝ}/\Pi )\ne 2,$ we obtain that

$$[\delta \left(\kappa \right),\Delta ({\kappa }^{\prime },{\kappa }_3)]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im .$$

Taking ${\kappa }_3$ by ${\kappa }^{\prime }$ in this expression, we obtain

$$[\delta \left(\kappa \right),\Delta ({\kappa }^{\prime },{\kappa }^{\prime })]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

That is

$$[\delta \left(\kappa \right),\delta \left({\kappa }^{\prime }\right)]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Using this expression in our hypothesis, we have

$$\left[\kappa ,{\kappa }^{\prime }\right]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Applying similar reasoning as in the proof of $\left(i\right)$ following (1), the desired result follows. □

Theorem 2.

Let ϝ be a ring, Π a semiprime ideal of $\mathsf{ϝ} and\Im$ a nonzero ideal of ϝ which contains Π and $char(\mathsf{ϝ}/\Pi )\ne 2.$ Suppose that ϝ admits $\Delta :\mathsf{ϝ}\times \mathsf{ϝ}\to \mathsf{ϝ}$ a symmetric bi-derivation with a trace of δ such that for all $\kappa ,{\kappa }^{\prime }\in \Im ,$

(i) $\delta (\kappa \circ {\kappa }^{\prime })\pm \kappa \circ {\kappa }^{\prime }\in \Pi ,$

(ii) $\delta \left(\kappa \right)\circ \delta \left({\kappa }^{\prime }\right)\pm \kappa \circ {\kappa }^{\prime }\in \Pi ,$

then $\Im /\Pi$ is commutative.

Proof. 

(i) We obtain

$$\delta (\kappa \circ {\kappa }^{\prime })\pm \kappa \circ {\kappa }^{\prime }\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

(3)

The linearization of (3) gives us

$$\begin{array}{cc}\hfill \Pi & \ni \delta (\kappa \circ ({\kappa }^{\prime }+{\kappa }_3))\pm \kappa \circ ({\kappa }^{\prime }+{\kappa }_3)\hfill \\ \hfill & =\delta (\kappa \circ {\kappa }^{\prime }+\kappa \circ {\kappa }_3)\pm \kappa \circ {\kappa }^{\prime }\pm \kappa \circ {\kappa }_3\hfill \\ \hfill & =\delta (\kappa \circ {\kappa }^{\prime })+\delta (\kappa \circ {\kappa }_3)+2\Delta (\kappa \circ {\kappa }^{\prime },\kappa \circ {\kappa }_3)\pm \kappa \circ {\kappa }^{\prime }\pm \kappa \circ {\kappa }_3.\hfill \end{array}$$

Applying the hypotheses, we obtain that

$$\Delta (\kappa \circ {\kappa }^{\prime },\kappa \circ {\kappa }_3)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im .$$

Replacing ${\kappa }_3$ by ${\kappa }^{\prime },$ we have

$$\delta (\kappa \circ {\kappa }^{\prime })\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Using this in our hypothesis, we see that

$$\kappa \circ {\kappa }^{\prime }\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

(4)

Substituting ${\kappa }^{\prime }{\kappa }_3,{\kappa }_3\in \Im$ for ${\kappa }^{\prime }$ and using this expression, we find that

$$\kappa [{\kappa }^{\prime },{\kappa }_3]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im$$

(5)

and so

$$\kappa \mathsf{ϝ}[{\kappa }^{\prime },{\kappa }_3]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im ,\mathsf{ϝ}\in \mathsf{ϝ}.$$

(6)

Replacing $\kappa {\kappa }^{\prime }$ by $\kappa ,$ we have

$$\kappa {\kappa }^{\prime }\mathsf{ϝ}[{\kappa }^{\prime },{\kappa }_3]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im ,\mathsf{ϝ}\in \mathsf{ϝ}.$$

Left multiplying by ${\kappa }^{\prime }$ the expression (6), we obtain

$${\kappa }^{\prime }\kappa \mathsf{ϝ}[{\kappa }^{\prime },{\kappa }_3]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im ,\mathsf{ϝ}\in \mathsf{ϝ}.$$

Subtracting the last two expressions, we arrive at

$$[{\kappa }^{\prime },\kappa ]\mathsf{ϝ}[{\kappa }^{\prime },{\kappa }_3]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im ,\mathsf{ϝ}\in \mathsf{ϝ}$$

and so

$$[{\kappa }^{\prime },{\kappa }_3]\mathsf{ϝ}[{\kappa }^{\prime },{\kappa }_3]\subseteq \Pi ,\mathrm{for}\mathrm{all}{\kappa }^{\prime },{\kappa }_3\in \Im .$$

Since $\Pi$ is semiprime ideal of $\mathsf{ϝ},$ we find that

$$[{\kappa }^{\prime },{\kappa }_3]\in \Pi ,\mathrm{for}\mathrm{all}{\kappa }^{\prime },{\kappa }_3\in \Im .$$

Applying the same techniques after (1) in the proof of Theorem 1 (i), we obtain the required results.

(ii) We obtain

$$\delta \left(\kappa \right)\circ \delta \left({\kappa }^{\prime }\right)\pm \kappa \circ {\kappa }^{\prime }\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

(7)

Taking ${\kappa }^{\prime }$ by ${\kappa }^{\prime }+{\kappa }_3,{\kappa }_3\in \Im$ in (7) and using this, $char(\mathsf{ϝ}/\Pi )\ne 2,$ we see that

$$\delta \left(\kappa \right)\circ \delta \left({\kappa }^{\prime }\right)+\delta \left(\kappa \right)\circ \delta \left({\kappa }_3\right)+2(\delta \left(\kappa \right)\circ \Delta ({\kappa }^{\prime },{\kappa }_3))\pm \kappa \circ {\kappa }^{\prime }\pm \kappa \circ {\kappa }_3\in \Pi$$

and so

$$\delta \left(\kappa \right)\circ \Delta ({\kappa }^{\prime },{\kappa }_3)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im .$$

Replacing ${\kappa }_3$ by ${\kappa }^{\prime },$ we write

$$\delta \left(\kappa \right)\circ \delta \left({\kappa }^{\prime }\right)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

By our hypothesis, we obtain that

$$\kappa \circ {\kappa }^{\prime }\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Using the same arguments after (4) in the proof of (i), we obtain the required results. □

Theorem 3.

Let ϝ be a ring, Π a semiprime ideal of $\mathsf{ϝ} and\Im$ a nonzero ideal of ϝ which contains $\Pi$, $char(\mathsf{ϝ}/\Pi )\ne 2.$ Suppose that ϝ admits $\Delta :\mathsf{ϝ}\times \mathsf{ϝ}\to \mathsf{ϝ}$ a symmetric bi-derivation with a trace of δ such that for all $\kappa ,{\kappa }^{\prime }\in \Im ,$

(i) $\delta (\kappa \circ {\kappa }^{\prime })\pm [\kappa ,{\kappa }^{\prime }]\in \Pi ,$

(ii) $\delta \left(\kappa \right)\circ \delta \left({\kappa }^{\prime }\right)\pm [\kappa ,{\kappa }^{\prime }]\in \Pi ,$

then $\Im /\Pi$ is commutative.

Proof. 

(i) Let assume that

$$\delta (\kappa \circ {\kappa }^{\prime })\pm [\kappa ,{\kappa }^{\prime }]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Linearizing this expression, we have

$$\delta (\kappa \circ {\kappa }^{\prime })+\delta (\kappa \circ {\kappa }_3)+2\Delta (\kappa \circ {\kappa }^{\prime },\kappa \circ {\kappa }_3)\pm [\kappa ,{\kappa }^{\prime }]\pm [\kappa ,{\kappa }_3]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im .$$

Using the hypothesis and $char(\mathsf{ϝ}/\Pi )\ne 2,$ we obtain

$$\Delta (\kappa \circ {\kappa }^{\prime },\kappa \circ {\kappa }_3)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im .$$

Replacing ${\kappa }_3$ by ${\kappa }^{\prime }$ in this expression, we arrive at

$$\Delta (\kappa \circ {\kappa }^{\prime },\kappa \circ {\kappa }^{\prime })\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im$$

and so

$$\delta (\kappa \circ {\kappa }^{\prime })\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

This equation gives us

$$[\kappa ,{\kappa }^{\prime }]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

by our hypothesis. Applying the same lines after (1) in the proof of Theorem 1 (i), we obtain the required results.

(ii) We obtain

$$\delta \left(\kappa \right)\circ \delta \left({\kappa }^{\prime }\right)\pm [\kappa ,{\kappa }^{\prime }]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Replacing ${\kappa }^{\prime }$ by ${\kappa }^{\prime }+{\kappa }_3,{\kappa }_3\in \Im ,$ we have

$$\delta \left(\kappa \right)\circ \delta \left({\kappa }^{\prime }\right)+\delta \left(\kappa \right)\circ \delta \left({\kappa }_3\right)+2(\delta \left(\kappa \right)\circ \Delta ({\kappa }^{\prime },{\kappa }_3))\pm [\kappa ,{\kappa }^{\prime }]\pm [\kappa ,{\kappa }_3]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im .$$

By our hypotheses, we obtain that

$$\delta \left(\kappa \right)\circ \Delta ({\kappa }^{\prime },{\kappa }_3)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im .$$

Taking ${\kappa }_3$ by ${\kappa }^{\prime }$ in the last expression, we have

$$\delta \left(\kappa \right)\circ \delta \left({\kappa }^{\prime }\right)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Using this in our hypothesis, we obtain

$$[\kappa ,{\kappa }^{\prime }]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Applying the same techniques after (1) in the proof of Theorem 1 (i), we obtain the required results. □

Theorem 4.

Let ϝ be a ring, Π a semiprime ideal of $\mathsf{ϝ} and\Im$ a nonzero ideal of ϝ which contains Π and $char(\mathsf{ϝ}/\Pi )\ne 2.$ Suppose that ϝ admits $\Delta :\mathsf{ϝ}\times \mathsf{ϝ}\to \mathsf{ϝ}$ a symmetric bi-derivation with a trace of δ such that for all $\kappa ,{\kappa }^{\prime }\in \Im ,$

(i) $\delta \left([\kappa ,{\kappa }^{\prime }]\right)\pm \kappa \circ {\kappa }^{\prime }\in \Pi ,$

(ii) $\left[\delta \left(\kappa \right),\delta \left({\kappa }^{\prime }\right)\right]\pm \kappa \circ {\kappa }^{\prime }\in \Pi$

then $\Im /\Pi$ is commutative.

Proof. 

(i) We obtain

$$\delta \left([\kappa ,{\kappa }^{\prime }]\right)\pm \kappa \circ {\kappa }^{\prime }\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Linearizing this expression and using the hypothesis, we have

$$\delta \left([\kappa ,{\kappa }^{\prime }]\right)+\delta \left([\kappa ,{\kappa }^{\prime }]\right)+2\Delta ([\kappa ,{\kappa }^{\prime }],[\kappa ,{\kappa }_3])\pm \kappa \circ {\kappa }^{\prime }\pm \kappa \circ {\kappa }_3\in \Pi$$

and so

$$2\Delta ([\kappa ,{\kappa }^{\prime }],[\kappa ,{\kappa }_3])\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im .$$

Since $char(\mathsf{ϝ}/\Pi )\ne 2$ and taking ${\kappa }_3$ by ${\kappa }^{\prime }$ in above expression, we find that

$$\delta \left([\kappa ,{\kappa }^{\prime }]\right)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

and so

$$\kappa \circ {\kappa }^{\prime }\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

By the same ways after Equation (4) in the proof of Theorem 2 (i), we complete the proof.

(ii) Assume that

$$\left[\delta \left(\kappa \right),\delta \left({\kappa }^{\prime }\right)\right]\pm \kappa \circ {\kappa }^{\prime }\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Replacing ${\kappa }^{\prime }$ by ${\kappa }^{\prime }+{\kappa }_3$ and using this, $char(\mathsf{ϝ}/\Pi )\ne 2,$ we see that

$$[\delta \left(\kappa \right),\Delta ({\kappa }^{\prime },{\kappa }_3)]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im .$$

Taking ${\kappa }_3$ by ${\kappa }^{\prime }$ in this expression, we obtain

$$[\delta \left(\kappa \right),\Delta ({\kappa }^{\prime },{\kappa }^{\prime })]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im$$

and so

$$[\delta \left(\kappa \right),\delta \left({\kappa }^{\prime }\right)]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

From the hypothesis, we obtain that

$$\kappa \circ {\kappa }^{\prime }\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Applying the same techniques after (4) in the proof of Theorem 2 (i), we obtain the required results. □

Theorem 5.

Let ϝ be a ring, Π a semiprime ideal of $\mathsf{ϝ} and\Im$ a nonzero ideal of ϝ which contains Π and $char(\mathsf{ϝ}/\Pi )\ne 2.$ Suppose that ϝ admits $\Delta :\mathsf{ϝ}\times \mathsf{ϝ}\to \mathsf{ϝ}$ a symmetric bi-derivation with a trace of δ such that for all $\kappa ,{\kappa }^{\prime }\in \Im ,$

(i) $\delta \left(\kappa \right)\delta \left({\kappa }^{\prime }\right)\pm \kappa {\kappa }^{\prime }\in \Pi ,$

(ii) $\delta \left(\kappa \right)\delta \left({\kappa }^{\prime }\right)\pm {\kappa }^{\prime }\kappa \in \Pi ,$

then $\Im /\Pi$ is commutative.

Proof. 

(i) We have

$$\delta \left(\kappa \right)\delta \left({\kappa }^{\prime }\right)\pm \kappa {\kappa }^{\prime }\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Taking ${\kappa }^{\prime }$ by ${\kappa }^{\prime }+{\kappa }_3,{\kappa }_3\in \Im ,$ we obtain

$$\delta \left(\kappa \right)(\delta \left({\kappa }^{\prime }\right)+2\Delta ({\kappa }^{\prime },{\kappa }_3)+\delta \left({\kappa }_3\right))\pm \kappa ({\kappa }^{\prime }+{\kappa }_3)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im .$$

By the hypothesis, we see that

$$2\delta \left(\kappa \right)\Delta ({\kappa }^{\prime },{\kappa }_3)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im .$$

Since $char(\mathsf{ϝ}/\Pi )\ne 2,$ we have

$$\delta \left(\kappa \right)\Delta ({\kappa }^{\prime },{\kappa }_3)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im .$$

Replacing ${\kappa }_3$ by ${\kappa }^{\prime }$ in the last expression, we obtain

$$\delta \left(\kappa \right)\delta \left({\kappa }^{\prime }\right)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

By the hypothesis, we arrive at

$$\kappa {\kappa }^{\prime }\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im$$

(8)

and so

$${\kappa }^{\prime }\kappa \in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Hence, we obtain that

$$[\kappa ,{\kappa }^{\prime }]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .,$$

Thus, we find that the $\Im /\Pi$ commutative ring by the same ways after (1) in the proof of Theorem 1 (i). This completes the proof.

(ii) We obtain

$$\delta \left(\kappa \right)\delta \left({\kappa }^{\prime }\right)\pm {\kappa }^{\prime }\kappa \in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Linearizing this expression and using the hypotheses, we have

$$\delta \left(\kappa \right)\Delta ({\kappa }^{\prime },{\kappa }_3)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im .$$

Replacing ${\kappa }_3$ by ${\kappa }^{\prime }$ in the last expression, we obtain

$$\delta \left(\kappa \right)\delta \left({\kappa }^{\prime }\right)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Using this in our hypothesis, we find that

$${\kappa }^{\prime }\kappa \in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Using the same arguments after (8) in the proof of (i), we obtain the required result. □

Theorem 6.

Let ϝ be a ring, Π a semiprime ideal of $\mathsf{ϝ},and\Im$ a nonzero ideal of ϝ which contains Π and $char(\mathsf{ϝ}/\Pi )\ne 2.$ Suppose that ϝ admits $\Delta :\mathsf{ϝ}\times \mathsf{ϝ}\to \mathsf{ϝ}$ a symmetric bi-derivation with a trace of δ such that for all $\kappa ,{\kappa }^{\prime }\in \Im ,$

(i) $\delta \left(\kappa \right)\delta \left({\kappa }^{\prime }\right)\pm [\kappa ,{\kappa }^{\prime }]\in \Pi ,$

(ii) $\delta \left(\kappa \right)\delta \left({\kappa }^{\prime }\right)\pm \kappa \circ {\kappa }^{\prime }\in \Pi$

then $\Im /\Pi$ is commutative.

Proof. 

(i) We obtain

$$\delta \left(\kappa \right)\delta \left({\kappa }^{\prime }\right)\pm [\kappa ,{\kappa }^{\prime }]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Taking ${\kappa }^{\prime }$ by ${\kappa }^{\prime }+{\kappa }_3,{\kappa }_3\in \Im$ in this expression and using this, we arrive at

$$\delta \left(\kappa \right)(\delta \left({\kappa }^{\prime }\right)+2\Delta ({\kappa }^{\prime },{\kappa }_3)+\delta \left({\kappa }_3\right))\pm [\kappa ,{\kappa }^{\prime }]\pm [\kappa ,{\kappa }_3]\in \Pi .$$

$$2\delta \left(\kappa \right)\Delta ({\kappa }^{\prime },{\kappa }_3)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im .$$

Since $char(\mathsf{ϝ}/\Pi )\ne 2,$ we obtain

$$\delta \left(\kappa \right)\Delta ({\kappa }^{\prime },{\kappa }_3)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im .$$

Replacing ${\kappa }_3$ by ${\kappa }^{\prime }$ in the last expression, we see that

$$\delta \left(\kappa \right)\delta \left({\kappa }^{\prime }\right)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

By the hypothesis, we arrive at

$$[\kappa ,{\kappa }^{\prime }]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

This gives that $\Im /\Pi$ commutative ring as above.

(ii) Assume that

$$\delta \left(\kappa \right)\delta \left({\kappa }^{\prime }\right)\pm \kappa \circ {\kappa }^{\prime }\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Linearizing this expression and using this, $char(\mathsf{ϝ}/\Pi )\ne 2,$ we arrive at

$$\delta \left(\kappa \right)\Delta ({\kappa }^{\prime },{\kappa }_3)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im .$$

Writing ${\kappa }_3$ by ${\kappa }^{\prime }$ in the last expression, we find that

$$\delta \left(\kappa \right)\delta \left({\kappa }^{\prime }\right)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Using this expression in our hypothesis, we arrive at

$$\kappa \circ {\kappa }^{\prime }\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

We conclude that $\Im /\Pi$ is a commutative ring using the same lines in the proof of Theorem 2. □

Theorem 7.

Let ϝ be a ring, Π a semiprime ideal of $\mathsf{ϝ} and\Im$ a nonzero ideal of ϝ which contains Π and $char(\mathsf{ϝ}/\Pi )\ne 2.$ Suppose that ϝ admits $\Delta :\mathsf{ϝ}\times \mathsf{ϝ}\to \mathsf{ϝ}$ a symmetric bi-derivation with a trace of δ such that for all $\kappa ,{\kappa }^{\prime }\in \Im ,$

(i) $\delta \left(\kappa {\kappa }^{\prime }\right)\pm \kappa {\kappa }^{\prime }\in \Pi ,$

(ii) $\delta \left(\kappa {\kappa }^{\prime }\right)\pm {\kappa }^{\prime }\kappa \in \Pi ,$

then $\Im /\Pi$ is commutative.

Proof. 

(i) By the hypothesis, we obtain

$$\delta \left(\kappa {\kappa }^{\prime }\right)\pm \kappa {\kappa }^{\prime }\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Taking ${\kappa }^{\prime }$ by ${\kappa }^{\prime }+{\kappa }_3,$${\kappa }_3\in \Im$ in this, we see that

$$\delta \left(\kappa {\kappa }^{\prime }\right)+\delta \left(\kappa {\kappa }_3\right)+2\Delta (\kappa {\kappa }^{\prime },\kappa {\kappa }_3)\pm \kappa {\kappa }^{\prime }\pm \kappa {\kappa }_3\in \Pi .$$

Using our hypothesis and $char(\mathsf{ϝ}/\Pi )\ne 2,$ we arrive at

$$\Delta (\kappa {\kappa }^{\prime },\kappa {\kappa }_3)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

(9)

Substituting ${\kappa }^{\prime }$ for ${\kappa }_3$ in the last expression, we have

$$\delta \left(\kappa {\kappa }^{\prime }\right)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Again, using the hypothesis, we find that

$$\kappa {\kappa }^{\prime }\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

By the similar arguments after (8), we obtain the required result.

(ii) Let us assume

$$\delta \left(\kappa {\kappa }^{\prime }\right)\pm {\kappa }^{\prime }\kappa \in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Using the linearization of this expression, we obtain

$$\delta \left(\kappa {\kappa }^{\prime }\right)+\delta \left(\kappa {\kappa }_3\right)+2\Delta (\kappa {\kappa }^{\prime },\kappa {\kappa }_3)\pm ({\kappa }^{\prime }+{\kappa }_3)\kappa \in \Pi .$$

Using the hypothesis and $char(\mathsf{ϝ}/\Pi )\ne 2,$ we obtain that

$$\Delta (\kappa {\kappa }^{\prime },\kappa {\kappa }_3)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im .$$

This expression is same as (9) in (i). Applying the same line as above, we conclude our result. □

Theorem 8.

Let ϝ be a ring, Π a semiprime ideal of $\mathsf{ϝ} and\Im$ a nonzero ideal of ϝ which contains Π and $char(\mathsf{ϝ}/\Pi )\ne 2.$ Suppose that ϝ admits $\Delta :\mathsf{ϝ}\times \mathsf{ϝ}\to \mathsf{ϝ}$ a symmetric bi-derivation with a trace of δ such that for all $\kappa ,{\kappa }^{\prime }\in \Im ,$

(i) $\delta \left(\kappa {\kappa }^{\prime }\right)\pm [\kappa ,{\kappa }^{\prime }]\in \Pi ,$

(ii) $\delta \left(\kappa {\kappa }^{\prime }\right)\pm \kappa \circ {\kappa }^{\prime }\in \Pi ,$

then $\Im /\Pi$ is commutative.

Proof. 

(i) We obtain

$$\delta \left(\kappa {\kappa }^{\prime }\right)\pm [\kappa ,{\kappa }^{\prime }]\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Replacing ${\kappa }^{\prime }$ by ${\kappa }^{\prime }+{\kappa }_3,{\kappa }_3\in \Im ,$ we have

$$\delta \left(\kappa {\kappa }^{\prime }\right)+\delta \left(\kappa {\kappa }_3\right)+2\Delta (\kappa {\kappa }^{\prime },\kappa {\kappa }_3)\pm [\kappa ,{\kappa }^{\prime }]\pm [\kappa ,{\kappa }_3]\in \Pi .$$

Applying our hypothesis, we arrive at

$$\Delta (\kappa {\kappa }^{\prime },\kappa {\kappa }_3)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im .$$

Using the same lines after (9), we complete the proof.

(ii) We assume that

$$\delta \left(\kappa {\kappa }^{\prime }\right)\pm \kappa \circ {\kappa }^{\prime }\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

By the linearization on ${\kappa }^{\prime },$ we obtain

$$\delta \left(\kappa {\kappa }^{\prime }\right)+\delta \left(\kappa {\kappa }_3\right)+2\Delta (\kappa {\kappa }^{\prime },\kappa {\kappa }_3)\pm \kappa \circ {\kappa }^{\prime }\pm \kappa \circ {\kappa }_3\in \Pi ,$$

and so

$$\Delta (\kappa {\kappa }^{\prime },\kappa {\kappa }_3)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im .$$

Appliying the same ways after (9), we obtain our result. □

Theorem 9.

Let ϝ be a ring, Π a semiprime ideal of $\mathsf{ϝ} and\Im$ a nonzero ideal of ϝ which contains Π and $char(\mathsf{ϝ}/\Pi )\ne 2.$ Suppose that ϝ admits $\Delta ,G:\mathsf{ϝ}\times \mathsf{ϝ}\to \mathsf{ϝ}$ symmetric bi-derivations with traces of $\delta ,g$ respectively, such that for all $\kappa ,{\kappa }^{\prime }\in \Im ,$

(i) $g\left(\kappa {\kappa }^{\prime }\right)+\delta \left(\kappa \right)\delta \left({\kappa }^{\prime }\right)\pm \kappa {\kappa }^{\prime }\in \Pi ,$

(ii) $g\left(\kappa {\kappa }^{\prime }\right)+\delta \left(\kappa \right)\delta \left({\kappa }^{\prime }\right)\pm {\kappa }^{\prime }\kappa \in \Pi ,$

then $\Im /\Pi$ is commutative.

Proof. 

(i) By our hypothesis, we obtain

$$g\left(\kappa {\kappa }^{\prime }\right)+\delta \left(\kappa \right)\delta \left({\kappa }^{\prime }\right)\pm \kappa {\kappa }^{\prime }\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Replacing ${\kappa }^{\prime }$ by ${\kappa }^{\prime }+{\kappa }_3,$${\kappa }_3\in \Im ,$ we arrive at

$$\begin{array}{c}g\left(\kappa {\kappa }^{\prime }\right)+g\left(\kappa {\kappa }_3\right)+2G(\kappa {\kappa }^{\prime },\kappa {\kappa }_3)+\delta \left(\kappa \right)(\delta \left({\kappa }^{\prime }\right)\hfill \\ +2\Delta ({\kappa }^{\prime },{\kappa }_3)+\delta \left({\kappa }_3\right))\pm \kappa {\kappa }^{\prime }\pm \kappa {\kappa }_3\in \Pi .\hfill \end{array}$$

Using the hypothesis and $char(\mathsf{ϝ}/\Pi )\ne 2,$ we have

$$G(\kappa {\kappa }^{\prime },\kappa {\kappa }_3)+\delta \left(\kappa \right)\Delta ({\kappa }^{\prime },{\kappa }_3)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Writing ${\kappa }_3$ by ${\kappa }^{\prime }$ in this expression, we arrive at

$$g\left(\kappa {\kappa }^{\prime }\right)+\delta \left(\kappa \right)\delta \left({\kappa }^{\prime }\right)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Again using our hypothesis, we obtain

$$\kappa {\kappa }^{\prime }\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Using the same arguments after (8), we obtain the required result.

(ii) We obtain

$$g\left(\kappa {\kappa }^{\prime }\right)+\delta \left(\kappa \right)\delta \left({\kappa }^{\prime }\right)\pm {\kappa }^{\prime }\kappa \in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Linearizing this expression on ${\kappa }^{\prime }$ and using the hypothesis, we obtain that

$$\begin{array}{c}g\left(\kappa {\kappa }^{\prime }\right)+g\left(\kappa {\kappa }_3\right)+2G(\kappa {\kappa }^{\prime },\kappa {\kappa }_3)+\delta \left(\kappa \right)(\delta \left({\kappa }^{\prime }\right)\\ +2\Delta ({\kappa }^{\prime },{\kappa }_3)+\delta \left({\kappa }_3\right))\pm ({\kappa }^{\prime }+{\kappa }_3)\kappa \in \Pi .\end{array}$$

and so

$$2G(\kappa {\kappa }^{\prime },\kappa {\kappa }_3)+2\delta \left(\kappa \right)\Delta ({\kappa }^{\prime },{\kappa }_3)\in \Pi .$$

Since $char(\mathsf{ϝ}/\Pi )\ne 2,$ we have

$$G(\kappa {\kappa }^{\prime },\kappa {\kappa }_3)+\delta \left(\kappa \right)\Delta ({\kappa }^{\prime },{\kappa }_3)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime },{\kappa }_3\in \Im .$$

Replacing ${\kappa }_3$ by ${\kappa }^{\prime }$ in the last expression, we obtain

$$g\left(\kappa {\kappa }^{\prime }\right)+\delta \left(\kappa \right)\delta \left({\kappa }^{\prime }\right)\in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

Again using our hypothesis, we see that

$${\kappa }^{\prime }\kappa \in \Pi ,\mathrm{for}\mathrm{all}\kappa ,{\kappa }^{\prime }\in \Im .$$

By the same ways after (8), we obtain our result. □

3. Open Problems

In this study, the subject of symmetric bi-derivations is discussed on noncommutative rings. Similarly, the subject of symmetric bi-derivations can be discussed on different structures. One of these topics is the subject of non-associative structures. Many authors have been investigating this subject in recent years (for example [24]). The subject of symmetric derivations can also be discussed on the structure of alternative rings.

4. Conclusions

In this work, we study the symmetric bi-derivations of a ring under the influence of a semiprime ideal without imposing any conditions on the ring. Furthermore, this work is one of the first to address this issue on symmetric derivations. The conditions examined here are those that exist in the literature used to prove that a ring is commutative. The findings contribute to the broader landscape of commutative theorems, providing new insights into the structural properties of rings with derivations. This research opens the way for further research on different algebraic structures, such as alternative rings, near rings, operator algebras, Banach algebras, non-commutative geometry, and other areas where ring theory plays a fundamental role.