Characterization generalized derivations of tensor products of nonassociative algebras

Ahmed Aboubakr

doi:10.1515/dema-2025-0115

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Characterization generalized derivations of tensor products of nonassociative algebras

Ahmed Aboubakr

Published/Copyright: April 25, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

Consider A and ℬ to be nonassociative unital algebras. Under the assumption that either one of them has finite dimensions or that both are finite dimensions, a generalized derivation is an additive map ℱ : A → A associated with a derivation d of A if ℱ ( u v ) = ℱ ( u ) v + u d ( v ) for all u , v ∈ A . The objective of this study is to characterize and elucidate the structure of a generalized derivation on the tensor product of nonassociative algebras. Specifically, we prove that if ℱ is a generalized derivation of A ⊗ ℬ associated with a derivation d of A ⊗ ℬ , then ℱ = ℒ u + d , where ℒ u is a left multiplication by u and u belongs to the left nucleus of A ⊗ ℬ (i.e., ℒ u r = u r for all r ∈ A ⊗ ℬ and u ∈ N l ( A ⊗ ℬ ) ). Moreover, every generalized derivation of A ⊗ ℬ can be represented as the sum of the derivations of the three categories: (i) w + a d u , where u , w ∈ N ( A ⊗ ℬ ) , (ii) ℒ z ⊗ f , where f is a derivation of ℬ and z ∈ Z ( A ) (the center of A ), and (iii) g ⊗ ℒ w , where g is a derivation of A and w ∈ Z ( ℬ ) .

Keywords: generalized derivation; derivation; nonassociative algebra; tensor product of algebras

MSC 2010: 17A36; 15A69

1 Introduction

Let A and ℬ be nonassociative algebras over a field K . We define a derivation d : A → A as an additive map satisfying the Leibniz rule: d ( r s ) = d ( r ) s + r d ( s ) , for all r , s ∈ A . In the study by Brešar [1], the following question was investigated: what properties characterize the derivations of the tensor product algebra A ⊗ ℬ ? Brešar established that if both A and ℬ are unital and at least one of them is finite-dimensional (or both are finite-dimensional), then every derivation d of A ⊗ ℬ can be expressed as the sum of three types of derivations:

a d u , where u belongs to the nucleus of A ⊗ ℬ ;
L z ⊗ f , where f is a derivation of ℬ and z belongs to the center A ; and
g ⊗ L w for g is a derivation of A and w is in Z ( ℬ ) belongs to the center ℬ .

Research literature examines the behavior of tensor products within nonassociative algebras from multiple angles. Studies into nonassociative algebras and their derivational traits constitute a vital area of modern algebraic study, offering significant insights into the analysis of intricate algebraic systems and their transformative processes. Recent research has shed light on the properties of tensor product algebras, biderivations, and derivations across various algebraic frameworks.

Prior to Brešar’s research, Block [[2], Theorem 7.1] conducted an analogous investigation, albeit with a notable distinction: the implementation of a less restrictive condition wherein A is unital. In Block’s framework, ℬ was assumed to possess both associative and commutative properties. Further developments of Block’s theorem, including significant extensions, emerged through Saeid Azam’s seminal contribution [3]. Azam explored the derivation algebra of tensor product Lie algebras, specifically examining their characteristics with respect to finite-order automorphisms while investigating their potential infinite-dimensionality.

In a specific scenario where A represents the associative matrix algebra M n ( F ) , Benkart and Osborn addressed this situation in detail [4, Corollary 4.9]. Finally, when both A and ℬ exhibit associativity, a characterization of derivations within A ⊗ ℬ emerges, contingent upon certain finiteness assumptions. This result is derived as a byproduct of investigations into Hochschild cohomology, as exemplified by [5, Corollary 3.4].

Brešar’s seminal work [6] proved that if the property that every Jordan { g , h } -derivation is a { g , h } -derivation holds in a unital algebra A , then so does in the algebra A ⊗ S for every commutative unital algebra S . In parallel, Kour’s [7] further refined this domain by showing that d 1 ⊕ d 2 is a simple derivation of A [ X , Y ] if and only if A [ X ] is d 1 -simple and A [ Y ] is d 2 -simple. They also showed that if d 1 and d 2 are positively homogeneous derivations and d 2 is a generalized triangular derivation, then d 1 ⊕ d 2 is the simple derivation of A [ X , Y ] if and only if d 1 is a simple derivation of A [ X ] and d 2 is a simple derivation of A [ Y ] .

Following Brešar’s research (2017), Brešar et al. [8] enhanced the understanding of Skolem-Noether algebras. An algebra S is defined as a Skolem-Noether algebra (abbreviated as SN algebra) if, for every central simple algebra R , every homomorphism R → R ⊗ S extends to an inner automorphism of R ⊗ S . They demonstrated that every semilocal algebra, and thus every finite-dimensional algebra, also satisfies this criterion. However, it is important to note that not every domain qualifies as an SN algebra.

Arfa et al. [9] provided significant insights into Hom-Jordan algebras and their induced ternary structures, extending understanding of the relationships between Hom-Jordan algebras and their induced ternary Hom-Jordan algebras. They gave some properties of the α K -generalized derivation algebra G D e r ( J ) of ternary Hom-Jordan algebras.

Chooi and Wong’s [10] research investigates commuting additive maps on tensor products of matrix algebras. Specifically, they study such maps over structures of the form ⊗ i = 1 k M n i , where M n i denotes the algebra of n i × n i matrices over a field F . They obtained a structural characterization of additive maps ψ : ⊗ i = 1 k M n i → ⊗ i = 1 k M n i , which satisfy some conditions.

Most recently, Benayadi and Oubba’s groundbreaking work [11] provides fundamental insights into nonassociative algebras of biderivation-type, particularly emphasizing Lie-admissible algebras’ structural complexities. Their research critically examines extensions of Lie algebras and explores specialized algebraic classes, including symmetric Leibniz algebras and Milnor algebras.

An additive map F : A → A is called a generalized derivation of A . If a derivation exists, d of A s.t., F ( u v ) = F ( u ) v + u d ( v ) , ∀ u , v ∈ A . The simple example of derivation is a linear map of the form u → r u + u s for some r , s ∈ A , which is called inner derivation, and generalized inner derivation, and the left centralizers (i.e., linear maps satisfying H ( u v ) = H ( u ) v for all u , v ∈ A ). Brešar introduced the notation of generalized derivations [12], and these maps were studied in both algebra and analysis [13–17].

This research represents a significant advancement in understanding the structural characteristics of generalized derivations within tensor product algebras, particularly focusing on nonassociative algebras. Our primary contribution is a detailed characterization of generalized derivations in tensor product algebras. We established a comprehensive structural representation of these mappings under specific conditions, namely, when at least one algebra is finite-dimensional or both are finitely generated. The main theorem provides a groundbreaking decomposition of generalized derivations. Our research systematically extends and refines previous work by mathematicians such as Brešar, Block, and Azam. Compared to Brešar’s 2017 [1] work on derivations of tensor products, our study offers a more comprehensive view of generalized derivations. Where Brešar primarily examined standard derivations, we expanded the scope to include generalized derivations, capturing a broader range of algebraic mappings.

Section 2 provides all necessary definitions and proves fundamental lemmas. Section 3 focuses on the presentation of the main results.

2 Preliminaries

Assume that A is a nonassociative algebra on K . For a , b , c ∈ A , then

( a , b , c ) = ( a b ) c − a ( b c ) .

The sets:

N l ( A ) = { n ∈ A ∣ ( n , A , A ) = 0 } is called the left nucleus of A .
N m ( A ) = { n ∈ A ∣ ( A , n , A ) = 0 } is called the medium nucleus of A .
N r ( A ) = { n ∈ A ∣ ( A , A , n ) = 0 } is defined as the right nucleus of A .
N ( A ) = { n ∈ A ∣ ( A , A , n ) = ( A , n , A ) = ( n , A , A ) = 0 } is defined as the nucleus of A .

A is associative if and only if N ( A ) = A . In our study, we will consider the case where S = A ⊗ ℬ , the tensor product of unital algebras A and ℬ . Hence, it is necessary to know that

N ( A ⊗ ℬ ) = N ( A ) ⊗ N ( ℬ ) ,

Der ( A ) defined the set of all derivations of A and G D e r ( A ) defined the set of all generalized derivations of A . Furthermore, for each a ∈ A , we define L a and R a : A ⟶ A by

L a ( r ) = a r , R a ( r ) = r a .

The additive mapping i d v : A ⟶ A defined by i d v = R v − L v for some v ∈ N ( A ) is a derivation known as the inner derivation of A given by v .

An additive map g d u , v : A ⟶ A is defined by g d u , v = L u + R v where u ∈ N l ( A ) and v ∈ N ( A ) . These maps are called generalized inner derivations because they extend the concept of inner derivations. Specifically, g d u , v is a generalized derivation associated with the inner derivation i d v .

Note that i d v ∈ Der ( A ) and g d u , v ∈ G D e r ( A ) if u ∈ N l ( A ) , v ∈ N ( A ) .

Remark 1

L u is a generalized derivation linked to 0 as soon as u ∈ N l ( A ) .
R v is a r-generalized derivation linked to 0 as soon as v ∈ N r ( A ) .
d ( 1 ) = 0 , where d is a derivation of A .
F ( 1 ) ∈ N l ( A ) since F is a generalized derivation of A .

Example 2.1

If u = a 1 ⊗ b 1 ∈ N l ( A ⊗ ℬ ) , v = a 2 ⊗ b 2 ∈ N ( A ⊗ ℬ ) and g d u , v is generalized inner derivation of A ⊗ ℬ linked to inner derivation i d v of A ⊗ ℬ , then

g d u , v = ∑ i = 1 2 ( L a i ⊗ L b i ) + i d v .

Lemma 2.2

If g = L u + i d v , where u ∈ N l ( A ⊗ ℬ ) , then g is a generalized inner derivation linked to i d v .

Proof

From our hypothesis,

(1) g ( r s ) = ( L u + i d v ) ( r s ) = L u ( r s ) + i d v ( r s ) = u r s + r s v − v r s .

On the other hand,

(2) g ( r ) s + r i d v ( s ) = ( L u + i d v ) ( r ) s + r i d v ( s ) = L u ( r ) s + i d v ( r ) s + r i d v ( s ) = u r s + ( r v − v r ) s + r ( s v − v s ) = u r s + r s v − v r s .

This proves that g is a generalized inner derivation linked to inner derivation i d v .□

Lemma 2.3

[18, Lemma 4.8] Let e 1 , … , e n ∈ U be linearly independent. v 1 , … , v n ∈ V are such that

∑ i = 1 n e i ⊗ v i = 0 ,

then each v i = 0 .

Lemma 2.4

[18, Lemma 4.9] Let e 1 , … , e n ∈ U be linearly independent. If v 1 , … , v n ∈ V are such that

∑ i = 1 n e i ⊗ v i = ∑ j = 1 m w j ⊗ z j

for some w 1 , … , w m ∈ U and z 1 , … , z m ∈ V , then each v i is a linear combination of z 1 , … , z m .

Lemma 2.5

[1, Theorem 3.1] Let A and ℬ be nonassociative unital algebras. Suppose that either at least one of A and ℬ is finite-dimensional or they both are finitely generated. Then, every derivation d of A ⊗ ℬ can be written as

d = i d v + ∑ j = 1 p L z j ⊗ f j + ∑ i = 1 q g i ⊗ L w i ,

where v ∈ N ( A ) ⊗ N ( ℬ ) , z j ∈ Z ( A ) , w i ∈ Z ( ℬ ) , f j ∈ Der ( A ) , and g i ∈ Der ( ℬ ) .

Lemma 2.6

Consider two nonassociative algebras A and ℬ . Let F be a generalized derivation of the tensor product A ⊗ ℬ , which is associated with a derivation d of A ⊗ ℬ . Furthermore, let { b i ∣ i ∈ I } be a basis of ℬ . Assume that ℬ has a unit element. For each element i in the set I, there exists a generalized derivation G i of A associated with a derivation d i of A. This implies that for every element x in A , the following holds:

(3) F ( x ⊗ 1 ) = ∑ i ∈ I G i ( x ) ⊗ b i ,

(4) d ( x ⊗ 1 ) = ∑ i ∈ I d i ( x ) ⊗ b i ,

and G i ( x ) = 0 = d i ( x ) for all but finitely many i ∈ I .

Proof

For any element x in A , there are uniquely determined elements G i ( x ) and d i ( x ) in A such that equations (3) and (4) are satisfied, and G i ( x ) = 0 = d i ( x ) for all but a finite number of elements i in I . The linearity of the maps F and d directly imply the linearity of the maps G i and d i , which both map from A to A . Additionally,

F ( x u ⊗ 1 ) = F ( ( x ⊗ 1 ) ( u ⊗ 1 ) ) = F ( x ⊗ 1 ) ( u ⊗ 1 ) + ( x ⊗ 1 ) d ( u ⊗ 1 )

yields

∑ i ∈ I ( G i ( x u ) − G i ( x ) u − x d i ( u ) ) ⊗ a i = 0 ,

which implies that G i ∈ GDer ( A ) is associated with d i ∈ Der ( A ) .□

Lemma 2.7

Under conditions in Lemma 2.6 if both A and ℬ have identity and either they are finitely generated or one of them is finite-dimensional then G i = 0 = d i for all but a finite number of elements i in I.

Proof

Let us assume that the collection a 1 , … , a m generates A . The set I ∘ , which consists of all i in I for which G i ( a j ) ≠ 0 for some j in the range of 1 to m , is finite. Similarly, the set I ∘ ′ , which consists of all i in I for which d i ( a j ) ≠ 0 for some j in the range of 1 to m , is also finite. It is evident that G i = 0 , ∀ i ∈ I \ I ∘ ∪ I ∘ ′ .□

3 Main result

In this theorem, we elucidate the structure of a generalized derivation on the tensor product of nonassociative algebras and present the main results of this work in the concluding corollary.

Theorem 3.1

Consider A and ℬ as nonassociative unital algebras. Assume that either A or ℬ is finite dimensions, or both are finitely generated. If F is a generalized derivation of A ⊗ ℬ linked to a derivation d of A ⊗ ℬ , then

F = L u + d , w h e r e u ∈ N l ( A ⊗ ℬ ) .

Proof

Take a basis { a j ∣ j ∈ J } of A and a basis { b i ∣ i ∈ I } of ℬ . Applying our assumption, either A is finitely generated or ℬ is finite-dimensional. By applying Lemma 2.7, we conclude that there exist finitely many G i ∈ GDer ( A ) , i = 1 , … , p linked to derivations g i ∈ Der ( A ) , i = 1 , … , p , respectively, such that,

(5) F ( x ⊗ 1 ) = ∑ i = 1 p G i ( x ) ⊗ b i , ∀ x ∈ A

(6) d ( x ⊗ 1 ) = ∑ i = 1 p g i ( x ) ⊗ b i , ∀ x ∈ A .

Analogously, there exist finitely many H j ∈ GDer ( ℬ ) , j = 1 , … , q linked to derivations h j ∈ Der ( ℬ ) , j = 1 , … , q such that

(7) F ( 1 ⊗ y ) = ∑ j = 1 q a j ⊗ H j ( y ) , ∀ y ∈ ℬ

(8) d ( 1 ⊗ y ) = ∑ j = 1 q a j ⊗ h j ( y ) , ∀ y ∈ ℬ .

Assume that there is at least one nonzero value for G i . We can suppose that { G 1 , … , G s } are linearly independent subsets of { G 1 , … , G p } without losing generality. By expressing each G i with i > s as a linear combination of G 1 , … , G s , it is obvious that equation (5) may be written as

(9) F ( x ⊗ 1 ) = ∑ i = 1 s G i ( x ) ⊗ n i , ∀ x ∈ A ,

where n i are linearly independent elements.

Similarly, by assuming that { H 1 , … , H t } is linearly independent subset of { H 1 , … , H q } , we can rewrite the relation as

(10) F ( 1 ⊗ y ) = ∑ j = 1 t m j ⊗ H j ( y ) , for all y ∈ ℬ ,

where m j are linearly independent elements.

Combining equations (9) and (10), we obtain

(11) F ( x ⊗ y ) = F ( ( x ⊗ 1 ) ( 1 ⊗ y ) ) = F ( x ⊗ 1 ) ( 1 ⊗ y ) + ( x ⊗ 1 ) d ( 1 ⊗ y ) = ∑ i = 1 s G i ( x ) ⊗ n i y + ∑ j = 1 q x a j ⊗ h j ( y ) , for all x ∈ A , y ∈ ℬ .

On the other hand,

(12) F ( x ⊗ y ) = F ( ( 1 ⊗ y ) ( x ⊗ 1 ) ) = F ( 1 ⊗ y ) ( x ⊗ 1 ) + ( 1 ⊗ y ) d ( x ⊗ 1 ) = ∑ j = 1 t m j x ⊗ H j ( y ) + ∑ i = 1 p g i ( x ) ⊗ y b i , ∀ x ∈ A , y ∈ ℬ .

Upon comparing these two expressions, we obtain the following result:

(13) ∑ i = 1 s G i ( x ) ⊗ n i y + ∑ j = 1 q x a j ⊗ h j ( y ) = ∑ j = 1 t m j x ⊗ H j ( y ) + ∑ i = 1 p g i ( x ) ⊗ y b i , for all x ∈ A , y ∈ ℬ .

In particular, letting y = 1 in equation (13) and since h j as a derivation vanishes on unity, we have

(14) ∑ i = 1 s G i ( x ) ⊗ n i = ∑ j = 1 t m j x ⊗ H j ( 1 ) + ∑ i = 1 p g i ( x ) ⊗ b i ∀ x ∈ A .

Right multiplication of (14) by 1 ⊗ y gives

(15) ∑ i = 1 s G i ( x ) ⊗ n i y = ∑ j = 1 t m j x ⊗ H j ( 1 ) y + ∑ i = 1 p g i ( x ) ⊗ b i y , ∀ x ∈ A , y ∈ ℬ .

On the other side, letting x = 1 , y = 1 in equation (13) and since h j and g i as derivations vanish on unity, we have

(16) ∑ i = 1 s G i ( 1 ) ⊗ n i = ∑ j = 1 t m j ⊗ H j ( 1 ) .

Since m j ∈ A , j = 1 , … , t are linearly independent, it follows from Lemma 2.4 that each H j ( 1 ) is a linear combination of n i ’s. Thus, there exist λ i j ∈ F such that

(17) H j ( 1 ) = ∑ i = 1 s λ i j n i .

Substituting equation (17) into equation (15), we obtain

(18) ∑ i = 1 s G i ( x ) ⊗ n i y = ∑ j = 1 t ∑ i = 1 s λ i j ( m j x ⊗ n i y ) + ∑ i = 1 p g i ( x ) ⊗ b i y for all x ∈ A , y ∈ ℬ .

Substituting equation (18) into equation (11), we have

(19) F ( x ⊗ y ) = ∑ j = 1 t ∑ i = 1 s λ i j ( m j x ⊗ n i y ) + ∑ i = 1 p g i ( x ) ⊗ b i y + ∑ j = 1 q x a j ⊗ h j ( y ) , ∀ x ∈ A , y ∈ ℬ .

On the other hand, since d is a derivation,

(20) d ( x ⊗ y ) = d ( ( x ⊗ 1 ) ( 1 ⊗ y ) ) = d ( x ⊗ 1 ) ( 1 ⊗ y ) + ( x ⊗ 1 ) d ( 1 ⊗ y )

Using both expressions of d , and substituting equations (6) and (8) into (20), we obtain

(21) d ( x ⊗ y ) = ∑ i = 1 p g i ( x ) ⊗ b i ( 1 ⊗ y ) + ( x ⊗ 1 ) ∑ j = 1 q a j ⊗ h j ( y ) = ∑ i = 1 p g i ( x ) ⊗ b i y + ∑ j = 1 q x a j ⊗ h j ( y ) , for all x ∈ A , y ∈ ℬ .

Accordingly, substituting equation (21) into the last two terms in equation (19), we have

(22) F ( x ⊗ y ) = ∑ j = 1 t ∑ i = 1 s λ i j ( m j x ⊗ n i y ) + d ( x ⊗ y ) , ∀ x ∈ A , y ∈ ℬ .

To establish the theorem, we shall demonstrate that m j ⊗ n i ∈ N r ( A ⊗ ℬ ) for all values of i and j . To accomplish this, we will compute the function F ( x ⊗ y z ) , where x ∈ A , and y , z ∈ ℬ . Therefore, using equation (11), we obtain

(23) F ( x ⊗ y z ) = ∑ i = 1 s G i ( x ) ⊗ n i ( y z ) + ∑ j = 1 q x a j ⊗ h j ( y z ) = ∑ i = 1 s G i ( x ) ⊗ n i ( y z ) + ∑ j = 1 q x a j ⊗ h j ( y ) z + ∑ j = 1 q x a j ⊗ y h j ( z ) , for all x ∈ A , y , z ∈ ℬ .

On the other hand, by the definition of generalized derivation F ,

(24) F ( x ⊗ y z ) = F ( ( x ⊗ y ) ( 1 ⊗ z ) ) = F ( x ⊗ y ) ( 1 ⊗ z ) + ( x ⊗ y ) d ( 1 ⊗ z ) .

Using both expressions of d in equation (8) and F equation (11) in (24), we obtain

(25) F ( x ⊗ y z ) = ∑ i = 1 s G i ( x ) ⊗ ( n i y ) ( 1 ⊗ z ) + ( x ⊗ y ) ∑ j = 1 q a j ⊗ h j ( z ) = ∑ i = 1 s G i ( x ) ⊗ ( n i y ) z + ∑ j = 1 q x a j ⊗ h j ( y ) z + ∑ j = 1 q x a j ⊗ y h j ( z ) , for all x ∈ A , y , z ∈ ℬ .

Upon comparing equations (23) and (25), we obtain

(26) ∑ i = 1 s G i ( x ) ⊗ ( n i ( y z ) − ( n i y ) z ) = 0 , ∀ x ∈ A , y , z ∈ ℬ .

This can be written as

(27) ∑ i = 1 s G i ⊗ ( L n i L y − L n i y ) ( x ⊗ z ) = 0 , ∀ x ∈ A , y , z ∈ ℬ .

Since G 1 , … , G s are linearly independent, Lemma 2.3 gives L n i L y − L n i y = 0 for all y ∈ ℬ and for every i , i.e., ( n i , ℬ , ℬ ) = 0 for every i ; thus, n i ∈ N l ( A ⊗ ℬ ) , i = 1 , … , s . In a similar way by computing F ( x z ⊗ y ) , we can prove that m j ∈ N l ( A ⊗ ℬ ) , j = 1 , … , t , which is the desired conclusion.□

Corollary 3.2

Consider A and ℬ to be unital algebras that are nonassociative. Assume that either A or ℬ is finite dimensions, or both are finitely generated. If F is a generalized derivation of A ⊗ ℬ linked to derivation d of A ⊗ ℬ , then

F = g d u , v + ∑ j = 1 p L z j ⊗ f j + ∑ i = 1 q g i ⊗ L w i ,

where u ∈ N l ( A ⊗ ℬ ) , v ∈ N ( A ⊗ ℬ ) , z j ∈ Z ( A ) , w i ∈ Z ( ℬ ) , f j ∈ Der ( A ) , and g i ∈ Der ( ℬ ) .

Proof

The result of Theorem 3.1 is

(28) F = L u + d , u ∈ N l ( A ⊗ ℬ ) .

From Lemma 2.5, d = i d v + ∑ j = 1 p L z j ⊗ f j + ∑ i = 1 q g i ⊗ L w i , in equation (28), we obtain

(29) F = L u + i d v + ∑ j = 1 p L z j ⊗ f j + ∑ i = 1 q g i ⊗ L w i , u ∈ N l ( A ⊗ ℬ ) .

Lemma 2.2 gives that L u + i d v is a generalized inner derivation, which completes our proof.□

4 Conclusion

This research marks a significant advancement in understanding the structural characteristics of generalized derivations within tensor product algebras, particularly focusing on nonassociative algebras. It builds upon existing research, offering new insights into the intricate properties of algebraic mappings. Key contributions include a comprehensive characterization of generalized derivations when one algebra is finite-dimensional or both are finitely generated, a groundbreaking structural decomposition theorem, and the extension of prior frameworks by researchers such as Brešar, Block, and Azam. Unlike studies that focus solely on associative algebras, this work delves into the most complex nonassociative structures, revealing deeper insights into algebraic mappings. Compared to Brešar’s 2017 analysis, this study provides a broader view by expanding the scope to include generalized derivations. Potential applications range from theoretical algebra to computational mathematics, offering strategies for analyzing complex algebraic systems and inspiring future research in specialized algebraic frameworks. Despite the constraints regarding finite dimensions, this work offers a precise description of generalized derivations, contributing significantly to our understanding and providing a blueprint for exploring complex algebraic systems.

afs00@fayoum.edu.eg

Acknowledgments

The authors extend their appreciation to the Deputyship for Research and Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number 445-9-948. The author would like to thank Professor Consuelo Martínez López for her guidance, insights, and support throughout this research.

Funding information: The authors extend their appreciation to the Deputyship for Research and Innovation, Ministry of Education in Saudi Arabia for funding this research work through the project number 445-9-948.
Author contributions: A. A. – conceptualization, methodology, writing – original draft, writing – review and editing, and validation. The author has read and agreed to the published version of the manuscript.
Conflict of interest: The author states no conflict of interest.
Data availability statement: Not applicable.

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Received: 2024-05-25

Revised: 2024-12-11

Accepted: 2025-02-24

Published Online: 2025-04-25

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/dema-2025-0115

Keywords for this article

generalized derivation; derivation; nonassociative algebra; tensor product of algebras

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