Generalized (τ, σ)-L-Derivations in Rings

Abstract

Let $\tau$ and $\sigma :X⟶X$ be automorphisms of an arbitrary associative ring X, and let L be a prime ideal of X. The main objective of this article is to combine the notions of generalized L-derivations and $(\tau ,\sigma )$-L-derivations by introducing and analyzing a novel additive mapping $\Pi :X\to X$ called a generalized $(\tau ,\sigma )$-L-derivation associated with a $(\tau ,\sigma )$-L-derivation $\pi$. Later, we will examine the algebraic properties of a factor ring $X/L$ under the influence of certain algebraic expressions containing this generalized $(\tau ,\sigma )$-L-derivation and lying in a prime ideal L. Through our main findings, we establish certain results under different conditions. It also provides various illustrative examples to show that our primeness hypotheses in various theorems are not exaggerated.

1. Introduction

Everywhere in this article, let X represent an associative ring with the center denoted by $Z\left(X\right)$. A ring is classified as a prime ring if for any elements $w,𝚥\in X$, the condition $wX𝚥=\left\{0\right\}$ implies that $w=0$ or $𝚥=0$, and semiprime if for any $w\in X$, $wXw=\left\{0\right\}$ implies that $w=0$. Equivalently, a ring X is classified as prime if for any two nonzero ideals $M,N$ in X, then $MN\ne \left\{0\right\}$, and semiprime if for any ideal M in X such that $M^2=\left\{0\right\}$, then $M=\left\{0\right\}$. Extending this notion, a proper ideal L is said to be prime if and only if for some $w,𝚥\in X$, $wX𝚥\subseteq L$, then either $w\in L$ or $𝚥\in L$. Equivalently, a proper ideal L in X is classified as prime if $MN\subseteq L$ for some ideals $M,N$ in X, then either $M\subseteq L$ or $N\subseteq L$. Furthermore, a ring is termed an integral domain if it is a commutative ring with unity and is devoid of zero divisors. For more details and examples of the previous concepts, please refer to the relevant literature. A Lie product is defined for some $w,𝚥\in X$ as $[w,𝚥]=w𝚥-𝚥w$. A ring X is said to be commutative if and only if the Lie product equals zero. A Jordan product is defined for some $w,𝚥\in X$ as $w\circ 𝚥=w𝚥+𝚥w$. The characteristic of a ring X is said to be different from 2 if for some $w\in X$, $2w=0$, then $w=0$, usually symbolized by $char\left(X\right)\ne 2$. In the case of $char\left(X\right)=2$, we can notice that the Jordan product $w\circ 𝚥=w𝚥+𝚥w=w𝚥+𝚥w-w𝚥+𝚥w=[w,𝚥]+2𝚥w$. Since $2𝚥w=0$, the last relation implies that $w\circ 𝚥=[w,𝚥]$ for all $w,𝚥\in X$. For all $w,𝚥,q\in X$, the following identities are valid and will be utilized implicitly throughout the current paper to aid in the proofs of our results:

$$\begin{gathered} [w𝚥,q]=w[𝚥,q]+[w,q]𝚥, \\ [w,𝚥q]=𝚥[w,q]+[w,𝚥]q, \\ w\circ (𝚥q)=(w\circ 𝚥)q-𝚥[w,q]=𝚥(w\circ q)+[w,𝚥]q, \\ (w𝚥)\circ q=w(𝚥\circ q)-[w,q]𝚥=(w\circ q)𝚥+w[𝚥,q]. \end{gathered}$$

A standard derivation $\pi :X⟶X$ is an additive mapping satisfying the condition $\pi (w𝚥)=\pi \left(w\right)𝚥+w\pi (𝚥)$ for every $w,𝚥\in X$. For a fixed $r\in X$, a map ${\pi }_r:X⟶X$ defined by ${\pi }_r\left(w\right)=[r,w]$ for all $w\in X$ is a derivation on X called an inner derivation induced by r. A generalized derivation $\Pi$ associated with a standard derivation $\pi$ is defined as an additive mapping from a ring X to itself, satisfying $\Pi (w𝚥)=\Pi \left(w\right)𝚥+w\pi (𝚥)$ for every $w,𝚥\in X$. In fact, the concept of derivations plays a crucial role, alongside other concepts such as fractional derivatives (including q-derivatives), in the generalization of theory of classical differentiation and its applications in various mathematical and physical contexts. For more details, please see [1,2].

By imposing two endomorphisms $\tau$ and $\sigma$ on X, a generalization of the concept of a derivation has been introduced as an $(\tau ,\sigma )$-derivation. This is an additive mapping $\pi$ of X that fulfills the equation $\pi (w𝚥)=\pi \left(w\right)\tau (𝚥)+\sigma \left(w\right)\pi (𝚥)$ for all $w,𝚥\in X$. Additionally, an additive mapping $\Pi$ is known as a generalized $(\tau ,\sigma )$-derivation if there exists an $(\tau ,\sigma )$-derivation $\pi$ such that $\Pi (w𝚥)=\Pi \left(w\right)\tau (𝚥)+\sigma \left(w\right)\pi (𝚥)$ for all $w,𝚥\in X$. It is important to note that if $\tau =\sigma =i{d}_X$, where $i{d}_X:X\to X$ is an identity mapping defined as $i{d}_X\left(w\right)=w$ for all $w\in X$, then the notion of a generalized derivation is obtained. This implies that the notion of a generalized $(\tau ,\sigma )$-derivation encompasses the notion of a generalized derivation. In recent decades, numerous researchers have examined the behavior of prime and semi-prime rings using certain equations that include appropriate additive mappings. These mappings can operate on the entire ring or on suitable subsets of it. For further exploration, examples and counterexamples can be found in [3,4,5], and their bibliographies. Instead of solely focusing on the primeness or semi-primeness of a ring X in previous studies, several researchers have opted to investigate a more general context by imposing a prime ideal $L\ne \left\{0\right\}$ of an arbitrary ring X. They then analyze the algebraic properties of the factor ring $X/L$ under the impact of various types of additive mappings that satisfy multiple algebraic identities involving a prime ideal L. For more detailed information on these findings, please refer to [6,7].

An L-derivation is a generalization of the notion of a derivation, defined as a map from X to X that satisfies the following properties: $\left(i\right)$ $\pi (w+𝚥)-\pi \left(w\right)-\pi (𝚥)\in L$ (L-additive), $\left(ii\right)$ $\pi (w𝚥)-\pi \left(w\right)𝚥-w\pi (𝚥)\in L$ for all $w,𝚥\in X$. A generalized L-derivation $\Pi$ associated with an L-derivation $\pi$ is defined as a map from X to X satisfying the following conditions for all $w,𝚥\in X$: $\left(i\right)$ $\Pi (w+𝚥)-(\Pi (w)+\Pi (𝚥\left)\right)\in L$, $\left(ii\right)$ $\Pi (w𝚥)-(\Pi (w)𝚥+w\pi (𝚥\left)\right)\in L$, where L is a prime ideal of X. It is clear that every generalized derivation is a generalized L-derivation, but the converse is not generally valid. A counterexample can be found in [8]. In fact, the equivalence between these two notions is achieved when $L=\left\{0\right\}$.

In their article [9], Mohssine et al. introduced the concept of an $(\alpha ,\tau )$-L-derivation $\pi$, where $\alpha$ and $\tau$ are automorphisms of a near-ring ℵ. They examined the effects of $\pi$ on the behavior of a factor near-ring $\aleph /L$ when $\pi$ satisfies certain algebraic identities involving an ideal L of ℵ. A year later, in [8], Sandhu et al. studied the relationship between a factor ring $X/L$ and certain differential identities involving an L-derivation, a generalized L-derivation, and an L-left multiplier. Recently, Hummdi et al. in [10] continued these studies with some differential identities involving a pair of generalized L-derivations $\Pi$ and $\mathsf{\Omega }$ associated with L-derivations $\pi$ and ð, respectively.

In light of the above, the study of the properties of algebraic structures of rings, whether they are prime, semi-prime, or factor rings, has become increasingly important through various generalized forms of derivations. The question that arises is the following: Can we extend the scope of these derivations by defining one that encompasses previous definitions, thereby automatically including previous studies? The answer to this question will be addressed in this article.

Encouraged by the notions of generalized L-derivations and $(\tau ,\sigma )$-L-derivations, this article introduces a novel additive mapping $\Pi :X⟶X$ called a generalized $(\tau ,\sigma )$-L-derivation associated with a $(\tau ,\sigma )$-L-derivation $\pi$, where $\tau$ and $\sigma$ are endomorphisms of a ring X. We then relate a ring X to a prime ideal L by imposing certain algebraic identities that ensure a generalized $(\tau ,\sigma )$-L-derivation $\Pi$, and investigate the relationship between the algebraic structure of a factor ring $X/L$. Furthermore, we explore various outcomes that arise under specific constraints. In addition, we provide a list of illustrative examples to demonstrate the necessity of our hypothesis in various theorems.

2. Preliminary Results

Since every ring is a near-ring, we can easily define the following with the aid of ([9], Definition 1):

Definition 1.

Let L be a prime ideal of a ring X, and let τ and $\sigma :X⟶X$ be endomorphisms of X. A mapping $\pi :X⟶X$ is said to be an $(\tau ,\sigma )$-L-derivation if the following criteria are fulfilled for all $w,𝚥\in X$:

$\left(i\right)$ $\pi (w+𝚥)-\pi \left(w\right)-\pi (𝚥)\in L$, and $\left(ii\right)$ $\pi (w𝚥)-\pi \left(w\right)\tau (𝚥)-\sigma \left(w\right)\pi (𝚥)\in L$.

To further explore the properties of the algebraic structure of a factor ring $X/L$ within the scope of our current study, we extend Definition 1 to encompass a generalized $(\tau ,\sigma )$-L-derivation associated with an $(\tau ,\sigma )$-L-derivation.

Definition 2.

Let L be a prime ideal of a ring X, and let τ and $\sigma :X⟶X$ be endomorphisms of X. A mapping $\Pi :X⟶X$ is said to be a generalized $(\tau ,\sigma )$-L-derivation if there exists an $(\tau ,\sigma )$-L-derivation $\pi :X⟶X$ such that the following criteria are fulfilled for all $w,𝚥\in X$:

$\left(A\right)$ $\Pi (w+𝚥)-\Pi \left(w\right)-\Pi (𝚥)\in L$, and $\left(B\right)$ $\Pi (w𝚥)-\Pi \left(w\right)\tau (𝚥)-\sigma \left(w\right)\pi (𝚥)\in L$.

Next, we will explore examples that illustrate the existence and comprehensiveness of the previous definition. But before we do that, let us introduce the following definition:

Definition 3.

Following [11,12], let $K_{2^n}=〈x_1,\dots ,x_n〉$ be a set with n-generator elements $x_1,\dots ,x_n$. This set consists of $2^n$ elements, i.e., $|K_{2^n}|=2^n$. In the following, the set $x_{g_i}$, where $i=1,\dots ,n$ will denote the generator elements of $K_{2^n}$. To define the operations of addition and multiplication on $K_{2^n}$, we consider the set of elements $x_{s_j}={\displaystyle \sum _{i=1}^{s_j}}{x}_{g_i}$, where $j=1,\dots ,n$, which are generated by $x_{g_i}$, where $i=1,\dots ,n$. This precisely defines the additive operation on $K_{2^n}$. For instance, if $n=2$, then $x_{g_i}=a,b$ where $i=1,2$ and $x_{s_j}=a+b=c,a+a=b+b=0$, so $|K_{2^2}|=|\langle a,b\rangle |=4$, which correspond to the ring $K_{2^2}$ in ([11], Example 2.1). Now, let us categorize two types of elements that generate $x_{s_j}$, namely when the number of elements that generate $x_{s_j}$ is even or odd. We will denote them as $x_{e_i}$ and $x_{o_i}$, respectively. The multiplication operation on $K_{2^n}$ for any $x_{g_i}$, $x_{s_j}$ is defined as follows:

$$\begin{array}{c}\hfill x_{g_i}·x_{s_j}=\left\{\begin{array}{ccc}0& if{x}_{s_j}=x_{e_i};\hfill & \\ \hfill x_{g_i}& if{x}_{s_j}=x_{o_i}.\hfill \end{array}\right.\end{array}$$

For further clarification, the preceding operations of addition and multiplication on $K_{2^n}$ can be represented via the following Cayley tables (

Table 1. Table of additive.

+	0	$x_1$	…	$x_n$	$x_{e_1}$	…	$x_{e_i}$	$x_{o_1}$	…	$x_{o_i}$
0	0	$x_1$	…	$x_n$	$x_{e_1}$	…	$x_{e_i}$	$x_{o_1}$	…	$x_{o_i}$
$x_1$	$x_1$	0	…	$x_{e_{m_1}}$	$x_{o_{j_1}}$	…	$x_{o_{t_1}}$	$x_{e_{i_1}}$	…	$x_{e_{t_1}}$
⋮	⋮	⋮	⋱	⋮	⋮	⋱	⋮	⋮	⋱	⋮
$x_n$	$x_n$	$x_{e_{m_1}}$	…	0	$x_{o_{j_n}}$	…	$x_{o_{t_n}}$	$x_{e_{i_n}}$	…	$x_{e_{t_n}}$
$x_{e_1}$	$x_{e_1}$	$x_{o_{j_1}}$	…	$x_{o_{j_n}}$	0	…	$x_{e_{s_1}}$	$x_{o_{p_1}}$	…	$x_{o_{h_1}}$
⋮	⋮	⋮	⋱	⋮	⋮	⋱	⋮	⋮	⋱	⋮
$x_{e_i}$	$x_{e_i}$	$x_{o_{t_1}}$	…	$x_{o_{t_n}}$	$x_{e_{s_1}}$	…	0	$x_{o_{p_n}}$	…	$x_{o_{h_n}}$
$x_{o_1}$	$x_{o_1}$	$x_{e_{i_1}}$	…	$x_{e_{i_n}}$	$x_{o_{p_1}}$	…	$x_{o_{p_n}}$	0	…	$x_{e_{v_1}}$
⋮	⋮	⋮	⋱	⋮	⋮	⋱	⋮	⋮	⋱	⋮
$x_{o_i}$	$x_{o_i}$	$x_{e_{t_1}}$	…	$x_{e_{t_n}}$	$x_{o_{h_1}}$	…	$x_{o_{h_n}}$	$x_{e_{v_1}}$	…	0

and

Table 2. Table of multiplicative.

$.$	0	$x_1$	…	$x_n$	$x_{e_1}$	…	$x_{e_i}$	$x_{o_1}$	…	$x_{o_i}$
0	0	0	…	0	0	…	0	0	…	0
$x_1$	0	$x_1$	…	$x_1$	0	…	0	$x_1$	…	$x_1$
⋮	⋮	⋮	⋱	⋮	⋮	⋱	⋮	⋮	⋱	⋮
$x_n$	0	$x_n$	…	$x_n$	0	…	0	$x_n$	…	$x_n$
$x_{e_1}$	0	$x_{e_1}$	…	$x_{e_1}$	0	…	0	$x_{e_1}$	…	$x_{e_1}$
⋮	⋮	⋮	⋱	⋮	⋮	⋱	⋮	⋮	⋱	⋮
$x_{e_i}$	0	$x_{e_i}$	…	$x_{e_i}$	0	…	0	$x_{e_i}$	…	$x_{e_i}$
$x_{o_1}$	0	$x_{o_1}$	…	$x_{o_1}$	0	…	0	$x_{o_1}$	…	$x_{o_1}$
⋮	⋮	⋮	⋱	⋮	⋮	⋱	⋮	⋮	⋱	⋮
$x_{o_i}$	0	$x_{o_i}$	…	$x_{o_i}$	0	…	0	$x_{o_i}$	…	$x_{o_i}$

). The indices $i,j,m,t,s,p,v$ and h are merely subscripts used to distinguish between different elements in $K_{2^n}$.

Then $(K_{2^n},+,.)$ is an algebra over the field ${\mathbb{Z}}_2$. This algebra is a non-commutative ring (since $x_{o_i}{x}_{e_i}=0$ while $x_{e_i}{x}_{o_i}=x_{e_i}$) without the multiplicative identity 1. Furthermore, $K_{2^n}$ is not a prime and the characteristic of $K_{2^n}$ is 2. Also, every $x_{o_i}$ is a left zero divisor, and every $x_{e_i}$ is a two-sided zero divisor. Consequently, $K_{2^n}$ is not integral domain. Moreover, we can note that the center of the ring $K_{2^n}$ is equal to zero and $K_{2^n}$ is not reduced (it has non-zero nilpotent elements, for instance $x_{e_{i_1}}{x}_{e_{i_1}}=0$).

Example 1.

Let $K_{2^n}$ be as above, and let $L=\left\{x_{e_i}\right\}$, where $i=1,\dots ,n$. Define $\Pi ,\pi ,\tau$ and $\sigma :K_{2^n}⟶K_{2^n}$ as follows:

$$\begin{array}{c}\hfill \tau \left(w\right)=\left\{\begin{array}{lll}0& ifw=x_{e_i},& \\ x_h,\mathit{for}\mathit{fixed}h& ifw=x_{o_i}\end{array}\right.;\sigma \left(w\right)=\left\{\begin{array}{lll}0& ifw=x_{e_i},& \\ x_m,\mathit{for}\mathit{fixed}m& ifw=x_{o_i}.\end{array}\right.\end{array}$$

and

$$\begin{array}{c}\hfill \Pi \left(w\right)=\left\{\begin{array}{lll}0& ifw=x_{e_i},& \\ x_t,\mathit{for}\mathit{fixed}t& ifw=x_{o_i}\end{array}\right.;\pi \left(w\right)=\left\{\begin{array}{lll}0& ifw=x_{e_i},& \\ x_{2q},\mathit{for}\mathit{fixed}q& ifw=x_{o_i}.\end{array}\right.\end{array}$$

Then it is easy to check that L is a prime ideal in $K_{2^n}$ and τ, σ are endomorphisms of $K_{2^n}$. Clearly, Π and π are L-additive mappings of $K_{2^n}$. Therefore, to prove that $(\Pi ,\pi )$ is a generalized $(\tau ,\sigma )$-L-derivation of $K_{2^n}$, it is sufficient to show that condition $\left(B\right)$ is satisfied in all the following cases for all $w,𝚥\in \{x_1,\dots ,x_n,x_{e_1},\dots ,x_{e_i},x_{o_1},\dots ,x_{o_i}\}$, where $i=1,\dots ,n$:

(a)

If $w=x_{e_i}$ and $𝚥=x_{e_i}$, then

$$\begin{array}{ccc}\hfill \Pi (w𝚥)-\Pi \left(w\right)\tau (𝚥)-\sigma \left(w\right)\pi (𝚥)& =& \Pi \left(x_{e_i}{x}_{e_i}\right)-\Pi \left(x_{e_i}\right)\tau \left(x_{e_i}\right)-\sigma \left(x_{e_i}\right)\pi \left(x_{e_i}\right)\hfill \\ & =& \Pi \left(0\right)-00-00\hfill \\ & =& 0\in L.\hfill \end{array}$$

(b)

If $w=x_{e_i}$ and $𝚥\ne x_{e_i}$, then

$$\begin{array}{ccc}\hfill \Pi (w𝚥)-\Pi \left(w\right)\tau (𝚥)-\sigma \left(w\right)\pi (𝚥)& =& \Pi (x_{e_i}𝚥)-\Pi \left(x_{e_i}\right)\tau (𝚥)-\sigma \left(x_{e_i}\right)\pi (𝚥)\hfill \\ & =& \Pi \left(x_{e_i}\right)-0x_h-0x_{2q}\hfill \\ & =& 0-0-0=0\in L.\hfill \end{array}$$

(c)

If $𝚥=x_{e_i}$ and $w\ne x_{e_i}$, then

$$\begin{array}{ccc}\hfill \Pi (w𝚥)-\Pi \left(w\right)\tau (𝚥)-\sigma \left(w\right)\pi (𝚥)& =& \Pi \left(w{x}_{e_i}\right)-\Pi \left(w\right)\tau \left(x_{e_i}\right)-\sigma \left(w\right)\pi \left(x_{e_i}\right)\hfill \\ & =& \Pi \left(0\right)-x_{t}0-x_{m}0\hfill \\ & =& 0-0-0=0\in L.\hfill \end{array}$$

(d)

If $w\ne x_{e_i}$ and $𝚥\ne x_{e_i}$, then

$$\begin{array}{ccc}\hfill \Pi (w𝚥)-\Pi \left(w\right)\tau (𝚥)-\sigma \left(w\right)\pi (𝚥)& =& \Pi \left(w\right)-x_t{x}_h-x_m{x}_{2q}\hfill \\ & =& x_t-x_t-0=0\in L.\hfill \end{array}$$

Therefore, we can conclude that Π is a generalized $(\tau ,\sigma )$-L-derivation on $K_{2^n}$, where π is an $(\tau ,\sigma )$-L-derivation associated with Π.

The following two examples illustrate the comprehensiveness of Definition 2 for the concepts of generalized $(\tau ,\sigma )$-derivation and generalized L-derivation.

Example 2.

Let $X=\mathbb{Z}\left[w\right]$ be a ring of polynomials with indeterminate w and integer coefficients, and let $L=<2w+1>$. Define the functions $\Pi ,\pi ,\tau ,\sigma :\mathbb{Z}\left[w\right]⟶\mathbb{Z}\left[w\right]$ as follows:

$\Pi \left(s\right(w\left)\right)=\chi \left(s\right(w\left)\right)<2w+1>$, where $\chi \left(s\right(w\left)\right)=g.c.d$ of coefficients of $s\left(w\right)$.

$\pi \left(s\right(w\left)\right)=\kappa \left(s\right(w\left)\right)<2w+1>$, where $\kappa \left(s\right(w\left)\right)=l.c.m$ of coefficients of $s\left(w\right)$.

$\tau \left(s\right(w\left)\right)=\mu \left(s\right(w\left)\right)$, where $\mu \left(s\right(w\left)\right)=$ maximum value of coefficients of $s\left(w\right)$.

$\sigma \left(s\right(w\left)\right)=𝚥\left(s\right(w\left)\right)$, where $𝚥\left(s\right(w\left)\right)=$ minimum value of coefficients of $s\left(w\right)$.

It can be easily checked that L is a prime ideal of X, τ and σ are endomorphisms of X. Additionally, we can verify that Π is a generalized $(\tau ,\sigma )$-L-derivation associated with an $(\tau ,\sigma )$-L-derivation π. However, Π is not a generalized $(\tau ,\sigma )$-derivation on X.

Example 3.

Let $X=\left\{\left(\begin{array}{cc}w& 𝚥\\ 0& w\end{array}\right)\right|w,𝚥\in \mathbb{R}\}$, and let $L=\left\{\left(\begin{array}{cc}0& 𝚥\\ 0& 0\end{array}\right)\right|𝚥\in \mathbb{R}\}$. Define the mapping $\Pi ,\pi ,\tau ,\sigma :X⟶X$ by

$$\Pi \left(\begin{array}{cc}w& 𝚥\\ 0& w\end{array}\right)=\left(\begin{array}{cc}-w& det\left(T\right)\\ 0& -w\end{array}\right),d\left(\begin{array}{cc}w& 𝚥\\ 0& w\end{array}\right)=\left(\begin{array}{cc}w& det\left(T\right)\\ 0& w\end{array}\right),$$

$$\tau \left(\begin{array}{cc}w& 𝚥\\ 0& w\end{array}\right)=\left(\begin{array}{cc}w& 0\\ 0& w\end{array}\right)\mathit{and}\sigma \left(\begin{array}{cc}w& 𝚥\\ 0& w\end{array}\right)=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right).$$

It can be easily verified that L is a prime ideal of X, τ and σ are endomorphisms of X. Clearly, Π and π are L-additive mappings of X. Therefore, to prove that $(\Pi ,\pi )$ is a generalized $(\tau ,\sigma )$-L-derivation of X, it is sufficient to show that the condition $\left(B\right)\Pi \left(TS\right)-\Pi \left(T\right)\tau \left(S\right)-\sigma \left(S\right)\pi \left(T\right)\in L$ is satisfied as in the following; for any $T=\left(\begin{array}{cc}x& 𝚥\\ 0& x\end{array}\right)$ and $S=\left(\begin{array}{cc}w& \mu \\ 0& w\end{array}\right)$ in X. We have

$$\begin{array}{ccc}\hfill \Pi \left(TS\right)& -& \Pi \left(T\right)\tau \left(S\right)-\sigma \left(S\right)\pi \left(T\right)\hfill \\ & =& \Pi \left(\left(\begin{array}{cc}x& 𝚥\\ 0& x\end{array}\right)\left(\begin{array}{cc}w& \mu \\ 0& w\end{array}\right)\right)-\Pi \left(\left(\begin{array}{cc}x& 𝚥\\ 0& x\end{array}\right)\right)\tau \left(\left(\begin{array}{cc}w& \mu \\ 0& w\end{array}\right)\right)-\sigma \left(\left(\begin{array}{cc}x& 𝚥\\ 0& x\end{array}\right)\right)\pi \left(\left(\begin{array}{cc}w& \mu \\ 0& w\end{array}\right)\right)\hfill \\ & =& \Pi \left(\begin{array}{cc}xw& x\mu +𝚥w\\ 0& xw\end{array}\right)-\left(\begin{array}{cc}-x& det\left(T\right)\\ 0& -x\end{array}\right)\left(\begin{array}{cc}w& 0\\ 0& w\end{array}\right)-\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)\left(\begin{array}{cc}w& det\left(S\right)\\ 0& w\end{array}\right)\hfill \\ & =& \left(\begin{array}{cc}-xw& det\left(TS\right)\\ 0& -xw\end{array}\right)-\left(\begin{array}{cc}-xw& det\left(T\right)w\\ 0& -xw\end{array}\right)-\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)\hfill \\ & =& \left(\begin{array}{cc}0& det\left(TS\right)-det\left(T\right)w\\ 0& 0\end{array}\right)\in L.\hfill \end{array}$$

Therefore, Π is a generalized $(\tau ,\sigma )$-L-derivation associated with a $(\tau ,\sigma )$-L-derivation π. Furthermore, it can be verified that Π is neither a generalized L-derivation nor a generalized $(\tau ,\sigma )$-derivation.

Remark 1.

In light of the previous examples, we can conclude the following:

(i)

If a prime ideal L equals zero, then a generalized $(\tau ,\sigma )$-L-derivation is equivalent to a generalized $(\tau ,\sigma )$-derivation. This also applies to an $(\tau ,\sigma )$-L-derivation.

(ii)

By setting $\tau =\sigma =i{d}_X$ in Definition 2, we directly obtain ([8], Definition 2).

Below, we present the next lemma that will be repeatedly employed to achieve our desired results. To prove it, only use arguments similar to the proof of ([13], Lemma 3). Therefore, the proof is omitted.

Lemma 1.

Let L be a prime ideal of a ring X, and let τ and $\sigma :X⟶X$ be epimorphisms of X. Then, a factor ring $X/L$ is a commutative integral domain if and only if $\left[\tau \right(w),\sigma (𝚥\left)\right]\in L$ holds for all $w,𝚥\in X$.

3. Main Result

In [14], Quadri et al. established the commutativity of a prime ring X with a characteristic other than two, which admits a generalized derivation $\Pi$ that fulfills any of the following identities: $\Pi \left(\right[w,𝚥\left]\right)=\pm [w,𝚥]$ or $\Pi (w\circ 𝚥)=\pm (w\circ 𝚥)$ for all $w,𝚥$ in a non-zero ideal M of X. Subsequently, Golbasi et al. [15] examined the potential implications of the aforementioned results by substituting a non-zero ideal M with a square closed Lie ideal $\Lambda$. In their work, Rahman et al. [16] proved that a derivation $\pi$ is L-commuting on an ideal M when an arbitrary ring X admits a nonzero generalized derivation $(\Pi ,\pi )$ that satisfies any of the following statements for every $w,𝚥\in M$: $\Pi \left(\right[w,𝚥\left]\right)\pm [w,\pi (𝚥\left)\right]\in L$, $\Pi (w\circ 𝚥)\pm (w\circ \pi (𝚥\left)\right)\in L$, $\Pi \left(\right[w,𝚥\left]\right)\pm (w\circ 𝚥)\in L$, $\Pi \left(\right[w,𝚥\left]\right)\pm \Pi (𝚥)w\in L$, $\Pi \left(\right[w,𝚥\left]\right)\pm \Pi \left(w\right)𝚥\in L$, $\Pi (w\circ 𝚥)\pm \Pi (𝚥)w\in L$, or $\Pi (w\circ 𝚥)\pm \Pi \left(w\right)𝚥\in L$, where L is a semiprime ideal of X. In the context of a generalized $(\tau ,\sigma )$-derivation $\Pi$ associated with an $(\tau ,\sigma )$-derivation $\pi$, Sandhu et al. [17] examined the relationship between $(\Pi ,\pi )$ and the behavior of a prime ring X that satisfies any of the identities $\Pi \left(\right[w,𝚥\left]\right)=\tau \left(w\right)\circ \Pi (𝚥)$, or $\Pi (w\circ 𝚥)=\left[\tau \right(w),\Pi (𝚥\left)\right]$ for every $w,𝚥$ in a Lie ideal $\Lambda$. The following study aims to build upon previous results by incorporating more general identities that involve a comprehensive additive mapping $\Pi$ called a generalized $(\tau ,\sigma )$-L-derivation associated with an $(\tau ,\sigma )$-L-derivation $\pi$, where $\tau$ and $\sigma$ are epimorphisms of an arbitrary ring X and L is a prime ideal of X. To be concise, let’s denote a generalized $(\tau ,\sigma )$-L-derivation $\Pi$ associated with an $(\tau ,\sigma )$-L-derivation $\pi$ as $(\Pi ,\pi )$. Next, we will investigate the connection between $(\Pi ,\pi )$ and the behavior of a factor ring $X/L$.

Theorem 1.

Assume that a ring X admits a generalized $(\tau ,\sigma )$-L-derivation $(\Pi ,\pi )$ that satisfies any of the following statements for every $w,𝚥\in X$:

$\left(i\right)$ $\Pi \left(\right[w,𝚥\left]\right)+[\Pi (w),\tau (𝚥\left)\right]\pm \tau \left(\right[w,𝚥\left]\right)\in L$, or

$\left(ii\right)$ $\Pi \left(\right[w,𝚥\left]\right)-[\Pi (𝚥),\tau (w\left)\right]\pm \tau \left(\right[w,𝚥\left]\right)\in L$. Then either $\pi \left(X\right)\subseteq L$ and $(2\Pi \pm \tau )\left(X\right)\subseteq L$, or $X/L$ is an integral domain.

Proof. 

$\left(i\right)$ Based on the given hypothesis, we have

$$\Pi \left(\right[w,𝚥\left]\right)+[\Pi (w),\tau (𝚥\left)\right]\pm \tau \left(\right[w,𝚥\left]\right)\in L\forall w,𝚥\in X.$$

(1)

If we replace w with $w𝚥$ in Equation (1) and apply it, we obtain

$$\sigma \left(\right[w,𝚥\left]\right)\pi (𝚥)+\left[\sigma \right(w),\tau (𝚥\left)\right]\pi (𝚥)+\sigma \left(w\right)\left[\pi \right(𝚥),\tau (𝚥\left)\right]\in L\mathrm{for}\mathrm{all}w,𝚥\in X.$$

(2)

By replacing w with $qw$ in Equation (2) and applying it, we get

$$\sigma \left(\right[q,𝚥\left]\right)\sigma \left(w\right)\pi (𝚥)+\left[\sigma \right(q),\tau (𝚥\left)\right]\sigma \left(w\right)\pi (𝚥)\in L\forall w,𝚥,q\in X.$$

The hypothesis that $\sigma$ is an epimorphism of X implies that $\left(\sigma \right([q,𝚥])+[\sigma \left(q\right),\tau (𝚥)\left]\right)X\pi (𝚥)\subseteq L\mathrm{for}\mathrm{all}𝚥,q\in X.$ The primeness of L implies that for each $𝚥\in X$, either $\sigma \left(\right[q,𝚥\left]\right)+\left[\sigma \right(q),\tau (𝚥\left)\right]\in L$ or $\pi (𝚥)\in L$. We define two subsets of X: $T=\{𝚥\in X|\sigma \left(\right[q,𝚥\left]\right)+\left[\sigma \right(q),\tau (𝚥\left)\right]\in L\mathrm{for}\mathrm{all}q\in X\}$ and $S=\{𝚥\in X|\pi (𝚥)\in L\}$. It is clear that both subsets T and S are additive subgroups of X, and $T\cup S=(X,+)$. By invoking Brauer’s trick, we can deduce that either $T=X$ or $S=X$. Consequently, we will analyze the following scenarios:

If $T=X$, then we have $\sigma \left(\right[q,𝚥\left]\right)+\left[\sigma \right(q),\tau (𝚥\left)\right]\in L$ for all $𝚥,q\in X$. By setting $q=𝚥$, the last relation becomes $\sigma \left(\right[q,𝚥\left]\right)\equiv 0\left(\mathrm{mod}L\right)$ and $\left[\sigma \right(q),\tau (𝚥\left)\right]\equiv \left[\sigma \right(𝚥),\tau (𝚥\left)\right]\left(\mathrm{mod}L\right)$. Thus, we have $\left[\sigma \right(q),\tau (𝚥\left)\right]\in L$ for every $𝚥\in X$. Using Lemma 1, we conclude that $X/L$ is an integral domain.

On the other scenario, if $S=X$, then for all $𝚥\in X$, we have $\pi (𝚥)\in L$. For any $ϵ\in X$, replace w with $wϵ$ in Equation (1) and use it, we deduce that

$$2\Pi \left(w\right)\tau \left(\right[ϵ,𝚥\left]\right)\pm \tau \left(w\right)\tau \left(\right[ϵ,𝚥\left]\right)\in L.$$

(3)

By replacing $ϵ$ by $ϵ\mu$ in Equation (3) and applying it, we can conclude that

$$(2\Pi \pm \tau )\left(w\right)X\left(\right[\tau \left(\mu \right),\tau (𝚥)\left]\right)\subseteq L\forall w,𝚥,\mu \in X.$$

Therefore, the primeness of L implies that either $(2\Pi \pm \tau )\left(w\right)\in L$ or $\left[\tau \right(\mu ),\tau (𝚥\left)\right]\in L$ $\forall w,𝚥,\mu \in X$. If $(2\Pi \pm \tau )\left(w\right)\in L$ for any $w\in X$, then $(2\Pi \pm \tau )\left(X\right)\subseteq L$. In the case of $\left[\tau \right(\mu ),\tau (𝚥\left)\right]\in L$ for any $𝚥,\mu \in X$, $X/L$ is an integral domain, as shown by Lemma 1.

$\left(ii\right)$ Now, if we consider the second basic hypothesis $\Pi \left(\right[w,𝚥\left]\right)-[\Pi (𝚥),\tau (w\left)\right]\pm \tau \left(\right[w,𝚥\left]\right)\in L$ for all $w,𝚥\in X$, then by using approaches analogous to those used in the proof of part $\left(i\right)$, the desired conclusions are justified. □

In the preceding theorem, if we set $w=𝚥$, we obtain an updated version of ([10], Lemma 3) in the context of a generalized $(\tau ,\sigma )$-L-derivation as follows:

Corollary 1.

If a ring X admits a generalized $(\tau ,\sigma )$-L-derivation $(\Pi ,\pi )$, then $[\Pi (w),\tau (w\left)\right]\in L\forall w\in X$ if and only if either $\pi \left(X\right)\subseteq L$ or $X/L$ is an integral domain.

Remark 2.

Under certain restrictions on Theorem 1, the following can be verified:

$\left(i\right)$ If $L=\left\{0\right\}$ and $\pi \ne 0$, then X is commutative.

$\left(ii\right)$ If $\tau =\sigma =i{d}_X$, then either $\pi \left(X\right)\subseteq L$ and $(2\Pi \pm i{d}_X)\left(X\right)\subseteq L$, or $X/L$ is an integral domain.

$\left(iii\right)$ If $\Pi =\pi$, then $X/L$ is an integral domain.

If the negative sign in the second term of Theorem 1 $\left(ii\right)$ is changed to a positive sign, a different conclusion is reached. This is demonstrated in the following theorem:

Theorem 2.

Assume that a ring X admits a generalized $(\tau ,\sigma )$-L-derivation $(\Pi ,\pi )$ such that for every $w,𝚥\in X$: $\Pi \left(\right[w,𝚥\left]\right)+[\Pi (𝚥),\tau (w\left)\right]\pm \tau \left(\right[w,𝚥\left]\right)\in L$. Then $X/L$ is an integral domain.

Proof. 

Let $\Pi \left(\right[w,𝚥\left]\right)+[\Pi (𝚥),\tau (w\left)\right]\pm \tau \left(\right[w,𝚥\left]\right)\in L$ for all $w,𝚥\in X$. Then we can easily verify that analogous techniques used to prove Theorem 1, with some slight variations, reduce Equation (3) to $\tau \left(w\right)\tau \left(\right[ϵ,𝚥\left]\right)\in L\forall w,𝚥,ϵ\in X$. Thus, for any $\mu \in X$ substituting $ϵ$ with $ϵ\mu$ in the last expression and using it, we find $\tau \left(w\right)X\tau \left(\right[\mu ,𝚥\left]\right)\subseteq L\forall w,𝚥,\mu \in X$. However, for any $w\in X$, $\tau \left(w\right)$ is not in L because L is a prime ideal of X. Therefore, we have $\tau \left(\right[\mu ,𝚥\left]\right)\in L\forall 𝚥,\mu \in X$. By using Lemma 1, we conclude that $X/L$ is an integral domain. □

Remark 3.

In Theorem 2, if $L=\left\{0\right\}$ or $\tau =\sigma =i{d}_X$, then both assumptions imply that $X/L$ is an integral domain.

The following example illustrates the necessity of assuming that P is prime in Theorems 1 and 2:

Example 4.

Let $X=\left\{\left[\begin{array}{ccc}0& \gamma \left(w\right)& ϵ\left(w\right)\\ 0& 0& \zeta \left(w\right)\\ 0& 0& 0\end{array}\right]\right|\gamma \left(w\right),ϵ\left(w\right),\zeta \left(w\right)\in \mathbb{Z}\left[w\right]\}$, $L=\left\{\left[\begin{array}{ccc}0& 0& 2ϵ\left(w\right)\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\right|ϵ\left(w\right)\in \mathbb{Z}\left[w\right]\}$. Define the mappings $\Pi ,\pi ,\tau ,\sigma :X⟶X$ as follows:

$$\Pi \left[\begin{array}{ccc}0& \gamma \left(w\right)& ϵ\left(w\right)\\ 0& 0& \zeta \left(w\right)\\ 0& 0& 0\end{array}\right]=\left[\begin{array}{ccc}0& 0& -\zeta \left(w\right)\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],\pi \left[\begin{array}{ccc}0& \gamma \left(w\right)& ϵ\left(w\right)\\ 0& 0& \zeta \left(w\right)\\ 0& 0& 0\end{array}\right]=\left[\begin{array}{ccc}0& 0& \gamma \left(w\right)\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],$$

and

$$\tau \left[\begin{array}{ccc}0& \gamma \left(w\right)& ϵ\left(w\right)\\ 0& 0& \zeta \left(w\right)\\ 0& 0& 0\end{array}\right]=\left[\begin{array}{ccc}0& 0& 2\zeta \left(w\right)\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],\sigma \left[\begin{array}{ccc}0& \gamma \left(w\right)& ϵ\left(w\right)\\ 0& 0& \zeta \left(w\right)\\ 0& 0& 0\end{array}\right]=\left[\begin{array}{ccc}0& 0& \zeta \left(w\right)\\ 0& 0& 0\\ 0& 0& 0\end{array}\right].$$

It is easy to verify that the ideal L is not a prime in X. This is because for any $a,b\in X$, where $a=\left[\begin{array}{ccc}0& 0& \zeta \left(w\right)\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]andb=\left[\begin{array}{ccc}0& 0& \gamma \left(w\right)\\ 0& 0& ϵ\left(w\right)\\ 0& 0& 0\end{array}\right]$, we can check that $aXb\subseteq L$ but neither $a\in L$ nor $b\in L$. Moreover, it can be verified that τ and σ are endomorphisms on X and $(\Pi ,\pi )$ is a generalized $(\tau ,\sigma )$-L-derivation associated with an $(\tau ,\sigma )$-L-derivation π that satisfies the identities imposed in Theorems 1 and 2 for every $w,𝚥\in X$. However, $X/L$ is noncommutative, $\pi \left(X\right)⊈L$, and $(2\Pi \pm \tau )\left(X\right)⊈L$. This confirms the invalidity of Theorems 1 and 2 without assuming that L is prime.

In the following theorem, we will use “anticommutator” instead of “commutator” in the first term of Theorem 1, and see what conclusions we can draw.

Theorem 3.

Assume that a ring X admits a generalized $(\tau ,\sigma )$-L-derivation $(\Pi ,\pi )$:

(i)

If $\Pi (w\circ 𝚥)+[\Pi (w),\tau (𝚥\left)\right]\pm \tau \left(\right[w,𝚥\left]\right)\in L$ for every $w,𝚥\in X$, then either

(1)

$\pi \left(X\right)\subseteq L$ and $(2\Pi \pm \tau )\left(X\right)\subseteq L$, or

(2)

$X/L$ is an integral domain and $\Pi \left(X\right)\subseteq L$, or

(3)

$X/L$ is an integral domain and $char(X/L)=2$.

(ii)

If $\Pi (w\circ 𝚥)+[\Pi (𝚥),\tau (w\left)\right]\pm \tau \left(\right[w,𝚥\left]\right)\in L$ for every $w,𝚥\in X$, then either

(1)

$\pi \left(X\right)\subseteq L$ and $(2\Pi \mp \tau )\left(X\right)\subseteq L$, or

(2)

$X/L$ is an integral domain and $\Pi \left(X\right)\subseteq L$, or

(3)

$X/L$ is an integral domain and $char(X/L)=2$.

Proof. 

$\left(i\right)$ Based on the initial hypothesis, we have

$$\Pi (w\circ 𝚥)+[\Pi (w),\tau (𝚥\left)\right]\pm \tau \left(\right[w,𝚥\left]\right)\in L\forall w,𝚥\in X.$$

(4)

By replacing w with $w𝚥$ in Equation (4) and applying it, we obtain

$$\sigma (w\circ 𝚥)\pi (𝚥)+\left[\sigma \right(w),\tau (𝚥\left)\right]\pi (𝚥)+\sigma \left(w\right)\left[\pi \right(𝚥),\tau (𝚥\left)\right]\in L\forall w,𝚥\in X.$$

(5)

In the previous equation, replacing w by $ϵw$ and using it, we obtain

$$\begin{array}{c}\hfill \sigma [ϵ,𝚥]\sigma \left(w\right)\pi (𝚥)-\left[\sigma \right(ϵ),\tau (𝚥\left)\right]\sigma \left(w\right)\pi (𝚥)\in L\mathrm{for}\mathrm{all}w,𝚥,ϵ\in X.\end{array}$$

By putting $ϵ=𝚥$ in the last equation, we can deduce that $\sigma [𝚥,𝚥]\sigma \left(w\right)\pi (𝚥)\equiv 0\left(\mathrm{mod}L\right)$ and $\left[\sigma \right(𝚥),\tau (𝚥\left)\right]\sigma \left(w\right)\pi (𝚥)\in L$. Therefore, we can conclude that $\left[\sigma \right(𝚥),\tau (𝚥\left)\right]\sigma \left(w\right)\pi (𝚥)\in L$ for all $w,𝚥\in X$. Since $\sigma$ is an epimorphism and L is prime, we can determine that for all $𝚥\in X$, either $\left[\sigma \right(𝚥),\tau (𝚥\left)\right]\in L$ or $\pi (𝚥)\in L$. Consequently, we will analyze the following two scenarios:

Scenario 1: If $\pi (𝚥)\in L$ for all $𝚥\in X$, then replacing w by $wϵ$ in Equation (4) and using it, we can infer $\Pi \left(w\right)\tau \left(\right[ϵ,𝚥\left]\right)+\Pi \left(w\right)\left[\tau \right(ϵ),\tau (𝚥\left)\right]\pm \tau \left(w\right)\tau \left(\right[ϵ,𝚥\left]\right)\in L$ for all $w,𝚥,ϵ\in X$. This can be expressed as $(2\Pi \pm \tau )\left(w\right)\left[\tau \right(ϵ),\tau (𝚥\left)\right]\in L$. Therefore, as discussed after Equation (3), we can deduce that either $X/L$ is an integral domain or $(2\Pi \pm \tau )\left(X\right)\subseteq L$. The commutativity $\left(\mathrm{mod}L\right)$ along with our assumption that $\pi \left(X\right)\subseteq L$ facilitates Equation (4) to $\Pi (w\circ 𝚥)\equiv 2\Pi \left(w\right)\tau (𝚥)\left(\mathrm{mod}L\right)$ and $[\Pi (w),\tau (𝚥\left)\right]\pm \tau \left(\right[w,𝚥\left]\right)\equiv 0\left(\mathrm{mod}L\right)$. Thus, $2\Pi \left(w\right)\tau (𝚥)\in L$ for all $w,𝚥\in X$. Substituting w with $w\mu$, for any $\mu \in X$, we obtain $2\Pi \left(w\right)X\tau (𝚥)\subseteq L\forall w,𝚥\in X$. But for any $𝚥\in X$, $\tau (𝚥)\notin L$ because $X/L$ has no non-zero divisors and L is prime. According to this, we have $2\Pi \left(w\right)\in L\forall w\in X$. Primeness of L forces either $char(X/L)=2$ or $\Pi \left(w\right)\in L$ for all $w\in X$. The last case implies that $\Pi \left(X\right)\subseteq L$.

Scenario 2: If $\left[\sigma \right(𝚥),\tau (𝚥\left)\right]\in L\forall 𝚥\in X$, then by using Lemma 1, we find $X/L$ is an integral domain. This reduces Equation (4) to $2\Pi (w𝚥)\in L$ for all $w,𝚥\in X$. Recalling the primeness of L, either $char(X/L)=2$ or $\Pi (w𝚥)\in L$. By combining the case $\Pi (w𝚥)\equiv 0\left(\mathrm{mod}L\right)$ with Definition 2 $\left(ii\right)$, we obtain $\Pi \left(w\right)\tau (𝚥)+\sigma \left(w\right)\pi (𝚥)\in L$ for all $w,𝚥\in X$. Substituting 𝚥 with $𝚥ϵ$ in the last statement and applying it, we conclude that $\sigma \left(w\right)X\pi \left(ϵ\right)\subseteq L$ for all $w,ϵ\in X$. The primeness of L together with our hypothesis that $\sigma$ is an epimorphism forces, $\sigma \left(w\right)$ is not in L, so we have $\pi \left(ϵ\right)\in L$ for all $ϵ\in X$. Applying this conclusion in Definition 2 $\left(ii\right)$, we deduce that $\Pi \left(w\right)\tau (𝚥)\in L\forall w,𝚥\in X$. Since L is a prime ideal of X, either $\Pi \left(w\right)\in L$ or $\tau (𝚥)\in L$ for all $w,𝚥\in X$. Our hypothesis that $\tau$ is an epimorphism of X, along with the fact that $L\ne X$, implies that for any $𝚥\in X$, $\tau (𝚥)\notin L$. Therefore, we can conclude that $\Pi \left(X\right)\subseteq L$.

$\left(ii\right)$ Analogous to the proof of part $\left(i\right)$, the required outcomes can be obtained when the assumed identity is $\Pi (w\circ 𝚥)+[\Pi (𝚥),\tau (w\left)\right]\pm \tau \left(\right[w,𝚥\left]\right)\in L$ for all $w,𝚥\in X$. □

The following corollary can be immediately derived by taking $\tau =\sigma =i{d}_X$ in Theorem 3:

Corollary 2.

Assume that a ring X admits a generalized L-derivation $(\Pi ,\pi )$:

(i)

If $\Pi (w\circ 𝚥)+[\Pi (w),𝚥]\pm [w,𝚥]\in L$ for every $w,𝚥\in X$, then either

(1)

$\pi \left(X\right)\subseteq L$ and $(2\Pi \pm i{d}_X)\left(X\right)\subseteq L$, or

(2)

$X/L$ is an integral domain and $\Pi \left(X\right)\subseteq L$, or

(3)

$X/L$ is an integral domain and $char(X/L)=2$.

(ii)

If $\Pi (w\circ 𝚥)+[\Pi (𝚥),w]\pm [w,𝚥]\in L$ for all $w,𝚥\in X$, then either

(1)

$\pi \left(X\right)\subseteq L$ and $(2\Pi \mp i{d}_X)\left(X\right)\subseteq L$, or

(2)

$X/L$ is an integral domain and $\Pi \left(X\right)\subseteq L$, or

(3)

$X/L$ is an integral domain and $char(X/L)=2$.

Remark 4.

Based on Theorem 3, the validity of the following situations can be verified:

$\left(i\right)$ If $L=\left\{0\right\}$ and $\pi \ne 0$, then X is commutative and $char\left(X\right)=2$.

$\left(ii\right)$ If $\Pi =\pi$, then either $X/L$ is an integral domain of characteristic 2 or $\pi \left(X\right)\subseteq L$ and $X/L$ is an integral domain.

Instead of $\Pi$, let’s consider $\pi$ in the second term of Theorem 3. We can then introduce the following proposition:

Proposition 1.

Assume that a ring X admits a generalized $(\tau ,\sigma )$-L-derivation $(\Pi ,\pi )$ that satisfies any one of the following statements for every $w,𝚥\in X$:

(i)

$\Pi (w\circ 𝚥)\pm \left[\pi \right(w),\tau (𝚥\left)\right]\pm \tau \left(\right[w,𝚥\left]\right)\in L$, or

(ii)

$\Pi (w\circ 𝚥)\mp \left[\pi \right(𝚥),\tau (w\left)\right]\pm \tau \left(\right[w,𝚥\left]\right)\in L$. Then either

(1)

$\pi \left(X\right)\subseteq L$ and $(\Pi \pm \tau )\left(X\right)\subseteq L$, or

(2)

$X/L$ is an integral domain and $\Pi \left(X\right)\subseteq L$, or

(3)

$X/L$ is an integral domain and $char(X/L)=2$.

Proof. 

Following the same reasoning as the proof of Theorem 3, with some slight adjustments, we can easily derive conclusions $\left(1\right)$, $\left(2\right)$, and $\left(3\right)$. Therefore, we will omit the proof. □

By changing the sign of the second term in identity $\left(ii\right)$ of Theorem 3 and applying similar arguments as in the proof of Theorem 3 with necessary modifications, we observe that conclusion $\left(1\right)$ no longer holds. This is formally stated in the following theorem:

Theorem 4.

Assume that a ring X admits a generalized $(\tau ,\sigma )$-L-derivation $(\Pi ,\pi )$. Then $\Pi (w\circ 𝚥)-[\Pi (𝚥),\tau (w\left)\right]\pm \tau \left(\right[w,𝚥\left]\right)\in L$ for all $w,𝚥\in X$ if and only if either

(i)

$X/L$ is an integral domain of characteristic two, or

(ii)

$X/L$ is an integral domain and $\Pi \left(X\right)\subseteq L$.

The necessity of assuming that L is prime in Theorems 3, 4 is illustrated by the following example:

Example 5.

In Example 4, we can note that the identities in Theorems 3, 4 and Proposition 1 satisfied, although $X/L$ is not an integral domain and $\Pi \left(X\right)⊈L$ or $char(X/L)\ne 2$, $\pi \left(X\right)⊈L$, $(2\Pi \pm \tau )\left(X\right)⊈P$. This emphasize the necessity of the primeness assumptions in Theorems 3, 4, and Proposition 1.

Theorem 5.

Assume that a ring X admits a generalized $(\tau ,\sigma )$-L-derivation $(\Pi ,\pi )$ that satisfies any one of the following statements for every $w,𝚥\in X$:

(i)

$\Pi \left(\right[w,𝚥\left]\right)+[\Pi (w),\tau (𝚥\left)\right]\pm \tau (w\circ 𝚥)\in L$, or

(ii)

$\Pi \left(\right[w,𝚥\left]\right)-[\Pi (𝚥),\tau (w\left)\right]\pm \tau (w\circ 𝚥)\in L$. Then either

(1)

$\pi \left(X\right)\subseteq L$ and $(2\Pi \pm \tau )\left(X\right)\subseteq L$, or

(2)

$X/L$ is an integral domain and $char(X/L)=2$.

Remark 5.

According to Theorem 5, the validity of the following statements can be verified:

$\left(i\right)$ If $L=\left\{0\right\}$ and $\pi \ne 0$, then X is commutative of characteristic two.

$\left(ii\right)$ If $\tau =\sigma =i{d}_X$, then either $\pi \left(X\right)\subseteq L$ and $(2\Pi \pm i{d}_X)\left(X\right)\subseteq L$, or $X/L$ is an integral domain of characteristic two.

$\left(iii\right)$ If $\Pi =\pi$, then $X/L$ is an integral domain of characteristic two.

In the following theorem, we will explore the implications when the “commutator” vanishes in all terms of the identities in Theorem 1.

Theorem 6.

Assume that a ring X admits a generalized $(\tau ,\sigma )$-L-derivation $(\Pi ,\pi )$ that satisfies one of the following statements for every $w,𝚥\in X$:

(i)

$\Pi (w\circ 𝚥)+\Pi \left(w\right)\circ \tau (𝚥)\pm \tau (w\circ 𝚥)\in L$, or

(ii)

$\Pi (w\circ 𝚥)-\Pi (𝚥)\circ \tau \left(w\right)\pm \tau (w\circ 𝚥)\in L$. Then either:

(1)

$\left(1\right)$ $\pi \left(X\right)\subseteq L$ and $(2\Pi \pm \tau )\left(X\right)\subseteq L$, or

(2)

$\left(2\right)$ $X/L$ is an integral domain and $char(X/L)=2$.

Proof. 

Based on the initial hypothesis, we have

$$\Pi (w\circ 𝚥)+\Pi \left(w\right)\circ \tau (𝚥)\pm \tau (w\circ 𝚥)\in L\mathrm{for}\mathrm{all}w,𝚥\in X.$$

(6)

By replacing w with $w𝚥$ in the last equation and using similar tactics and arguments as in previous discussions, with necessary minor modifications, we can conclude that either $X/L$ is an integral domain or $\pi \left(X\right)\subseteq L$. To continue the proof, we will analyze the implications of these conclusions on our initial hypothesis in the following two scenarios:

Scenario 1: In the context of the commutativity $\left(\mathrm{mod}L\right)$ property of $X/L$, Equation (6) can be expressed as

$$2(\Pi (w𝚥)+\Pi (w\left)\tau \right(𝚥)\pm \tau (w𝚥\left)\right)\in L.$$

(7)

By replacing 𝚥 with $𝚥ϵ$ in the last equation and applying it, we get $2\sigma (w𝚥)\pi \left(ϵ\right)\in L$ for all $w,𝚥,ϵ\in X$. Since $\sigma$ is an epimorphism of X, we can infer that $2\left(\sigma \right(w\left)\right)X\pi \left(ϵ\right)\subseteq L\forall w,ϵ\in X$. The primeness of L implies that either $char(X/L)=2$ or $\pi \left(ϵ\right)\in L$ for all $ϵ\in X$. The latter case simplifies Equation (7) to $2(2\Pi (w)\pm \tau (w\left)\right)\tau (𝚥)\in L\forall w,𝚥\in X$. Using the primeness of L, we can infer that either $2\Pi \left(w\right)\pm \tau \left(w\right)\in L\forall w\in X$ or $2\tau (𝚥)\in L$ for each $𝚥\in X$. However, $\tau (𝚥)$ is not a zero divisor in $X/L$ for any $𝚥\in X$. Therefore, we can conclude that either $(2\Pi \pm \tau )\left(X\right)\subseteq L$ or $char(X/L)=2$.

Scenario 2: Considering $\pi \left(X\right)\subseteq L$, if we replace w with $wϵ$ in Equation (6) and use it, we obtain $\Pi \left(w\right)\tau \left(\right[ϵ,𝚥\left]\right)+\Pi \left(w\right)\left[\tau \right(ϵ),\tau (𝚥\left)\right]\pm \tau \left(w\right)\tau \left(\right[ϵ,𝚥\left]\right)\in L$ for all $w,𝚥,ϵ\in X$. Based on our previous discussions, we can conclude that either $(2\Pi \pm \tau )\left(X\right)\subseteq L$ or $X/L$ is an integral domain. If the latter case applies, we can simply apply the same strategies as in scenario 1 to reach the desired conclusion.

$\left(ii\right)$ Assuming that $\Pi (w\circ 𝚥)-\Pi (𝚥)\circ \tau \left(w\right)\pm \tau (w\circ 𝚥)\in L$ for all $w,𝚥\in X$, we can verify the proof by proceeding with arguments and techniques analogous to the proof of part $\left(i\right)$, with some necessary modifications. □

Remark 6.

Based on Theorem 6, we can verify the following assertions:

$\left(i\right)$ If a ring X is prime and $\pi \ne 0$, then X is commutative of characteristic two.

$\left(ii\right)$ If $\tau =\sigma =i{d}_X$, then either $\pi \left(X\right)\subseteq L$ and $(2\Pi \pm i{d}_X)\left(X\right)\subseteq L$, or $X/L$ is an integral domain and $char(X/L)=2$.

$\left(iii\right)$ If $\Pi =\pi$, then $X/L$ is an integral domain of characteristic two.

If we use the positive sign $(+)$ instead of the negative sign $(-)$ in the second term of identity $\left(ii\right)$ in Theorem 6 and apply similar strategies in the proof with necessary modifications, verifying the validity of the conclusion of the following theorem becomes straightforward:

Theorem 7.

Assume that a ring X admits a generalized $(\tau ,\sigma )$-L-derivation $(\Pi ,\pi )$ that satisfies the following statement for every $w,𝚥\in X$: $\Pi (w\circ 𝚥)+\Pi (𝚥)\circ \tau \left(w\right)\pm \tau (w\circ 𝚥)\in L$. Then $X/L$ is an integral domain of characteristic two.

Remark 7.

In Theorem 7, if we assume that a ring X is prime or $\tau =\sigma =i{d}_X$, then both assumptions imply that X is commutative.

The importance of assuming that the ideal P is prime in Theorems 5–7 is illustrated by the following example:

Example 6.

Let $X=\{x{e}_{21}+y{e}_{31}+2x{e}_{32}|x,y,z\in \mathbb{H}\}$, where $\mathbb{H}$ is a Hamiltonian ring, and let $L=\left\{2y{e}_{31}\right\}$. Define $(\Pi ,\pi ):X⟶X$ by

$$\begin{array}{c}\hfill \Pi (x{e}_{21}+y{e}_{31}+2x{e}_{32})=2x{e}_{32},\mathit{with}\pi (x{e}_{21}+y{e}_{31}+2x{e}_{32})=x{e}_{21},\end{array}$$

and

$$\begin{array}{c}\hfill \tau (x{e}_{21}+y{e}_{31}+2x{e}_{32})=x{e}_{31},\mathit{with}\sigma (x{e}_{21}+y{e}_{31}+2x{e}_{32})=x{e}_{21},\end{array}$$

It is easy to verify that Π is a generalized $(\tau ,\sigma )$-L-derivation associated with an $(\tau ,\sigma )$-L-derivation π. We can also note that the identities in Theorems 5–7 are satisfied. However, $X/L$ is neither commutative nor of characteristic 2, $\pi \left(X\right)⊈L$ and $(2\Pi \pm \tau )\left(X\right)⊈L$. Since $\left(x{e}_{21}\right)X\left(2x{e}_{32}\right)\in L$, but neither $x{e}_{21}\in L$ nor $2x{e}_{32}\in L$, then L is not prime. Therefore, the hypothesis that L is prime in Theorems 5–7 is necessary.

Example 7.

Taking into account the considerations in Example 4, we can verify that $(\Pi ,\pi )$ still satisfies the specified identities imposed in Theorems 5–7. However, upon examining the conclusions of these theorems, it is revealed that $X/L$ is not commutative of characteristic 2, $\pi \left(X\right)⊈L$, and $(2\Pi \pm \tau )\left(X\right)⊈L$. This confirms the invalidity of Theorems 5–7 without assuming that L is prime.

Theorem 8.

Assume that a ring X admits a generalized $(\tau ,\sigma )$-L-derivation $(\Pi ,\pi )$. Then one of the following statements is true for every $w,𝚥\in X$:

(i)

$\Pi \left(\right[w,𝚥\left]\right)\pm \Pi \left(w\right)\pi (𝚥)\pm \tau \left(\right[w,𝚥\left]\right)\in L$,

(ii)

$\Pi \left(\right[w,𝚥\left]\right)\pm \Pi (𝚥)\pi \left(w\right)\pm \tau \left(\right[w,𝚥\left]\right)\in L$ if and only if either

(1)

$\pi \left(X\right)\subseteq L$ and $(\Pi \pm \tau )\left(X\right)\subseteq L$, or

(2)

$X/L$ is an integral domain and $\pi \left(X\right)\subseteq L$.

Proof. 

Based on the initial hypothesis, we have

$$\Pi \left(\right[w,𝚥\left]\right)\pm \Pi \left(w\right)\pi (𝚥)\pm \tau \left(\right[w,𝚥\left]\right)\in L\mathrm{for}\mathrm{all}w,𝚥\in X.$$

(8)

Replacing 𝚥 with $𝚥ϵ$ in Equation (8) and using it, we obtain

$$\Pi (𝚥)\tau [w,ϵ]+\sigma (𝚥)\pi [w,ϵ]+\sigma \left(\right[w,𝚥\left]\right)\pi \left(ϵ\right)\pm \Pi \left(w\right)\sigma (𝚥)\pi \left(ϵ\right)\pm \tau (𝚥)\tau [w,ϵ]\in L.$$

(9)

By replacing $ϵ$ with $ϵw$ in the last equation and using it, we get

$$\sigma (𝚥)\sigma \left(\right[w,ϵ\left]\right)\pi \left(w\right)+\sigma \left(\right[w,𝚥\left]\right)\sigma \left(ϵ\right)\pi \left(w\right)\pm \Pi \left(w\right)\sigma (𝚥ϵ)\pi \left(w\right)\in L.$$

(10)

Replacing 𝚥 by $q𝚥$ in Equation (10) and using it, we obtain

$$\left(\sigma \right([w,q])+[\Pi \left(w\right),\sigma \left(q\right)\left]\right)\sigma (𝚥)\sigma \left(ϵ\right)\pi \left(w\right)\in L.$$

Since $\sigma$ is an epimorphism, the last equation becomes

$$\left(\sigma \right([w,q])+[\Pi \left(w\right),\sigma \left(q\right)\left]\right)X\sigma \left(ϵ\right)\pi \left(w\right)\subseteq L,\forall w,ϵ,q\in X.$$

The primeness of L forces that $\forall w\in X$ either $\left(\sigma \right([w,q])+[\Pi \left(w\right),\sigma \left(q\right)\left]\right)\in L$ or $\sigma \left(ϵ\right)\pi \left(w\right)\in L$. Define the following two subsets of X: $T=\{w\in X|\left(\sigma \right([w,q])+[\Pi \left(w\right),\sigma \left(q\right)\left]\right)\in L\mathrm{for}\mathrm{all}q\in X\}$ and $S=\{w\in X|\sigma \left(ϵ\right)\pi \left(w\right)\in L\mathrm{for}\mathrm{all}ϵ\in X\}$. It is evident that both subsets T and S are additive subgroups of X and $T\cup S=(X,+)$. Recalling Brauer’s trick, we have either $T=X$ or $S=X$. If $T=X$, then $\left(\sigma \right([w,q])+[\Pi \left(w\right),\sigma \left(q\right)\left]\right)\in L\forall w,q\in X$. By setting $w=q$, the last equation becomes $\sigma \left(\right[w,q\left]\right)\equiv 0\left(\mathrm{mod}L\right)$ and $[\Pi (w),\sigma (q\left)\right]\equiv [\Pi (w),\sigma (w\left)\right]\left(\mathrm{mod}L\right)$. This yields that $[\Pi (w),\sigma (w\left)\right]\in L\forall w\in X$. By applying Corollary 1, we obtain either $X/L$ is an integral domain or $\pi \left(X\right)\subseteq L$. In the case of $S=X$, we have $\sigma \left(ϵ\right)\pi \left(w\right)\in L$ for all $w,ϵ\in X$. Since L is prime and $\sigma$ is an epimorphism of X, we can deduce $\pi \left(X\right)\subseteq L$. Consequently, we will analyze the following two scenarios:

Scenario 1: Given $\pi \left(X\right)\subseteq L$, this reduce Equation (9) to $(\Pi (𝚥)\pm \tau (𝚥\left)\right)\tau \left(\right[w,ϵ\left]\right)\in L$ for all $w,𝚥,ϵ\in X$. By setting $w=w\mu$ and utilizing the fact that $\tau$ is an epimorphism, we can infer that $(\Pi \pm \tau )(𝚥)X\tau \left(\right[\mu ,ϵ\left]\right)\subseteq L\forall 𝚥,\mu ,ϵ\in X$. Since L is prime, we can deduce that either $(\Pi \pm \tau )(𝚥)\in L$ or $\tau \left(\right[\mu ,ϵ\left]\right)\subseteq L$ for all $𝚥,\mu ,ϵ\in X$. In the first case, it implies that $(\Pi \pm \tau )\left(X\right)\subseteq L$. In the second case, we can conclude that $X/L$ is an integral domain by applying Lemma 1.

Scenario 2: If $X/L$ is an integral domain, then Equation (9) becomes $\Pi \left(w\right)\tau (𝚥)\pi \left(ϵ\right)\in L\forall w,𝚥,ϵ\in X$. By replacing w with $wϵ$ in the last equation and using it, we obtain $\sigma \left(w\right)\pi \left(ϵ\right)\sigma (𝚥)\pi \left(ϵ\right)\in L$ for all $w,𝚥,ϵ\in X$. As $\sigma$ is an epimorphism and L is prime, we can conclude that $\pi \left(X\right)\subseteq L$.

$\left(ii\right)$ By following similar arguments and techniques as used in the proof of part $\left(i\right)$, we can derive the required conclusions when the imposed identity $\Pi \left(\right[w,𝚥\left]\right)\pm \Pi (𝚥)\pi \left(w\right)\pm \tau \left(\right[w,𝚥\left]\right)\in L$ for all $w,𝚥\in X$. □

Remark 8.

In light of Theorem 8, the following observations can be verified:

$\left(i\right)$ If X is prime, then either X is commutative and $\pi =0$ or $\pi =0$ and $\Pi =\pm \tau$.

$\left(ii\right)$ If $(\Pi ,\pi )$ is a generalized L-derivation, then either $\pi \left(X\right)\subseteq L$ and $(\Pi \pm i{d}_X)\left(X\right)\subseteq L$, or $X/L$ is an integral domain and $\pi \left(X\right)\subseteq L$.

$\left(iii\right)$ If $\Pi =\pi$, then $\pi \left(X\right)\subseteq L$ and $X/L$ is an integral domain.

Theorem 9.

Assume that a ring X admits a generalized $(\tau ,\sigma )$-L-derivation $(\Pi ,\pi )$. Then one of the following statements is true for every $w,𝚥\in X$:

(i)

$\Pi (w\circ 𝚥)\pm \Pi \left(w\right)\pi (𝚥)\pm \tau \left(\right[w,𝚥\left]\right)\in L$, or

(ii)

$\Pi (w\circ 𝚥)\pm \Pi (𝚥)\pi \left(w\right)\pm \tau \left(\right[w,𝚥\left]\right)\in L$ if and only if either

(1)

$\pi \left(X\right)\subseteq L$ and $(\Pi \pm \tau )\left(X\right)\subseteq L$, or

(2)

$\pi \left(X\right)\subseteq L$ and $X/L$ is an integral domain of characteristic two, or

(3)

$X/L$ is an integral domain and $\Pi \left(X\right)\subseteq L$.

Proof. 

$\left(i\right)$ Based on the initial hypothesis, we have

$$\Pi (w\circ 𝚥)\pm \Pi \left(w\right)\pi (𝚥)\pm \tau \left(\right[w,𝚥\left]\right)\in L\mathrm{for}\mathrm{all}w,𝚥\in X.$$

(11)

By simulating the tactics and arguments of the proof of Theorem 8 with some necessary modifications, we establish the following two scenarios:

Scenario 1: If $\pi \left(X\right)\subseteq L$, then as discussed in scenario 1 of Theorem 8, we obtain either $X/L$ is a commutative integral domain or $(\Pi \pm \tau )\left(X\right)\subseteq L$. In the second case, we have $\pi \left(X\right)\subseteq L$ and $(\Pi \pm \tau )\left(X\right)\subseteq L$, which is the desired conclusion $\left(1\right)$. In the first case, commutativity $\left(\mathrm{mod}L\right)$ of $X/L$ with the assumption $\pi \left(X\right)\subseteq L$ reduces Equation (11) to $\Pi (w\circ 𝚥)\equiv 2\Pi (w𝚥)\left(\mathrm{mod}L\right)$ for all $w,𝚥\in X$. Consequently, as discussed previously, we obtain either $\Pi \left(X\right)\subseteq L$ or $char(X/L)=2$.

Scenario 2: The commutativity $\left(\mathrm{mod}L\right)$ of $X/L$ simplifies Equation (11) to $\Pi (2w𝚥)\pm \Pi \left(w\right)\pi (𝚥)\in L$ and $\pm \tau \left(\right[w,𝚥\left]\right)\equiv 0\left(\mathrm{mod}L\right)\forall w,𝚥\in X$. Therefore, we can deduce that $2\Pi (w𝚥)\pm \Pi \left(w\right)\pi (𝚥)\in L$. Substituting w with $wϵ$ in the previous expression and using it, we can conclude that $\sigma \left(w\right)\left(2\sigma \right(𝚥\left)\pi \right(ϵ)\pm \pi (ϵ\left)\pi \right(𝚥\left)\right)\in L\forall w,𝚥,ϵ\in X$. Since L is prime, for any $w\in X$, we have $\sigma \left(w\right)\notin L$. Thus, the last relation becomes $2\sigma (𝚥)\pi \left(ϵ\right)\pm \pi \left(ϵ\right)\pi (𝚥)\in L\forall 𝚥,ϵ\in X$. Substituting 𝚥 with $q𝚥$ in the previous expression and using it, we can deduce that $\pi \left(ϵ\right)X\pi (𝚥)\subseteq L$ for all $𝚥,ϵ\in X$. Therefore, primeness of L implies that $\pi \left(X\right)\subseteq L$. Consequently, as discussed above, we can find that either $\Pi \left(X\right)\subseteq L$ or $char(X/L)=2$.

$\left(ii\right)$ Assuming that $\Pi (w\circ 𝚥)\pm \Pi (𝚥)\pi \left(w\right)\pm \tau \left(\right[w,𝚥\left]\right)\in L$ for all $w,𝚥\in X$, we can verify the proof by proceeding with arguments and techniques analogous to the proof of part $\left(i\right)$, with some necessary modifications. □

Remark 9.

As an application of Theorem 9, we can verify the validity of the following:

$\left(i\right)$ If a ring X is prime with a characteristic other than two and if $\Pi \ne 0$, then $\Pi =0$ and $\Pi =\pm \tau$.

$\left(ii\right)$ By setting $\tau =\sigma =i{d}_X$ and $char(X/L)\ne 2$, then either $\pi \left(X\right)\subseteq L$ and $(\Pi \pm i{d}_X)\left(X\right)\subseteq L$, or $X/L$ is an integral domain and $\Pi \left(X\right)\subseteq L$.

Analogous to the proof of Theorem 9, with some necessary precautions, we can verify the following theorem:

Theorem 10.

Assume that a ring X admits a generalized $(\tau ,\sigma )$-L-derivation $(\Pi ,\pi )$. Then one of the following statements is true for every $w,𝚥\in X$:

(i)

$\Pi (w\circ 𝚥)\pm \Pi \left(w\right)\pi (𝚥)\pm \tau (w\circ 𝚥)\in L$

(ii)

$\Pi (w\circ 𝚥)\pm \Pi (𝚥)\pi \left(w\right)\pm \tau (w\circ 𝚥)\in L$ if and only if either

(1)

$\pi \left(X\right)\subseteq L$ and $(\Pi \pm \tau )\left(X\right)\subseteq L$, or

(2)

$\pi \left(X\right)\subseteq L$ and $X/L$ is an integral domain of characteristic two.

Corollary 3.

If a prime ring X admits a generalized $(\tau ,\sigma )$-derivation $(\Pi ,\pi )$ such that $\Pi \ne \pm \tau$, then one of the following statements is true for every $w,𝚥\in X$:

(i)

$\Pi (w\circ 𝚥)\pm \Pi \left(w\right)\pi (𝚥)\pm \tau (w\circ 𝚥)=0$

(ii)

$\Pi (w\circ 𝚥)\pm \Pi (𝚥)\pi \left(w\right)\pm \tau (w\circ 𝚥)=0$ if and only if $\pi =0$ and X is commutative of characteristic equal to two.

Corollary 4.

Assume a ring X such that $char(X/L)\ne 2$. If X admits a generalized L-derivation $(\Pi ,\pi )$, then one of the following statements is true for every $w,𝚥\in X$:

(i)

$\Pi (w\circ 𝚥)\pm \Pi \left(w\right)\pi (𝚥)\pm (w\circ 𝚥)\in L$

(ii)

$\Pi (w\circ 𝚥)\pm \Pi (𝚥)\pi \left(w\right)\pm (w\circ 𝚥)\in L$ if and only if either

(1)

$\pi \left(X\right)\subseteq L$ and $(\Pi \pm i{d}_X)\left(X\right)\subseteq L$, or

(2)

$X/L$ is an integral domain and $(\Pi \pm i{d}_X)\left(X\right)\subseteq L$.

In the following example, we will define mappings and a non-prime ideal on the ring $K_{2^n}$ in Definition 3. We will then prove that these mappings satisfy the conditions of Theorem 8, but the conclusion ultimately fails. Therefore, we can conclude that the primeness condition in this theorem cannot be ignored.

Example 8.

Let $X=K_{2^n}$, τ, σ and π are as in the Definition 3, and let $L=\left\{0\right\}$. Define $\Pi :K_{2^n}⟶K_{2^n}$ as

$$\begin{array}{c}\hfill \Pi \left(w\right)=\left\{\begin{array}{lll}0& ifw=x_{e_i},& \\ x_{2t},\mathit{for}\mathit{fixed}t& ifw=x_{o_i}.\end{array}\right.\end{array}$$

Then using arguments similar to those used in Example 1, it can be easily verified that Π is a generalized $(\tau ,\sigma )$-L-derivation associated with an $(\tau ,\sigma )$-L-derivation π.

Now let’s check that $(\Pi ,\pi )$ satisfies the identities $\left(i\right)$ $\Pi \left(\right[w,𝚥\left]\right)\pm \Pi \left(w\right)\pi (𝚥)\pm \tau \left(\right[w,𝚥\left]\right)\in L$, $\left(ii\right)$ $\Pi \left(\right[w,𝚥\left]\right)\pm \Pi (𝚥)\pi \left(w\right)\pm \tau \left(\right[w,𝚥\left]\right)\in L$ for all $w,𝚥\in X$.

(i)

We have $\Pi \left(\right[w,𝚥\left]\right)\pm \Pi \left(w\right)\pi (𝚥)\pm \tau \left(\right[w,𝚥\left]\right)\in L$ for all $w,𝚥\in X$. Therefore, it is sufficient to discuss the following cases:

(a)

If $w=x_{e_i}$ and $𝚥=x_{e_i}$, then

$$\begin{array}{ccc}\hfill \Pi \left(\right[w,𝚥\left]\right)\pm \Pi \left(w\right)\pi (𝚥)\pm \tau \left(\right[w,𝚥\left]\right)& =& \Pi \left(0\right)\pm \Pi \left(x_{e_i}\right)\pi \left(x_{e_i}\right)\pm \tau \left(0\right)\hfill \\ & =& 0\pm 00\pm 0=0\in L.\hfill \end{array}$$

(b)

If $w=x_{e_i}$ and $𝚥\ne x_{e_i}$, then

$$\begin{array}{ccc}\hfill \Pi \left(\right[w,𝚥\left]\right)\pm \Pi \left(w\right)\pi (𝚥)\pm \tau \left(\right[w,𝚥\left]\right)& =& \Pi (x_{e_i}𝚥-𝚥x_{e_i})\pm \Pi \left(x_{e_i}\right)x_{2q}\pm \tau (x_{e_i}𝚥-𝚥x_{e_i})\hfill \\ & =& \Pi (x_{e_i}-0)\pm 0x_{2q}\pm \tau (x_{e_i}-0)\hfill \\ & =& 0\pm 0\pm 0=0\in L.\hfill \end{array}$$

(c)

If $𝚥=x_{e_i}$ and $w\ne x_{e_i}$, then

$$\begin{array}{ccc}\hfill \Pi \left(\right[w,𝚥\left]\right)\pm \Pi \left(w\right)\pi (𝚥)\pm \tau \left(\right[w,𝚥\left]\right)& =& \Pi (w{x}_{e_i}-x_{e_i}w)\pm \Pi \left(w\right)\pi \left(x_{e_i}\right)\pm \tau (w{x}_{e_i}-x_{e_i}w)\hfill \\ & =& \Pi (0-x_{e_i})\pm x_{2t}0\pm \tau (0-x_{e_i})\hfill \\ & =& \Pi \left(x_{e_i}\right)\pm 0\pm \tau \left(x_{e_i}\right)\hfill \\ & =& 0\in L.\hfill \end{array}$$

(d)

If $w\ne x_{e_i}$ and $𝚥\ne x_{e_i}$, then

$$\begin{array}{ccc}\hfill \Pi \left(\right[w,𝚥\left]\right)\pm \Pi \left(w\right)\pi (𝚥)\pm \tau \left(\right[w,𝚥\left]\right)& =& \Pi (w𝚥-𝚥w)\pm \Pi \left(w\right)\pi (𝚥)\pm \tau (w𝚥-𝚥w)\hfill \\ & =& \Pi (w-𝚥)\pm x_{2t}{x}_{2j}\pm \tau (w-𝚥)\hfill \\ & =& \Pi \left(w\right)-\Pi (𝚥)\pm 0\pm \left(\tau \right(w)-\tau (𝚥\left)\right)\hfill \\ & =& x_{2t}-x_{2t}\pm (x_h-x_h)=0\in L.\hfill \end{array}$$

Therefore, in all cases the identity $\Pi \left(\right[w,𝚥\left]\right)\pm \Pi \left(w\right)\pi (𝚥)\pm \tau \left(\right[w,𝚥\left]\right)\in L$ is satisfied for all $w,𝚥\in X$. However, upon examining the conclusion of Theorem 8, we find that $X/L$ is isomorphic to X, which is not commutative and $\pi \left(X\right)⊈L$. However, L is not a prime ideal in X, since for some fixed elements m and l, we can see that $x_{2m}X{x}_{2l}\subseteq L$, but $x_{2m}\notin L$ and $x_{2l}\notin L$. This confirms that the primeness assumption of L in Theorem 8 cannot be neglected.

(ii)

Similarly, the necessity of the primeness hypothesis of L in Theorem 8 for the identity $\Pi \left(\right[w,𝚥\left]\right)\pm \Pi (𝚥)\pi \left(w\right)\pm \tau \left(\right[w,𝚥\left]\right)\in L$ for all $w,𝚥\in X$ can be verified.

Remark 10.

If we consider X, τ, σ, and $(\Pi ,\pi )$ as in Example 8 and follow similar strategies as above, then it can be verified that $(\Pi ,\pi )$ satisfies the identities in Proposition 1, Theorems 9 and 10. However, when we examine the conclusions of Proposition 1, Theorems 9 and 10, we find that $X/L$ is not commutative, $char(X/L)=2$, $\pi \left(X\right)⊈L$,$(\Pi \pm \tau )\left(X\right)⊈L$, and $\Pi \left(X\right)⊈L$. Hence, we can conclude that the primeness hypothesis of L in Proposition 1, Theorems 9 and 10 cannot be overcome.

The following remark indicates that Examples 4 and 6 remain valid in asserting that the primeness condition of L in Theorems 8–10 cannot be ignored.

Remark 11.

Taking into account the considerations in Example 4 or Example 6, we can similarly verify that $(\Pi ,\pi )$ still satisfies the specified identities in Theorems 8–10 for every $w,𝚥\in X$. However, the conclusions drawn from these theorems do not hold. This discrepancy arises from the fact that L is not a prime in X. Consequently, the assumption of primeness for L in Theorems 8–10 can not be disregarded.

4. Conclusions

Exploring derivations and their generalizations has played a pivotal role in understanding the properties of the algebraic structure of rings. Rather than discussing the properties of prime or semiprime rings, which have been covered in many recent studies, this article delves into this rich algebraic landscape in a broader context by introducing a novel additive mapping $\Pi$ on an arbitrary ring X. This mapping is called a generalized $(\tau ,\sigma )$-L-derivation associated with an $(\tau ,\sigma )$-L-derivation $\pi$. Afterwards, we assume a ring X that admits certain equations involving $(\Pi ,\pi )$ and discuss the behavior of a factor ring $X/L$, where L is a prime ideal in X. The importance of this study stems from the following reasons: $\left(i\right)$ Many of the findings in existing literature are straightforward under certain constraints. $\left(ii\right)$ Generalized $(\tau ,\sigma )$-L-derivations impose certain structural consequences on $X/L$, that these identities allow one to deduce integrality and commutativity, and that the framework unifies several derivation based approaches under a common setting.

5. Future Studies

This research provides a broad foundation for further development in various directions. For example, it can be assumed that the ideal L is semiprime, the ring X is non-associative, or that $(\Pi ,\pi )$ is a generalized $(\tau ,\sigma )$-L-semi-derivation, or a multiplicative generalized $(\tau ,\sigma )$-L-derivation, etc. Additionally, the potential effects of these identities on Lie algebras, non-associative algebras, or rings with involutions can be explored.