On the Structure of Quotient Rings R/P via Identities with Multiplicative (Generalized) Derivations

Abstract

This work investigates the structure of an arbitrary ring R that contains a two-sided ideal I and a prime ideal P satisfying the condition $P⊊I$. Our analysis centers on the consequences of several identities that involve three multiplicative (generalized) derivations, denoted by ${\Theta }_1, {\Theta }_2, {\Theta }_3:R\to R$. These are associated with maps ${\theta }_1, {\theta }_2, {\theta }_3:R\to R$, which are not presumed to be additive or to be derivations themselves. The study further incorporates a non-zero derivation $\Delta$ along with two arbitrary, potentially non-additive, maps ${\Gamma }_1, {\Gamma }_2:R\to R$. We establish conditions under which these identities lead to significant structural properties of the ring. To underscore the importance of our assumptions, we construct an example demonstrating that the primeness hypothesis on the ideal P is indispensable for our main conclusions.

1. Introduction

Let R represent an arbitrary ring and $Z\left(R\right)$ its center. For any elements $x, y\in R$, the notation $[x, y]$ will stand for the commutator, defined as $xy-yx$, while $x\circ y$ will denote the anti-commutator, given by $xy+yx$.

The notion of a generalized derivation extends the concept of a standard derivation. Specifically, an additive mapping $F:R\to R$ is classified as a generalized derivation if it is associated with a derivation $d:R\to R$ such that the rule $F\left(xy\right)=F\left(x\right)y+xd\left(y\right)$ is satisfied for all $x, y\in R$.

A significant body of research in ring theory has focused on how the presence of derivations and generalized derivations satisfying certain identities can force a ring to be commutative. An overview of such results can be found in sources like [1,2]. For instance, in the context of a prime ring R with a non-zero two-sided ideal I, Ashraf et al. [1] investigated conditions like $F\left(xy\right)-xy\in Z\left(R\right)$ and $F\left(x\right)F\left(y\right)-xy\in Z\left(R\right)$ for all $x, y\in I$, demonstrating that they lead to the commutativity of R. Building on this, Tiwari et al. [2] considered more elaborate identities involving two generalized derivations, F and G, such as $G\left(xy\right)\pm F\left(x\right)F\left(y\right)\pm xy\in Z\left(R\right)$ and $G\left(xy\right)\circ F\left(y\right)F\left(x\right)\pm yx\in Z\left(R\right)$ for all $x, y$ in an ideal I.

Related work has explored identities involving commutators and anti-commutators. For example, Bell and Daif [3] showed for a semiprime ring R that if $\left[d\right(x), d(y\left)\right]=[x, y]$ for all elements in a non-zero ideal U, then U must be central. Along similar lines, Ashraf et al. [4] examined identities like $d\left(x\right)\circ F\left(y\right)=0$ and $\left[d\right(x), F(y\left)\right]=[x, y]$ on an ideal I.

There is a growing interest in extending these types of results to a more general class of mappings. A key development in this area was the introduction of multiplicative (generalized) derivations by Dhara and Ali in [5]. A mapping $F:R\to R$, which is not required to be additive, is a multiplicative (generalized) derivation if $F\left(xy\right)=F\left(x\right)y+xg\left(y\right)$ holds for all $x, y\in R$, where g is any map from R to R. This broader definition includes standard and generalized derivations as special cases. The following example demonstrates the existence of this type of non-additive derivations in rings. Let $R=C[0, 1]$ denote the ring of all continuous functions with real or complex values (please refer to [5]). Define a map $F:R⟶R$ as follows:

$$F\left(g\left(x\right)\right)=\left\{\begin{array}{ccc}\hfill g\left(x\right)log\left|g\right(x\left)\right|, & \mathrm{if}g\left(x\right)\ne 0,& \\ 0,\hfill & otherwise.\end{array}\right.$$

It is easy to verify that F satisfies $F\left(gh\right)=F\left(g\right)h+gF\left(h\right)$ for all $g, h\in C[0, 1]$. Therefore, F is a derivation of R. However, F is not additive. The study of such maps in prime and semiprime rings is an active field of research, with contributions found in [5,6,7,8,9,10].

Recent studies have continued this line of inquiry. Dhara et al. [11] explored the action of three multiplicative (generalized) derivations on square closed Lie ideals, considering complex identities like $d\left(x\right)F\left(y\right)+G\left(y\right)d\left(x\right)\pm \left(E\right(x)y+yT(x\left)\right)=0$.

Recent studies have continued this line of inquiry by generalizing classical results to the context of quotient rings. For instance, Almahdi et al. [12] generalized Posner’s theorem, demonstrating that if a derivation satisfies certain conditions relative to a prime ideal P, then it either maps the ring into P or the quotient ring $R/P$ is commutative. In a similar spirit, Khan et al. [13] extended Herstein’s theorem to rings with involution, studying pairs of derivations on prime ideals and establishing conditions that force the commutativity of $R/P$ or constrain the derivations themselves. For further studies related to differential identities on rings relative to prime or semiprime ideals, see [14,15,16,17,18].

Drawing motivation from these findings, the present article extends the investigation to a more general context. We work with an arbitrary ring R, a prime ideal P, and a two-sided ideal I of R where $P⊊I$. Within this framework, we let ${\Theta }_1, {\Theta }_2, {\Theta }_3:R\to R$ be three multiplicative (generalized) derivations, whose associated maps ${\theta }_1, {\theta }_2, {\theta }_3$ are not necessarily additive or derivations. Further, we include a non-zero derivation $\Delta$ and two arbitrary maps ${\Gamma }_1, {\Gamma }_2$. This paper is dedicated to studying the consequences of the following identities for all $x, y\in I$: (i) $\Delta \left(x\right){\Theta }_1\left(y\right)+{\Theta }_2\left(y\right)\Delta \left(x\right)\pm ({\Gamma }_1\left(x\right)y+y{\Gamma }_2\left(x\right))\in P$, (ii) ${\Theta }_3\left(xy\right)+{\Theta }_2\left(y\right){\Theta }_1\left(x\right)\pm ({\Gamma }_1\left(y\right)x+x{\Gamma }_2\left(y\right))\in P$, (iii) ${\Gamma }_1\left(xy\right)+{\Theta }_1\left(x\right)y\pm (yx+xy)\in P$, (iv) ${\Theta }_1\left(x\right){\Theta }_1\left(y\right)+{\Gamma }_1\left(x\right)y\pm yx\in P$, (v) $\Delta \left(x\right)\Delta \left(y\right)+{\Gamma }_1\left(x\right)y+{\Theta }_1\left(yx\right)\in P$. Additionally, we provide examples to confirm that the hypotheses within our results are essential.

2. Preliminaries

We begin by recalling some fundamental definitions and lemmas that are crucial for the development of the subsequent sections.

Fact 1.

Let R be a ring with a prime ideal P, and let I be an ideal satisfying $P⊊I$. If $aIb\subseteq P$ for $a, b\in R$, then $a\in P$ or $b\in P$. In particular, $aI\subseteq P$ or $Ia\subseteq P$ implies $a\in P$.

Lemma 1

([19] Corollary 2.6). Let R be a ring with a prime ideal P and an ideal I such that $P⊊I$. If Δ is a derivation on R that satisfies $[\Delta (x), x]\in P$ for every $x\in I$, then either $\Delta \left(R\right)\subseteq P$ or the factor ring $R/P$ is commutative.

Lemma 2.

Suppose P is a prime ideal in a ring R and I is an ideal with $P⊊I$. If for certain fixed elements $a, b\in I$, the relation $axb+bxa\in P$ holds for all $x\in I$, then either $a\in P$ or $b\in P$.

Proof. 

Similar to the proof of ([20] Lemma 3.6), we can reach the desired conclusion. □

Lemma 3

([19] Lemma 2.3). Given a ring R, a prime ideal P, and an ideal I with $P⊊I$, the condition $[I, I]\subseteq P$ is sufficient to conclude that the quotient ring $R/P$ is commutative.

Lemma 4.

Let R be a ring, P a prime ideal, I a two-sided ideal with $P⊊I$, and $\Theta :R\to R$ a multiplicative (generalized) derivation associated with θ, where θ is an arbitrary self-map of R. If

$$\Theta \left(xy\right)-x\Theta \left(y\right)\in P(\forall x, y\in I),$$

then

$$\Theta \left(rs\right)-r\Theta \left(s\right)\in P(\forall r, s\in R).$$

Hence, Θ is a multiplicative right centralizer modulo P (and similarly on the left).

Proof. 

Since I is two-sided, for any $x, y\in I$ and $r\in R$, we have $xr, ry\in I$. Applying the hypothesis to $\left(xr\right)y$ and $x\left(ry\right)$ gives

$$\Theta \left(\right(xr\left)y\right)-xr\Theta \left(y\right)\in P,\Theta \left(x\right(ry\left)\right)-x\Theta \left(ry\right)\in P.$$

Subtracting and using the primality of P yields

$$\Theta \left(ry\right)-r\Theta \left(y\right)\in P(\forall r\in R,y\in I).$$

For any $s\in R$ and $y\in I$, we have $sy\in I$. Putting this in place of y in the previous equation gives

$$\Theta \left(r\right(sy\left)\right)-r\Theta \left(sy\right)\in P.$$

Writing $\Theta \left(r\right(sy\left)\right)=\Theta \left(\right(rs\left)y\right)$ and expanding the definition of $\Theta$ shows

$$\left(\Theta \left(rs\right)-r\Theta \left(s\right)\right)y\in P(\forall y\in I).$$

The primality of P again forces $\Theta \left(rs\right)-r\Theta \left(s\right)\in P$ for all $r, s\in R$. □

Lemma 5.

Let R be a ring with prime ideal P. If $\Theta :R\to R$ is a multiplicative (generalized) derivation modulo P, i.e.,

$$\Theta \left(xy\right)-\Theta \left(x\right)y-x\theta \left(y\right)\in P(\forall x, y\in R),$$

then θ is itself a multiplicative derivation modulo P on R.

Proof. 

Compare $\Theta \left(x\right(yz\left)\right)$ and $\Theta \left(\right(xy\left)z\right)$, substitute via the defining congruence, and cancel the common terms to obtain

$$x\left(\theta \left(yz\right)-\theta \left(y\right)z-y\theta \left(z\right)\right)\in P.$$

Primality of P yields the desired relation on $\theta$. □

Remark 1.

For a ring R with a prime ideal P and a two-sided ideal I where $P⊊I$, let Θ be a multiplicative (generalized) derivation associated with a map θ on $R.$ The following two conditions are equivalent:

(i)

$\Theta \left(xy\right)-\Theta \left(x\right)y\in P(\forall x, y\in I).$

($ii$)

$\theta \left(I\right)\subseteq P.$

Moreover, either condition forces $\theta \left(R\right)\subseteq P$.

Proof. 

$\left(i\right)\Rightarrow \left(ii\right)$ Suppose $\Theta \left(xy\right)-\Theta \left(x\right)y\in P$ for all $x, y\in I$. Given that $\Theta$ is a multiplicative (generalized) derivation, which means

$$\Theta \left(xy\right)=\Theta \left(x\right)y+x\theta \left(y\right).$$

Therefore, we can conclude that $x\theta \left(y\right)\in P$. Since $I⊈P$ and P is prime, this forces $\theta \left(y\right)\in P$ for all $y\in I$, i.e., $\theta \left(I\right)\subseteq P$.

$\left(ii\right)\Rightarrow \left(i\right)$ Conversely, if $\theta \left(I\right)\subseteq P$, then for all $x, y\in I$

$$\Theta \left(xy\right)-\Theta \left(x\right)y=x\theta \left(y\right)\in xP\subseteq P.$$

Hence, $\Theta \left(xy\right)-\Theta \left(x\right)y\in P.$

Finally, to show that $\theta \left(R\right)\subseteq P$, pick any $r\in R$ and $y\in I$. From Lemma 5, we have

$$\theta \left(ry\right)-\theta \left(r\right)y-r\theta \left(y\right)\in P.$$

Since $ry\in I$, we have $\theta \left(ry\right)\in P$ and $r\theta \left(y\right)\in P$, so $\theta \left(r\right)y\in P$. As $I⊈P$ and P is prime, it follows that $\theta \left(r\right)\in P$. Thus, $\theta \left(R\right)\subseteq P$. □

3. Main Results

We are now in a position to prove our main results. Each theorem in this section establishes a key structural property of the ring based on the identities involving multiplicative (generalized) derivations.

Theorem 1.

Let R be a ring with a prime ideal P and a two-sided ideal I such that $P⊊I$. Consider two multiplicative (generalized) derivations, ${\Theta }_1$ and ${\Theta }_2$, with associated maps ${\theta }_1$ and ${\theta }_2$, respectively, along with two arbitrary maps ${\Gamma }_1, {\Gamma }_2$ and a derivation Δ on $R.$ If the identity

$$\Delta \left(x\right){\Theta }_1\left(y\right)+{\Theta }_2\left(y\right)\Delta \left(x\right)\pm ({\Gamma }_1\left(x\right)y+y{\Gamma }_2\left(x\right))\in P$$

is satisfied for all $x, y\in I$, then one of the following assertions holds:

(i) 

The derivation Δ maps R into P, both $[{\Gamma }_1\left(I\right), R]$ and $[{\Gamma }_2\left(I\right), R]$ are contained in P, and the sum $({\Gamma }_1+{\Gamma }_2)$ maps I into P.

(ii) 

The factor ring $R/P$ is commutative, and the sum of the associated maps, $({\theta }_1+{\theta }_2)$, maps R into P.

(iii) 

The map ${\theta }_1$ sends R into P, and ${\Theta }_1$ acts as a multiplicative left centralizer while ${\Theta }_2$ acts as a multiplicative right centralizer, both modulo P on R.

Proof. 

We begin with the given identity

$$\Delta \left(x\right){\Theta }_1\left(y\right)+{\Theta }_2\left(y\right)\Delta \left(x\right)\pm ({\Gamma }_1\left(x\right)y+y{\Gamma }_2\left(x\right))\in P,\forall x, y\in I.$$

(1)

If we replace y with $yt$ for all $t\in I$, the identity becomes

$$\Delta \left(x\right)\{{\Theta }_1\left(y\right)t+y{\theta }_1\left(t\right)\}+\{{\Theta }_2\left(y\right)t+y{\theta }_2\left(t\right)\}\Delta \left(x\right)\pm ({\Gamma }_1\left(x\right)yt+yt{\Gamma }_2\left(x\right))\in P.$$

(2)

By post-multiplying relation (1) by t and subtracting the result from (2), we obtain

$$\Delta \left(x\right)y{\theta }_1\left(t\right)+{\Theta }_2\left(y\right)[t, \Delta \left(x\right)]+y{\theta }_2\left(t\right)\Delta \left(x\right)\pm y[t, {\Gamma }_2\left(x\right)]\in P,\forall x, y, t\in I.$$

(3)

The next step is to substitute $uy$ for y in (3), where $u\in I$, we get

$$\begin{array}{cc}\hfill & \Delta \left(x\right)uy{\theta }_1\left(t\right)+{\Theta }_2\left(uy\right)[t, \Delta \left(x\right)]+uy{\theta }_2\left(t\right)\Delta \left(x\right)\pm uy[t, {\Gamma }_2\left(x\right)]\in P.\hfill \end{array}$$

(4)

Now, pre-multiplying (3) by u and subtracting it from (4) yields

$$[\Delta \left(x\right), u]y{\theta }_1\left(t\right)+({\Theta }_2\left(uy\right)-u{\Theta }_2\left(y\right))[t, \Delta \left(x\right)]\in P,\forall x, y, t, u\in I.$$

(5)

Let us replace t by $tw$ in (5) for all $w\in I$. After expansion, and then subtracting the result of post-multiplying (5) by w, we isolate the following relation

$$[\Delta \left(x\right), u]yt{\theta }_1\left(w\right)+({\Theta }_2\left(uy\right)-u{\Theta }_2\left(y\right))t[w, \Delta \left(x\right)]\in P,\forall x, y, t, u, w\in I.$$

(6)

For any $q, v\in I$, we choose t in (6) to be the element $[q, \Delta (v\left)\right]t.$ This substitution, combined with an application of (5) to simplify an emerging term, leads to

$$[\Delta \left(x\right), u]y[q, \Delta \left(v\right)]t{\theta }_1\left(w\right)-[\Delta \left(v\right), u]y{\theta }_1\left(q\right)t[w, \Delta \left(x\right)]\in P.$$

(7)

In this last expression, if we substitute $\Delta \left(v\right)u$ for u and use (7) to simplify, we are left with $[\Delta \left(x\right), \Delta \left(v\right)]uy[\Delta \left(v\right), q]t{\theta }_1\left(w\right)\in P$. From the primeness of P and Fact 1, we can deduce that for any $v\in I$, either $[\Delta (x), \Delta (v\left)\right]\in P$ for all $x\in I$, or $[\Delta (v), q]\in P$ for all $q\in I$, or ${\theta }_1\left(I\right)\subseteq P$. Let $T_1=\{v\in I\mid [\Delta \left(x\right), \Delta \left(v\right)]\in P\forall x\in I\}$ and $T_2=\{v\in I\mid [\Delta \left(v\right), I]\subseteq P\}$. Since I is an additive group and cannot be the union of two proper subgroups, we must have $I=T_1$ or $I=T_2$. This creates a case distinction.

Case 1: Suppose $[\Delta (v), q]\in P$ for all $q, v\in I$. This implies $[\Delta (v), v]\in P$, and by Lemma 1, either $\Delta \left(R\right)\subseteq P$ or $R/P$ is commutative. If $\Delta \left(R\right)\subseteq P$, the initial identity (1) reduces to ${\Gamma }_1\left(x\right)y+y{\Gamma }_2\left(x\right)\in P$. A standard argument of replacing y with $yt$ shows that $[{\Gamma }_2\left(I\right), I]\subseteq P$, which can be extended to show $[{\Gamma }_2\left(I\right), R]\subseteq P$. Similarly, $[{\Gamma }_1\left(I\right), R]\subseteq P$. The identity then simplifies to $({\Gamma }_1\left(x\right)+{\Gamma }_2\left(x\right))y\in P$, which by primeness implies $({\Gamma }_1+{\Gamma }_2)\left(I\right)\subseteq P$. If, on the other hand, $R/P$ is commutative, then from (3), we get $y({\theta }_1\left(t\right)+{\theta }_2\left(t\right))\Delta \left(x\right)\in P$. Since we can find x such that $\Delta \left(x\right)\notin P$ (otherwise we are in the previous subcase), primeness implies $y({\theta }_1\left(t\right)+{\theta }_2\left(t\right))\in P$, and again, ${\theta }_1\left(t\right)+{\theta }_2\left(t\right)\in P$ for all $t\in I$. This can be extended to show $({\theta }_1+{\theta }_2)\left(R\right)\subseteq P$.

Case 2: Suppose ${\theta }_1\left(I\right)\subseteq P$. Then relation (6) simplifies to $({\Theta }_2\left(uy\right)-u{\Theta }_2\left(y\right))t[w, \Delta \left(x\right)]\in P$. By primeness, either $[I, \Delta (I\left)\right]\subseteq P$ (which leads to Case 1) or ${\Theta }_2\left(uy\right)-u{\Theta }_2\left(y\right)\in P$ for all $u, y\in I$. The latter means ${\Theta }_2$ is a multiplicative right centralizer on I, and thus on R by Lemma 4. Since ${\theta }_1\left(I\right)\subseteq P$, by Remark 1, ${\Theta }_1$ is a multiplicative left centralizer. This corresponds to conclusion (iii).

Case 3: Suppose $[\Delta (x), \Delta (v\left)\right]\in P$ for all $x, v\in I$. Replacing x with $xv$ gives $\Delta \left(x\right)[v, \Delta (v\left)\right]+[x, \Delta (v\left)\right]\Delta \left(v\right)\in P$. A further substitution of $vx$ for x leads to $\Delta \left(v\right)x[v, \Delta (v\left)\right]+[v, \Delta (v\left)\right]x\Delta \left(v\right)\in P$. By Lemma 2, this means $[v, \Delta (v\left)\right]\in P$, reducing this to Case 1. □

Assuming that $P=\left\{0\right\}$, meaning that the ring R is prime, the following two remarks connect Theorem 1 with the classical results of Herstein and Posner as follows:

Remark 2.

If we consider the constraints ${\Gamma }_1={\Gamma }_2=0$, ${\Theta }_2=-{\Theta }_1$, and ${\Theta }_1=\Delta$ in Equation (1), then we can simply obtain Herstein’s theorem [21]. This theorem states that if Δ is a non-zero derivation on a prime ring R with a characteristic other than 2, and the condition $[\Delta (x), \Delta (y\left)\right]=0$ is satisfied for all $x, y\in R$, then R is commutative.

Remark 3.

By setting ${\Gamma }_1={\Gamma }_2=0$, ${\Theta }_2=-{\Theta }_1$, ${\Theta }_1=I_{id}$ (identity map on R) and $y=x$, these constraints simplify Equation (1) to $[\Delta (x), x]\in P=\left\{0\right\}\in Z\left(R\right).$ Therefore, we can easily obtain Posner’s theorem [22], which states that if Δ is a non-zero derivation on a prime ring R, and the condition $[\Delta (x), x]\in Z\left(R\right)$ is satisfied for all $x\in R$, then R is commutative.

In ([10] Theorem 4), Tiwari et al. showed that $[{\theta }_2\left(x\right), \left(x\right)]=0$ for every $x\in I$, where I is a non-zero ideal in a semiprime ring R that admits multiplicative (generalized) derivations ${\Theta }_1$ and ${\Theta }_2$ associated with maps ${\theta }_1$ and ${\theta }_2$, respectively. These derivations must satisfy the identity ${\Theta }_1\left(xy\right)+{\Theta }_2\left(y\right){\Theta }_2\left(x\right)+[x, \Gamma \left(y\right)]=0$ for all $x, y\in I$, where $\Gamma$ is any map of R. The following two theorems, without the semiprime ring restriction, discuss potential conclusions when considering a more general identity that involves a prime ideal, two maps, and three multiplicative (generalized) derivations.

Theorem 2.

Let R be a ring with a prime ideal P and a two-sided ideal I such that $P⊊I$. Suppose ${\Theta }_1, {\Theta }_2, {\Theta }_3$ are three multiplicative (generalized) derivations with associated maps ${\theta }_1, {\theta }_2, {\theta }_3$, respectively, and let ${\Gamma }_1, {\Gamma }_2$ be arbitrary maps. If, for all $x, the relation y\in I$

$${\Theta }_3\left(xy\right)+{\Theta }_1\left(y\right){\Theta }_2\left(x\right)\pm ({\Gamma }_1\left(y\right)x+x{\Gamma }_2\left(y\right))\in P$$

holds, then it follows that the commutator $[{\Theta }_1\left(x\right), {\theta }_2\left(x\right)]$ is in P for every $x\in I.$

Proof. 

For all $x, y\in I$, we have

$$\begin{array}{c}\hfill {\Theta }_3\left(xy\right)+{\Theta }_1\left(y\right){\Theta }_2\left(x\right)\pm ({\Gamma }_1\left(y\right)x+x{\Gamma }_2\left(y\right))\in P.\end{array}$$

(8)

Let us analyze the result of replacing x by $xy$ in (8). The expansion of ${\Theta }_3\left(x{y}^2\right)$ and ${\Theta }_2\left(xy\right)$ is given by ${\Theta }_3\left(xy\right)y+xy{\theta }_3\left(y\right)$ and ${\Theta }_2\left(x\right)y+x{\theta }_2\left(y\right)$, respectively. The full expression for the substitution $x\to xy$ can be written as

$$\begin{array}{cc}\hfill & \{{\Theta }_3\left(xy\right)+{\Theta }_1\left(y\right){\Theta }_2\left(x\right)\pm ({\Gamma }_1\left(y\right)x+x{\Gamma }_2\left(y\right))\}y\hfill \\ \hfill & +xy{\theta }_3\left(y\right)+{\Theta }_1\left(y\right)x{\theta }_2\left(y\right)\pm x[y, {\Gamma }_2\left(y\right)]\in P.\hfill \end{array}$$

By our hypothesis, we are left with

$$\begin{array}{c}\hfill xy{\theta }_3\left(y\right)+{\Theta }_1\left(y\right)x{\theta }_2\left(y\right)\pm x[y, {\Gamma }_2\left(y\right)]\in P.\end{array}$$

(9)

In this relation, replace x with $rx$, where $r\in R$. Then, left-multiply (9) by r and subtract the two results to eliminate other terms, leaving $[{\Theta }_1\left(y\right), r]x{\theta }_2\left(y\right)\in P$ for all $x, y\in I$ and $r\in R$. By substituting $xt$ for x (for $t\in R$), we can show that $[{\Theta }_1\left(y\right), r]x[t, {\theta }_2\left(y\right)]\in P$. Making the specific choices $r={\theta }_2\left(y\right)$ and $t={\Theta }_1\left(y\right)$, we get $[{\Theta }_1\left(y\right), {\theta }_2\left(y\right)]x[{\Theta }_1\left(y\right), {\theta }_2\left(y\right)]\in P$. It follows that

$$\begin{array}{c}\hfill [{\Theta }_1\left(y\right), {\theta }_2\left(y\right)]sx[{\Theta }_1\left(y\right), {\theta }_2\left(y\right)]s\in P,\end{array}$$

where $s\in I$. By Fact 1, it follows that $[{\Theta }_1\left(y\right), {\theta }_2\left(y\right)]s\in P$ for all $y, s\in I$. It means $[{\Theta }_1\left(y\right), {\theta }_2\left(y\right)]I\subseteq P$ for all $y\in I$. Moreover, $[{\Theta }_1\left(y\right), {\theta }_2\left(y\right)]RI\subseteq P$ for all $y\in I$. By primeness, $[{\Theta }_1\left(y\right), {\theta }_2\left(y\right)]\subseteq P$ for all $y\in I$. This completes the proof. □

We now present another result derived from the same identity, this time considering the case where ${\theta }_1={\theta }_2$.

Theorem 3.

Let R be a ring containing a prime ideal P and a two-sided ideal I where $P⊊I$. Let ${\Theta }_1, {\Theta }_2, {\Theta }_3$ be multiplicative (generalized) derivations with associated maps ${\theta }_2, {\theta }_2, {\theta }_3$, respectively, and let ${\Gamma }_1, {\Gamma }_2$ be arbitrary maps. If the identity

$${\Theta }_3\left(xy\right)+{\Theta }_1\left(y\right){\Theta }_2\left(x\right)\pm ({\Gamma }_1\left(y\right)x+x{\Gamma }_2\left(y\right))\in P$$

holds for all $x, y\in I$, then $x[{\theta }_2\left(x\right), x]\in P$ for all $x\in I$.

Proof. 

Denote

$$E(x, y)={\Theta }_3\left(xy\right)+{\Theta }_1\left(y\right){\Theta }_2\left(x\right)\pm \left({\Gamma }_1\left(y\right)x+x{\Gamma }_2\left(y\right)\right),$$

so $E(x, y)\in P$ for all $x, y\in I$. Let $z\in I$. Replacing x by $xz$ in $E(x, y)$ and subtracting $E(x, y)z$ gives

$${\Theta }_3\left(xzy\right)-{\Theta }_3\left(xy\right)z+{\Theta }_1\left(y\right)x{\theta }_2\left(z\right)\pm x[z, {\Gamma }_2\left(y\right)]\in P.$$

In this relation, first set $y\mapsto yz$, then set $x\mapsto zx$, and subtract the two outcomes. The ${\Theta }_1$–terms cancel, leaving

$$\begin{array}{cc}\hfill & \left\{{\Theta }_3\left(xzyz\right)-{\Theta }_3\left(xyz\right)z\right\}-\left\{{\Theta }_3\left(zxzy\right)-{\Theta }_3\left(zxy\right)z\right\}\hfill \\ \hfill +& y{\theta }_2\left(z\right)x{\theta }_2\left(z\right)\pm \left(x[z, {\Gamma }_2\left(yz\right)]-zx[z, {\Gamma }_2\left(y\right)]\right)\in P.\hfill \end{array}$$

Taking $y=z$ and using the multiplicativity of ${\Theta }_3$, this simplifies to

$$\begin{array}{cc}\hfill & x{z}^2{\theta }_3\left(z\right)-zxz{\theta }_3\left(z\right)+z{\theta }_2\left(z\right)x{\theta }_2\left(z\right)\hfill \\ \hfill \pm & \left(x[z, {\Gamma }_2\left(z^2\right)]-zx[z, {\Gamma }_2\left(z\right)]\right)\in P.\hfill \end{array}$$

Finally, substituting $x\mapsto zx$ and then subtracting its left-z-multiple isolates $[z{\theta }_2\left(z\right), z]x{\theta }_2\left(z\right)\in P.$ It follows that $[z{\theta }_2\left(z\right), z]x[z{\theta }_2\left(z\right), z]\in P.$ By Fact 1, $z[{\theta }_2\left(z\right), z]\in P\mathrm{for}\mathrm{all}z\in I.$ □

Theorem 4.

Let ${\Theta }_1$ be a multiplicative (generalized) derivation on a ring R, and let ${\Gamma }_1$ be an arbitrary map. For a prime ideal P and a two-sided ideal I of R with $P⊊I$, if the condition

$${\Gamma }_1\left(xy\right)+{\Theta }_1\left(x\right)y\pm (yx+xy)\in P$$

holds for all $x, y\in I$, then the quotient ring $R/P$ must be commutative and the map ${\theta }_1$ must map R into P. As a direct result, ${\Theta }_1$ acts as a multiplicative left centralizer modulo P on R.

Proof. 

The initial assumption is

$${\Gamma }_1\left(xy\right)+{\Theta }_1\left(x\right)y\pm (yx+xy)\in P,\forall x, y\in I.$$

(10)

We are comparing the results of two substitutions. First, we replace x with $xz$ for $z\in I$. Second, we replace y with $zy$ in the original identity. The expansions are

$$\begin{array}{c}{\Gamma }_1\left(xzy\right)+{\Theta }_1\left(x\right)zy+x{\theta }_1\left(z\right)y\pm (yxz+xzy)\in P,\end{array}$$

(11)

$$\begin{array}{c}{\Gamma }_1\left(xzy\right)+{\Theta }_1\left(x\right)zy\pm (zyx+xzy)\in P.\end{array}$$

(12)

Subtracting (12) from (11) isolates $x{\theta }_1\left(z\right)y\pm (yxz-zyx)\in P$, which is $x{\theta }_1\left(z\right)y\pm [yx, z]\in P$. If we replace y by $yx$ in this relation, then subtract the result of right-multiplying the relation by x, we get $yx[x, z]\in P$. The primeness of P implies $x[x, z]\in P$ for all $x, z\in I$. A standard argument follows: replace z by $zt$ to get $xz[x, t]\in P$, which means $xI[x, t]\subseteq P$. This implies that for any $x\in I$, either $x\in P$ or $[x, I]\subseteq P$. Since $I⊈P$, it must be that $[I, I]\subseteq P$. By Lemma 3, $R/P$ is commutative. With commutativity, the relation $x{\theta }_1\left(z\right)y\pm [yx, z]\in P$ reduces to $x{\theta }_1\left(z\right)y\in P$. This implies $I{\theta }_1\left(I\right)I\subseteq P$, and by primeness, ${\theta }_1\left(I\right)\subseteq P$. By Remark 1, this gives ${\theta }_1\left(R\right)\subseteq P$. The consequence that ${\Theta }_1$ is a left centralizer follows from Lemma 4. □

Theorem 5.

For a ring R equipped with a prime ideal P and a two-sided ideal I satisfying $P⊊I$, let ${\Theta }_1$ be a multiplicative (generalized) derivation with its associated map ${\theta }_1$. If for an arbitrary map ${\Gamma }_1$, the identity

$${\Theta }_1\left(x\right){\Theta }_1\left(y\right)+{\Gamma }_1\left(x\right)y\pm yx\in P$$

is satisfied for all $x, y\in I$, then the quotient ring $R/P$ is commutative and ${\theta }_1\left(R\right)\subseteq P$. This, in turn, implies that ${\Theta }_1$ is a multiplicative left centralizer modulo P on R.

Proof. 

We start with the relation

$${\Theta }_1\left(x\right){\Theta }_1\left(y\right)+{\Gamma }_1\left(x\right)y\pm yx\in P,\forall x, y\in I.$$

(13)

Substituting $yz$ for y (where $z\in I$) and expanding ${\Theta }_1\left(yz\right)$, we obtain

$${\Theta }_1\left(x\right){\Theta }_1\left(y\right)z+{\Theta }_1\left(x\right)y{\theta }_1\left(z\right)+{\Gamma }_1\left(x\right)yz\pm yzx\in P.$$

(14)

Subtracting the result of right-multiplying (13) by z from (14) yields

$${\Theta }_1\left(x\right)y{\theta }_1\left(z\right)\pm y[z, x]\in P.$$

(15)

In this new expression, we replace y with $ty$ ($t\in I$), and then subtract the result of left-multiplying (15) by t. This isolates $[{\Theta }_1\left(x\right), t]y{\theta }_1\left(z\right)\in P$. This means $[{\Theta }_1\left(x\right), I]I{\theta }_1\left(I\right)\subseteq P$. Since P is a prime ideal, we have two cases.

Case 1: ${\theta }_1\left(I\right)\subseteq P$. The relation (15) simplifies to $y[z, x]\in P$, which implies $I[I, I]\subseteq P$. As $I⊈P$, we must have $[I, I]\subseteq P$, so $R/P$ is commutative by Lemma 3. Also, from ${\theta }_1\left(I\right)\subseteq P$, it follows that ${\theta }_1\left(R\right)\subseteq P$ by Remark 1.

Case 2: $[{\Theta }_1\left(x\right), I]\subseteq P$ for all $x\in I$. This means ${\Theta }_1\left(x\right)$ commutes with all elements of I modulo P. From (15), we have ${\Theta }_1\left(x\right)y{\theta }_1\left(z\right)\equiv \mp y[z, x](modP)$. Since $y{\theta }_1\left(z\right)\in I$, we can commute it with ${\Theta }_1\left(x\right)$, so $y{\theta }_1\left(z\right){\Theta }_1\left(x\right)\equiv \mp y[z, x](modP)$. This implies $y({\theta }_1\left(z\right){\Theta }_1\left(x\right)\pm [z, x])\in P$, and by primeness, ${\theta }_1\left(z\right){\Theta }_1\left(x\right)\pm [z, x]\in P$. Replacing x by $xz$ in this relation leads to ${\theta }_1\left(z\right)x{\theta }_1\left(z\right)\in P$, which means ${\theta }_1\left(z\right)\in P$. Thus, ${\theta }_1\left(I\right)\subseteq P$, and we are back in Case 1.

Both cases lead to the same conclusions, proving the theorem. □

In [23], Bell and Kappe proved that a derivation $\Delta$ is trivial on a prime ring R when the identity $\Delta \left(xy\right)=\Delta \left(y\right)\Delta \left(x\right)$ holds for every $x, y$ in a non-zero right ideal I in R. This result has garnered widespread attention in various contexts by multiple researchers. Building on these results, the following theorem aims to extend the previous theorem to include any map, derivation, and a multiplicative (generalized) derivation operating on a two-sided ideal along with a prime ideal in an arbitrary ring, as outlined below:

Theorem 6.

Let R be a ring, a prime ideal P, and a two-sided ideal I of R. Suppose Δ is a derivation on R and ${\Theta }_1$ is a multiplicative (generalized) derivation with its associated map ${\theta }_1$. For an arbitrary map ${\Gamma }_1$, if the relation

$$\Delta \left(x\right)\Delta \left(y\right)+{\Gamma }_1\left(x\right)y+{\Theta }_1\left(yx\right)\in P$$

holds for every $x, y\in I$, then precisely one of the following two conditions is true:

(i) 

The derivation Δ maps all of R into P. In this case, ${\Theta }_1$ is forced to be a multiplicative right centralizer modulo P on R.

(ii) 

The factor ring $R/P$ is commutative.

Proof. 

The given condition is

$$\Delta \left(x\right)\Delta \left(y\right)+{\Gamma }_1\left(x\right)y+{\Theta }_1\left(yx\right)\in P,\forall x, y\in I.$$

(16)

As in prior proofs, we substitute $yz$ for y and subtract the original relation right-multiplied by z. This yields

$$\Delta \left(x\right)y\Delta \left(z\right)+{\Theta }_1\left(y\right)zx+y{\theta }_1\left(zx\right)-{\Theta }_1\left(yx\right)z\in P.$$

(17)

Using the definition of ${\Theta }_1$ to expand ${\Theta }_1\left(yx\right)$, this becomes

$$\Delta \left(x\right)y\Delta \left(z\right)+{\Theta }_1\left(y\right)[z, x]+y{\theta }_1\left(zx\right)-y{\theta }_1\left(x\right)z\in P.$$

(18)

Again, we replace y by $ty$ and subtract the left-multiplication of (18) by t to get

$$[\Delta \left(x\right), t]y\Delta \left(z\right)+({\Theta }_1\left(ty\right)-t{\Theta }_1\left(y\right))[z, x]\in P,\forall x, y, z, t\in I.$$

(19)

Setting $z=x$ in this relation, we obtain $[\Delta (x), t]y\Delta \left(x\right)\in P$. By substituting $yt$ for y and using the primeness of P, we deduce that $[\Delta (x), t]\in P$ for all $x, t\in I$. By Lemma 1, this implies that either $\Delta \left(R\right)\subseteq P$ or $R/P$ is commutative.

If $R/P$ is commutative, we have conclusion (ii). If $\Delta \left(R\right)\subseteq P$, then Equation (19) becomes $({\Theta }_1\left(ty\right)-t{\Theta }_1\left(y\right))[z, x]\in P$. Since P is prime, either $[I, I]\subseteq P$ (which implies $R/P$ is commutative, taking us to conclusion (ii)) or ${\Theta }_1\left(ty\right)-t{\Theta }_1\left(y\right)\in P$. The latter means ${\Theta }_1$ is a multiplicative right centralizer on I, and by Lemma 4, on R. This establishes conclusion (i). □

The following example illustrates that the assumption of primeness on the ideal P is a necessary hypothesis for our main results.

Example 1.

Let the ring R be the set of matrices of the form $R=\left\{\left(\begin{array}{ccc}0& a& b\\ 0& 0& c\\ 0& 0& 0\end{array}\right):a, b, c\in \mathbb{Z}\right\}$. In this ring, the zero ideal $P=\left\{0\right\}$ is not a prime ideal. To see this, consider the matrix units $e_{12}=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$ and $e_{23}=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right)$. We have $e_{23}R{e}_{12}=\left\{0\right\}\subseteq P$, but neither $e_{12}$ nor $e_{23}$ is in P. Furthermore, the ring R is not commutative.

Let the ideal I be defined as $I=\left\{\left(\begin{array}{ccc}0& 0& b\\ 0& 0& 0\\ 0& 0& 0\end{array}\right):b\in \mathbb{Z}\right\}$. We now define a set of mappings on R. For any matrix $M=\left(\begin{array}{ccc}0& a& b\\ 0& 0& c\\ 0& 0& 0\end{array}\right)\in R$, let:

$${\Theta }_1\left(M\right)={\Theta }_2\left(M\right)={\Theta }_3\left(M\right):=\left(\begin{array}{ccc}0& 0& b\\ 0& 0& c^2\\ 0& 0& 0\end{array}\right),$$

$${\theta }_1\left(M\right)={\theta }_2\left(M\right)={\theta }_3\left(M\right)=\Delta \left(M\right):=\left(\begin{array}{ccc}0& 0& b\\ 0& 0& c\\ 0& 0& 0\end{array}\right).$$

Let ${\Gamma }_1$ and ${\Gamma }_2$ be arbitrary maps. It is straightforward to verify that Δ is a derivation and that each ${\Theta }_i$ is a multiplicative (generalized) derivation associated with the map ${\theta }_i$.

With these definitions, one can check that the premises for Theorems 1, 4, 5, and 6 are all met for this choice of R, I, and $P=\left\{0\right\}$. Nevertheless, the conclusions of these theorems fail to hold. For instance, $R/P\cong R$ is not commutative, and $\Delta \left(R\right)$ is not contained in P. This demonstrates that the requirement for P to be a prime ideal is indispensable for these theorems.

Remark 4.

We will reuse the same ring R from Example 1 to demonstrate that the primeness condition is also necessary for Theorem 2. Let $I=R$ and $P=\left\{0\right\}$. We define the maps as follows: ${\Theta }_3=0$, ${\Gamma }_2=0$, ${\Theta }_2={\theta }_2$, ${\Theta }_1\left(x\right)=x$ for all $x\in R$, and we choose the sign in the identity to be positive, with ${\Gamma }_1\left(x\right)=-x$. With these choices, the central identity in Theorem 2 states

$${\Theta }_3\left(xy\right)+{\Theta }_1\left(y\right){\Theta }_2\left(x\right)+({\Gamma }_1\left(y\right)x+x{\Gamma }_2\left(y\right))\in P,$$

reduces to the condition $y{\theta }_2\left(x\right)-yx=y({\theta }_2\left(x\right)-x)\in P$, i.e., $y({\theta }_2\left(x\right)-x)=\left\{0\right\}$. For any matrices $N=\left(\begin{array}{ccc}0& a^{\prime }& b^{\prime }\\ 0& 0& c^{\prime }\\ 0& 0& 0\end{array}\right)$ and $M=\left(\begin{array}{ccc}0& a& b\\ 0& 0& c\\ 0& 0& 0\end{array}\right)$ in R, the condition becomes

$$N·\left({\theta }_2\left(M\right)-M\right)=\left(\begin{array}{ccc}0& a^{\prime }& b^{\prime }\\ 0& 0& c^{\prime }\\ 0& 0& 0\end{array}\right)·\left(\left(\begin{array}{ccc}0& 0& b\\ 0& 0& c\\ 0& 0& 0\end{array}\right)-\left(\begin{array}{ccc}0& a& b\\ 0& 0& c\\ 0& 0& 0\end{array}\right)\right)=\left\{0\right\},$$

which is satisfied. However, the conclusion of Theorem 2 requires that

$[{\Theta }_1\left(x\right), {\theta }_2\left(x\right)]\in P$. A direct calculation for a generic matrix M shows

$$[{\Theta }_1\left(M\right), {\theta }_2\left(M\right)]=[M, {\theta }_2\left(M\right)]=\left(\begin{array}{ccc}0& -ac& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right).$$

This resulting matrix is not in $P=\left\{0\right\}$ for most choices of a and c. Thus, the conclusion of Theorem 2 is not satisfied, confirming that the primeness hypothesis is essential for it as well.

Future Research and Open Questions: The results presented in this paper establish important connections between specific operator identities and the structure of a ring relative to a prime ideal. This work naturally leads to several potential avenues for future investigation:

(i)

Generalizing the Ring Structure: A primary question is whether similar results can be obtained under different assumptions for the ring R. For instance, it would be valuable to explore these identities in the context of semiprime rings, rings with involution, or specific classes of topological rings such as Banach algebras.

($ii$)

Exploring Different Mappings: This study focused on multiplicative

(generalized) derivations. A logical next step would be to investigate analogous identities for other related mappings, such as biderivations, skew derivations, or more general forms of ($\alpha$,$\beta$) derivations to see if similar structural constraints emerge.

($iii$)

Weakening Conditions on Ideals: The hypotheses of our theorems rely on a two-sided ideal I where $P⊊I$. It remains an open question whether these conclusions hold under weaker conditions. Future work could examine these identities on one-sided ideals (i.e., left or right ideals) or on other significant subsets of the ring, such as Lie ideals.

($iv$)

Modifying the Identity Constraints: The identities in this paper all result in membership within the prime ideal P. It would be interesting to analyze what happens if the outcome of these identities is constrained in other ways, for example, if they are equal to zero, lie in the center $Z\left(R\right)$, or satisfy certain annihilator conditions.

4. Conclusions

In this paper, we have conducted a systematic investigation into the consequences of several novel identities within an arbitrary ring R containing a two-sided ideal I and a prime ideal P. Our analysis focused on identities involving a combination of multiplicative (generalized) derivations, a non-zero derivation, and other arbitrary maps.

The main contributions of this work demonstrate that these specific algebraic relations impose significant constraints on the structure of the ring. Our findings consistently show that the satisfaction of these identities forces the factor ring $R/P$ to be commutative or compels the involved mappings (such as $\Delta$ or the associated maps ${\theta }_i$) to behave in a highly specific manner, often vanishing modulo P. The results hold even when the associated maps ${\theta }_i$ are not assumed to be additive or derivations, and the maps ${\Gamma }_i$ are entirely arbitrary, highlighting the power of the identities themselves.

Furthermore, by constructing a specific counterexample using a ring of matrices, we have rigorously established that the primeness of the ideal P is an essential hypothesis for our theorems. Without this condition, the conclusions may not hold. This work extends the extensive body of research connecting differential identities to the commutativity and structure of rings, broadening the scope to more general mappings and a more abstract setting relative to prime ideals.