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On Left Derivations and Related Mappings
Abstract
Let R be a ring and X be a left R-module. The purpose of this paper is to investigate additive mappings D1: R → X and D2: R → X that satisfy D1(ab) = aD1(b) + bD1(a), a, b ∈ R (left derivation) and D2(a2) = 2aD2(a), a ∈ R (Jordan left derivation). We show, by the rather weak assumptions, that the existence of a nonzero Jordan left derivation of R into X implies R is commutative. This result is used to prove two noncommutative extensions of the classical Singer-Wermer theorem.
... In [4], Brešar and Vukman introduce the concepts of left derivations and Jordan left derivations. Suppose that M is a left R-module. ...
... Suppose that M is a left R-module. An additive mapping δ from R into M is called a left derivation if δ(AB) = Aδ(B) + Bδ(A) for each A, B in R; and δ is called a Jordan left derivation if δ(A 2 ) = 2Aδ(A) for every A in R. Brešar and Vukman [4] prove that if R is a prime ring and M is a 6-torsion free left R-module, then the existence of nonzero Jordan left derivations from R into M implies that R is a commutative ring. Deng [7] shows that [4,Theorem 2.1] is still true when M is only 2-torsion free. ...
... An additive mapping δ from R into M is called a left derivation if δ(AB) = Aδ(B) + Bδ(A) for each A, B in R; and δ is called a Jordan left derivation if δ(A 2 ) = 2Aδ(A) for every A in R. Brešar and Vukman [4] prove that if R is a prime ring and M is a 6-torsion free left R-module, then the existence of nonzero Jordan left derivations from R into M implies that R is a commutative ring. Deng [7] shows that [4,Theorem 2.1] is still true when M is only 2-torsion free. ...
Let be a ring, be a -bimodule and m,n be two fixed nonnegative integers with . An additive mapping from into is called an \emph{(m,n)-Jordan derivation} if for every A in . In this paper, we prove that every (m,n)-Jordan derivation from a -algebra into its Banach bimodule is zero. An additive mapping from into is called a (m,n)-Jordan derivable mapping at W in if for each A and B in with AB=BA=W. We prove that if is a unital -bimodule with a left (right) separating set generated algebraically by all idempotents in , then every (m,n)-Jordan derivable mapping at zero from into is identical with zero. We also show that if and are two unital algebras, is a faithful unital -bimodule and is a generalized matrix algebra, then every (m,n)-Jordan derivable mapping at zero from into itself is equal to zero.
... The concepts of a left derivation, (an additive mapping d 1 : R → X satisfying d 1 (ab) = ad 1 (b) + bd 1 (a), ∀ a ∈ R) and that of a Jordan left derivation (additive mapping d 2 : R → X such that d 2 (a 2 ) = 2ad 2 (a), a ∈ R) were introduced by Bresar and Vukman in [54]. In the past three decades, there has been a significant amount of work around Jordan left derivations and related mappings, as these are in a close connection with the so-called commuting mappings. ...
... Theorem 3.1. [54] "Let R be a ring and X be a 2-torsion free and 3-torsion free left R − module. Suppose that aRx = 0 with a ∈ R, x ∈ X implies that either a = 0 or x = 0. ...
... In this direction Ashraf and Nadeem [16] in 2000 obtained a general result which established that under appropriate restrictions on a Lie ideal U of a 2-torsion free prime ring, every Jordan left derivation on U turns out to be a left derivation on U (see [3,10] for more recent results). Bresar and Vukman [54] proved that every Jordan left derivation on a noncommutative 2-torsion free and 3-torsion free prime ring is identically zero. But the condition of 3-torsion free can be omitted, this was achieved by Hosseini; Let's end this section with a conjecture given by Vukman; ...
In this overview article, we provide a historical account on derivations, Jordan derivations, (α, β)-derivations, left derivations, pre-derivations, homoderivations, nilpotent derivations, and other variants, drawing from the contributions of multiple researchers. Additionally, we delve into recent findings and suggest potential avenues for future investigation in this area. Furthermore, we offer pertinent examples to illustrate that the assumptions underlying various results are indeed necessary and not redundant.
... for each T ∈ A. Taking d ′ = 1 2 d, the proof is complete. For the identity (F3) with n = 2, Brešar and Vukman gave an adequate description of f and g on a class of algebras in [6]. Precisely, in case B(X ) is the algebra of all bounded linear operators on a Banach space X , if additive mappings f , g which map B(X ) into X or into B(X ) satisfy f (A) − A 2 g(A −1 ) = 0 for each invertible operator A ∈ B(X ), then f and g are of the form f (A) = −g(A) = Af (I) for each A ∈ B(X ). ...
... Precisely, in case B(X ) is the algebra of all bounded linear operators on a Banach space X , if additive mappings f , g which map B(X ) into X or into B(X ) satisfy f (A) − A 2 g(A −1 ) = 0 for each invertible operator A ∈ B(X ), then f and g are of the form f (A) = −g(A) = Af (I) for each A ∈ B(X ). Comparing the result in [6] with Lemma 3.6, the forms of the mappings f and g differ only in a Jordan left derivation. This difference arises from the fact that every Jordan left derivation from B(X ) into X or into B(X ) is equal to zero (see [6,Corollary 1.5]). ...
... Comparing the result in [6] with Lemma 3.6, the forms of the mappings f and g differ only in a Jordan left derivation. This difference arises from the fact that every Jordan left derivation from B(X ) into X or into B(X ) is equal to zero (see [6,Corollary 1.5]). ...
The purpose of this paper is to characterize several classes of functional identities involving inverses with related mappings from a unital Banach algebra over the complex field into a unital -bimodule . Let N be a fixed invertible element in , M be a fixed element in , and n be a positive integer. We investigate the forms of additive mappings f, g from into satisfying one of the following identities: \begin{equation*} \begin{aligned} &f(A)A- Ag(A) = 0\\ &f(A)+ g(B)\star A= M\\ &f(A)+A^{n}g(A^{-1})=0\\ &f(A)+A^{n}g(B)=M \end{aligned} \qquad \begin{aligned} &\text{for each invertible element}~A\in\mathcal{A}; \\ &\text{whenever}~ A,B\in\mathcal{A}~\text{with}~AB=N;\\ &\text{for each invertible element}~A\in\mathcal{A}; \\ &\text{whenever}~ A,B\in\mathcal{A}~\text{with}~AB=N, \end{aligned} \end{equation*} where is either the Jordan product or the Lie product .
... [14]). After that, much attention was paid to various types of derivations (see for example [2,3,4,6,7,17]). In the last decade, some people tried to generalize these results to various types of semirings that play an important role in theoretical computer science (cf. ...
... In this note, which is a continuation of our previous article [1], we generalize the result of Brešar and Vukman (cf. [7]) that the existence of a non-zero left Jordan derivation forces a prime ring to be commutative. Namely, we prove that the existence of such a derivation on a prime, 2-and 3-torsion free M A-semiring forces the commutativity of this semiring (Theorem 4.3). ...
... A special role plays a left derivation defined as an additive mapping f : S → S such that f (xy) = xf (y) + yf (x) for all x, y ∈ S (cf. [3,5,7]). An additive mapping f : S → S satisfying the identity ...
We determine conditions under which a left Jordan derivation defined on an MA-semiring S is a left derivation on this semiring and prove when a left Jordan derivation on S implies the commutativity of S.
... They proved that the existence of a nonzero left derivation of a prime ring of characteristic not 2 implies the commutativity of the ring. After that many new results have been established on Jordan left derivations over different rings and algebras [1,4,8,10,12,13]. Recently, in 2016, Brešar introduced {g, h}-derivation and studied over semiprime algebras and tensor product of algebras [6]. ...
... Motivated by left derivation [4] and {g, h}-derivation [6], we introduce left {g, h}derivation over A as follows: Example 2.3. Let A = T 2 (C), X = x 1 x 2 0 x 3 ∈ A and f, g, h : A → A are defined by f (X) = 5x 1 7x 1 + 6x 2 0 6x 3 , g(X) = x 1 2x 1 + 3x 2 0 3x 3 and h(X) = 4x 1 5x 1 + 3x 2 0 3x 3 respectively. ...
In this article, left {g, h}-derivation and Jordan left {g, h}-derivation on algebras are introduced. It is shown that there is no Jordan left {g, h}-derivation over and , for g not equal to h. Examples are given which show that every Jordan left -derivation over , and are not left -derivations. Moreover, we characterize left -derivation and Jordan left -derivation over , and respectively. Also, we prove the result of Jordan left -derivation to be a left -derivation over tensor products of algebras as well as for algebra of polynomials.
... There are some other attempts to defined generalized Jordan derivations in the literature, According to [5], an additive (in this note we shall assume linearity) mapping T from a Banach algebra A into a left A-module M is called a Jordan left derivation if T (a 2 ) = 2aT (a) for every a ∈ A . It is shown in [5] that the existence of non-zero Jordan left derivations from a prime ring R into a 6-torsion free left R-module implies that R is a commutative ring. ...
... There are some other attempts to defined generalized Jordan derivations in the literature, According to [5], an additive (in this note we shall assume linearity) mapping T from a Banach algebra A into a left A-module M is called a Jordan left derivation if T (a 2 ) = 2aT (a) for every a ∈ A . It is shown in [5] that the existence of non-zero Jordan left derivations from a prime ring R into a 6-torsion free left R-module implies that R is a commutative ring. Vukman introduced in [44] the notion of (m, n)-Jordan derivation. ...
Let M be a Banach bimodule over an associative Banach algebra A , and let F : A → M be a linear mapping. Three main uses of the term generalized derivation are identified in the available literature, namely,
(a) F is a generalized derivation of the first type if there exists a derivation d : A → M satisfying F(ab) = F(a)b + ad(b) for all a, b ∈ A.
(b) F is a generalized derivation of the second type if there exists an element ξ ∈ M * * satisfying F(ab) = F(a)b + aF(b) − aξ b for all a, b ∈ A.
(c) F is a generalized derivation of the third type if there exist two (non-necessarily linear) mappings G, H : A → M satisfying F(ab) = G(a)b + aH(b) for all a, b ∈ A.
There are examples showing that these three definitions are not, in general, equivalent. Despite that the first two notions are well studied when A is a C*-algebra, it is not known if the three notions are equivalent under these special assumptions. In this note we prove that every generalized derivation of the third type whose domain is a C*-algebra is automatically continuous. We also prove that every (continuous) generalized derivation of the third type from a C*-algebra A into a general Banach A-bimodule is a generalized derivation of the first and second type. In particular, the three notions coincide in this case. We also explore the possible notions of generalized Jordan derivations on a C*-algebra and establish some continuity properties for them.
... Algebra, functional analysis and quantum physics are related with concept of derivation. An additive mapping such that , for all is called left derivation of , [3] it is clear that the concepts of derivation and left derivation are identical whenever is commutative. In 1987 the concept of a symmetric bi-derivation has been introduced by Maksa in [4], a bi-additive mapping is said to be bi-derivation if , for all . ...
... An additive subgroup of is called Lie ideal if whenever , then [ ] , [2]. A Lie ideal of is called a square closed Lie ideal of if , for all , [3]. A map is called commuting (resp. ...
... This result was subsequently, refined and extended by a number of authors. In [9], Bresar and Vukman showed that a prime ring must be commutative if admits a nonzero left derivation. Furthermore, Bresar and Vukman [9] studied the notions of a * -derivation and a Jordan * -derivations of . ...
... In [9], Bresar and Vukman showed that a prime ring must be commutative if admits a nonzero left derivation. Furthermore, Bresar and Vukman [9] studied the notions of a * -derivation and a Jordan * -derivations of . In [2], Asma et al. generalized some identities on additive maps with * -rings. ...
Let be an associative ring with involution *. An additive map λ → λ * of into itself is called an involution if the following conditions are satisfied (i)(λµ) * = µ * λ * , (ii)(λ *) * = λ for all λ, µ ∈. A ring equipped with an involution is called an *-ring or ring with involution. The aim of the present paper is to establish some results on *-α-derivations in *-rings and investigate the commutativity of prime *-rings admitting *-α-derivations on satisfying certain identities also prove that if admits a reverse *-α-derivation δ of , then α ∈ Z() and some related results have also been discussed. 1. Preliminaries Last few decades, several authors have investigated the relationship between the commutativity of the ring and certain specific types of derivations of. The first result in this direction is due to Posner [3] who proved that if a ring admits a nonzero derivation δ such that [δ(λ), λ] ∈ Z() for all λ ∈ , then is commutative. This result was subsequently, refined and extended by a number of authors. In [9], Bre-sar and Vukman showed that a prime ring must be commutative if admits a nonzero left derivation. Furthermore, Bresar and Vukman [9] studied the notions of a *-derivation and a Jordan *-derivations of. In [2], Asma et al. generalized some identities on additive maps with *-rings. Recently, many authors have obtained commutativity theorems for prime and semiprime rings admitting derivation, generalized.
... The concepts of a left derivation and a Jordan left derivation were introduced by Brešar and Vukman in [3]. In the past three decades, there has been considerable interest for Jordan left derivations and related mappings (see, e.g., [1,3,18]) which are in a close connection with so-called commuting mappings. ...
... The concepts of a left derivation and a Jordan left derivation were introduced by Brešar and Vukman in [3]. In the past three decades, there has been considerable interest for Jordan left derivations and related mappings (see, e.g., [1,3,18]) which are in a close connection with so-called commuting mappings. The main motivation comes from the Posner's fundamental result which states that if a prime ring admits a commuting nonzero derivation, then it must be commutative (see [14]). ...
Let A be an algebra, and let I be a semiprime ideal of A. Suppose thatd : A → A is a Jordan left derivation such that d(I) ⊆ I.We prove that if dim{d(a)+I : a ⋲ A} ≤ 1, then d(A) ⊆ I. Additionally, we consider several consequences of this result.
... An additive subgroup of is called Lie ideal if whenever ∈ , ∈ then [ , ] ∈ [7]. A Lie ideal of is called a square closed Lie ideal of if ∈ , for all ∈ [3]. A square closed Lie ideal of such that ⊈ is called an admissible Lie ideal of [11]. ...
... Recently [1] defined the concept of * --derivation in prime * -rings and semiprime * -rings. Many authors have proved the commutativity of prime and semiprime rings admitting derivation ( [11], [3]). In the present paper the commuting and centralizing of symmetric reverse * -n-derivation of Lie ideal are studied under certain conditions and on the other hand the commutativity of prime * -ring with symmetric reverse * --derivations that satisfying certain identities and some regarding results have also been discussed. ...
In this paper, the commuting and centralizing of symmetric reverse ∗- -derivation on Lie ideal are studied and the commutativity of prime ∗-ring with the concept of symmetric reverse ∗- -derivations are proved under certain conditions.
... An In 1988, the Singer-Wermer theorem was generalized by removing the boundedness of a derivation (see [1]), which was called as the Singer-Wermer conjecture. In 1990, Brešar and Vukman [2] introduced the concept of left derivations. They based upon the rather weak assumptions to exhibit that the existence of a nonzero Jordan left derivation in a 2-torsion free and 3-torsion free ring R implies that R is commutative. ...
Let R R be a unital associative ring. Our motivation is to prove that left derivations in column finite matrix rings over R R are equal to zero and demonstrate that a left derivation d : T → T d:{\mathcal{T}}\to {\mathcal{T}} in the infinite upper triangular matrix ring T {\mathcal{T}} is determined by left derivations d j {d}_{j} in R ( j = 1 , 2 , … ) R\left(j=1,2,\ldots ) satisfying d ( ( a i j ) ) = ( b i j ) d\left(\left({a}_{ij}))=\left({b}_{ij}) for any ( a i j ) ∈ T \left({a}_{ij})\in {\mathcal{T}} , where b i j = d j ( a 11 ) , i = 1 , 0 , i ≠ 1 . {b}_{ij}=\left\{\begin{array}{ll}{d}_{j}\left({a}_{11}),& i=1,\\ 0,& i\ne 1.\end{array}\right. The similar results about Jordan left derivations are also obtained when R R is 2-torsion free.
... Over the last few decades several authors studied derivations in rings, semigroups, semirings and investigated the relationship between the commutativity of algebraic structures and the existence of specified derivations of algebraic structure. In 1990, Bresar and Vukman [2] established that a prime ring must be commutative if it admits a nonzero left derivation.Over the last few decades, several authors have investigated the relationship between the commutativity of ring R and the existence of certain specified derivations of R: The first result in this direction is due to Posner [10] in 1957. In the year Kim [3], [4] studied right derivation and generalized derivation of incline algebra. ...
In this paper, we introduce the concept of right derivation on ordered semirings and we study some of the properties of right derivation on ordered semirings.
... In the same year, Vukman [29] proved that if f is an additive map on a noncommutative division ring of characteristic not 2 satisfying the identity f (x) = −x 2 f (x −1 ), then f = 0. In 1990, Brešar and Vukman [3] investigated the identity f (x) = x 2 g(x −1 ) on the algebra of all bounded linear operators of a Banach space. We refer the reader to [10,11] for related history. ...
... In the same year, Vukman [28] proved that if f is an additive map on a noncommutative division ring of characteristic not 2 satisfying the identity f (x) = −x 2 f (x −1 ), then f = 0. In 1990, Brešar and Vukman [3] investigated the identity f (x) = x 2 g(x −1 ) on the algebra of all bounded linear operators of a Banach space. We refer the reader to [10,11] for related history. ...
We study the functional identity G(x)f (x) = H(x) on a division ring D, where f : D → D is an additive map and G(X) = 0, H(X) are generalized polynomials in the variable X with coefficients in D. Precisely, it is proved that either D is finite-dimensional over its center or f is an elementary operator. Applying the result and its consequences, we prove that if D is a noncommutative division ring of characteristic not 2, then the only solution of additive maps f, g on D satisfying the identity f (x) = x n g(x −1) with n = 2 a positive integer is the trivial case, that is, f = 0 and g = 0. This extends Catalano and Merchán's result in 2023 to get a complete solution.
... Lee et al. [14,15] obtained that any Jordan * -derivation on a noncommutative prime ring A with involution ' * ' is X-inner unless dim C AC = 4 and char(A) = 2. Lee and Wong [13] proved that any Jordan τ-derivation of a noncommutative prime ring A is X-inner if either A is not a GPI-ring or A is a PI-ring except when char(A) = 2 and dim C AC = 4. Very recently Lin [16] showed that when A is a prime GPI-ring but not a PI-ring, then any Jordan τ-derivation of A is X-inner if either τ is of the second kind or both char(A) = 2 and τ is of the first kind with deg τ 2 = 2. Vukman [22] proved that if m and n are distinct positive integers such that A is a 2mn(m + n)|m − n|-torsion free prime ring, then any nonzero (m, n)-Jordan derivation F : A → A is a derivation and A is commutative unless char(A) = 2 or 3. Ali and Fošner [1] generalized this result and proved that if A is a 6mn(m + n)|m − n|-torsion free prime ring and G : A → A is a nonzero generalized (m, n)-Jordan derivation, then G is a derivation and A is commutative provided that char(A) = 0 or char(A) > 3. Recently, Bennis, Dhara and Fahid [4, Theorem 1.3] characterized nonzero generalized (m, n)-Jordan derivations in semiprime rings and obtained that if A is a 6mn(m + n)|m − n|-torsion free semiprime ring, then any nonzero generalized (m, n)-Jordan derivation on A is a derivation and maps A into Z(A). For other results see [6][7][8]17]. ...
Let 𝒜 {\mathcal{A}} be a prime ring equipped with an involution ‘ * {*} ’ of order 2 and let m ≠ n {m\neq n} be some fixed positive integers such that 𝒜 {\mathcal{A}} is 2 m n ( m + n ) | m - n | {2mn(m+n)|m-n|} -torsion free. Let 𝒬 m s ( 𝒜 ) {\mathcal{Q}_{ms}(\mathcal{A})} be the maximal symmetric ring of quotients of 𝒜 {\mathcal{A}} and consider the mappings ℱ {\mathcal{F}} and 𝒢 : 𝒜 → 𝒬 m s ( 𝒜 ) {\mathcal{G}:\mathcal{A}\to\mathcal{Q}_{ms}(\mathcal{A})} satisfying the relations
( m + n ) ℱ ( a 2 ) = 2 m ℱ ( a ) a * + 2 n a ℱ ( a ) (m+n)\mathcal{F}(a^{2})=2m\mathcal{F}(a)a^{*}+2na\mathcal{F}(a)
and
( m + n ) 𝒢 ( a 2 ) = 2 m 𝒢 ( a ) a * + 2 n a ℱ ( a ) (m+n)\mathcal{G}(a^{2})=2m\mathcal{G}(a)a^{*}+2na\mathcal{F}(a)
for all a ∈ 𝒜 {a\in\mathcal{A}} . Using the theory of functional identities and the structure of involutions on matrix algebras, we prove that if ℱ {\mathcal{F}} and 𝒢 {\mathcal{G}} are additive, then 𝒢 = 0 {\mathcal{G}=0} . We also show that, in case ‘ * * ’ is any nonidentity anti-automorphism, the same conclusion holds if either ‘ * {*} ’ is not identity on 𝒵 ( 𝒜 ) {\mathcal{Z}(\mathcal{A})} or 𝒜 {\mathcal{A}} is a PI-ring.
... Sinclair [13] proved that every continuous Jordan derivation of a semisimple Banach algebra is a derivation; for derivations and Jordan derivations on group algebras see [1,2,12]. Jordan left derivations have been introduced and studied by Brešar and Vukman [4]. They proved that there is no nonzero Jordan left derivation of noncommutative prime rings with a suitable characteristic. ...
In this paper, we study the types of Jordan derivations of a Banach algebra A with a right identity e. We show that if eA is commutative and semisimple, then every Jordan derivation of A is a derivation. In this case, Jordan derivations map A into the radical of A. We also prove that every Jordan triple left (right) derivation of A is a Jordan left (right) derivation. Finally, we investigate the range of Jordan left derivations and establish that every Jordan left derivation of A maps A into eA.
... Sinclair [13] proved that every continuous Jordan derivation of a semisimple Banach algebra is a derivation; for derivations and Jordan derivations on group algebras see [1,2,12]. Jordan left derivations have been introduced and studied by Brešar and Vukman [4]. They proved that there is no nonzero Jordan left derivation of noncommutative prime rings with a suitable characteristic. ...
In this paper, we study the types of Jordan derivations of a Banach algebra A with a right identity e. We show that if eA is commutative and semisimple, then every Jordan derivation of A is a derivation. In this case, Jordan derivations map A into the radical of A. We also prove that every Jordan triple left (right) derivation of A is a Jordan left (right) derivation. Finally, we investigate the range of Jordan left derivations and establish that every Jordan left derivation of A maps A into eA.
... The notion of derivation is useful in studying the structures, properties of algebraic systems and has important role in characterizing algebraic structures. Bresar and Vukman established that a prime ring must be commutative if it admits a nonzero left derivation [2] in 1990. The first result in this direction is due to Posner [19] in 1957. ...
In this paper, we introduce the notion of a derivation of ordered semirings, study some properties of derivations of ordered semirings, derivations of fuzzy ideals and relations between derivation and homomorphism of ordered semirings. We prove that if µ is a fuzzy prime ideal and d is an onto derivation of idempotent ordered semiring M with identity then d(µ) and d −1 (µ) are fuzzy prime ideals of M.
... Jordan left derivation) if d(rs) = rd(s) + sd(r) (resp. d(r 2 ) = 2rd(r)) holds for all r, s ∈ A. The concepts of left derivations and Jordan left derivations were introduced by Breşar et al. in [7], and it was shown that if a prime ring R of characteristic different from 2 and 3 admits a nonzero Jordan left derivation, then R must be commutative. Obviously, every left derivation is a Jordan left derivation, but the converse need not be true in general (see [9,Example 1.1]). ...
We will extend in this paper some results about commutativity of Jordan ideals proved in [2] and [6]. However, we will consider left derivations instead of derivations, which is enough to get good results in relation to the structure of near-rings. We will also show that the conditions imposed in the paper cannot be removed.
... There are many works dealing with the commutativity of prime and semi prime rings admitting certain types of derivations (see [3][4][5][6]8,12,17]). Ali [2] defined symmetric * -bi derivation, a symmetric left (resp. ...
Let R be a *-ring. In this paper we introduce the notion of generalized *-n-derivation in R. An additive mapping x → x * of R into itself is called an involution on R if it satisfies the conditions: (i) (x *) * = x,(ii) (xy) * = y * x * for all x, y ∈ R. A ring R equipped with an involution * is called a *-ring. It is shown that if a prime *-ring R admits a nonzero generalized *-n-derivation F (resp. a reverse generalized *-n-derivation) equipped with a *-n-derivation derivation (resp. a reverse *-n-derivation) D, then R is commutative. Further, some related properties of generalized *-n-derivation in a semiprime *-ring have also been investigated.
... The semiring 26 theory is useful to many areas of mathematics and theoretical computer science. 27 There are several authors investigated the relationship between the commutativ-28 ity of a ring R and the existence of certain specified derivations of R. Bresar 29 and Vukman [1] established that a prime ring admits a nonzero derivation in (ii) (a + b)cd = acd + bcd, a(b + c)d = abd + acd, ab(c + d) = abc + abd. ...
... In 1990, Brešar and Vukman [2], introduced the concept of left derivation and Jordan left derivation, respectively. An additive map D : R → R is said to be a left derivation if D(xy) = xD(y) + yD(x), for all x, y ∈ R and a Jordan left derivation if D(x 2 ) = 2xD(x), for all x ∈ R. ...
Let m and n be positive integers such that m = n, and R, an mn(m + n)|m − n|-torsion free semiprime ring. We prove that every generalized (m, n)-Jordan derivation over R is a derivation which maps R into its center.
... This result was subsequently, refined and extended by a number of authors. In [6], Bresar and Vuckman showed that a prime ring must be commutative if it admits a nonzero left derivation. Recently, many authors have obtained commutativity theorems for prime and semiprime rings admitting derivation, generalized derivation. ...
In this paper, we introduce the notion of α-semiderivation on prime rings, and we try to extend some results for derivations of rings or near-rings to a more general case for α-semiderivations of prime rings.
... This result was subsequently, refined and extended by a number of authors. In [7], Bresar and Vuckman showed that a prime ring must be commutative if it admits a nonzero left derivation. Recently, many authors have obtained commutativity theorems for prime and semiprime rings admitting derivation, generalized derivation. ...
In this paper, we investigate the commutativity of prime rings admitting multipliers of R satisfying certain identities and some related results have also been discussed.
... After that, in 1988 M. Bresar and J. Vukman gave an alternative proof of the result in [1]. In [2], M. Bresar and J. Vukman defined left derivation from a ring to a left R module X and they showed the existence of a nonzero Jordan left derivation of R into X implies R is commutative. The concept of a symmetric bi-derivation has been introduced and studied by Maksa in [8] (see also in [9]). ...
A symmetric bi additive mapping D, on a prime ring R is called skew symmetric bi-Jordan derivation if it satisfies the following condition associated with the automorphism of of R:% \begin{equation*} D( x^{2},z) =\alpha(x)D(x,z) +D(x,z)x \end{equation*}% The purpose of this paper is to prove some results concerning skew symmetric bi-Jordan derivation, as a generalization of symmetric bi-Jordan derivation.
... Recently, the author along with Ajda Fošner [17] have investigated the same problem for (σ, τ)-derivations from a C * -algebra A into a Banach A-module M. It should be mentioned that Herstein theorem has been fairly generalized by Beidar et al [4]. Vukman [10,24,25] used some interesting functional identities to characterize derivations and left derivations. Now, we come to the part where we talk about the achievements of this article. ...
The main purpose of this research is to characterize generalized (left) derivations and Jordan (*,*)-derivations on Banach algebras and rings using some functional identities. Let A be a unital semiprime Banach algebra and let F,G : A ? A be linear mappings satisfying F(x) =-x2G(x-1) for all x ? Inv(A), where Inv(A) denotes the set of all invertible elements of A. Then both F and G are generalized derivations on A. Another result in this regard is as follows. Let A be a unital semiprime algebra and let n > 1 be an integer. Let f,g : A ? A be linear mappings satisfying f (an) = nan-1g(a) = ng(a)an-1 for all a ? A. If g(e) ? Z(A), then f and g are generalized derivations associated with the same derivation on A. In particular, if A is a unital semisimple Banach algebra, then both f and 1 are continuous linear mappings. Moreover, we define a (*,*)-ring and a Jordan (*,*)-derivation. A characterization of Jordan (*,*)-derivations is presented as follows. Let R be an n!-torsion free (*,*)-ring, let n > 1 be an integer and let d : R ? R be an additive mapping satisfying d(an) = ?nj =1 a*n-jd(a)a* j-1 for all a ? R. Then d is a Jordan (*,*)-derivation on R. Some other functional identities are also investigated.
... (if ( x [1]. Bresar and Vukman [2] introduced the concept of left derivation .We refer the readers to several references [3,4,5,6] for results concerning Jordan left derivations. A linear mapping from onto is called Jordan left centralizers (JLC) (resp., left centralizers (LC)) if ( A linear map from onto is called a Jordan centralizer (JC) if satisfies (ab+ba)= ( ) +b (a)= (b)a+a (b) [7]. ...
We define skew matrix gamma ring and describe the constitution of Jordan left centralizers and derivations on skew matrix gamma ring on a -ring. We also show the properties of these concepts.
... Derivation is a very interesting and important area of research in the theory of algebraic structures in mathematics. Several authors [7][8][9][10][11] have studied derivations in rings and near rings. Jun and Xin [12] applied the notions of derivations in ring and near-ring theory to BCI-algebras, and they also introduced a new concept called a regular Derivation in BCI-algebra. ...
... Derivation is a very interesting and important area of research in the theory of algebraic structures in mathematics. Several authors [7][8][9][10][11] have studied derivations in rings and near rings. Jun and Xin [12] applied the notions of derivations in ring and near-ring theory to BCI-algebras, and they also introduced a new concept called a regular Derivation in BCI-algebra. ...
In this manuscript we insert the concept of derivations in associative PU-algebras and discuss some of its important results such that we prove that for a mapping being a (Left, Right) or (Right, Left)-derivation of an associative PU-algebra then such a mapping is one-one. If a mapping is regular then it is identity. If any element of an associative PU-algebra satisfying the criteria of identity function then such a map is identity. We also prove some useful properties for a mapping being (Left, Right)-regular derivation of an associative PU-algebra and (Right, Left)-regular derivation of an associative PU-algebra. Moreover we prove that if a mapping is regular (Left, Right)-derivation of an associative PU-algebra then its Kernel is a subalgebra. We have no doubt that the research along this line can be kept up, and indeed, some results in this manuscript have already made up a foundation for further exploration concerning the further progression of PU-algebras. These definitions and main results can be similarly extended to some other algebraic systems such as BCH-algebras, Hilbert algebras, BF-algebras, J-algebras, WS-algebras, CI-algebras, SU-algebras, BCL-algebras, BP-algebras and BO-algebras, Z-algebras and so forth. The main purpose of our future work is to investigate the fuzzy derivations ideals in PU-algebras, which may have a lot of applications in different branches of theoretical physics and computer science.
... -bimodule. In 2004, C. Park[10] gave a characterization of the stability of derivations from a Banach algebra into its Banach bimodule.In 1990, M. Brešar and J. Vukman[11] introduced the concepts of left derivations and Jordan left ...
In this paper, we study the Hyers-Ulam-Rassias stability of (m,n) -Jordan derivations. As applications, we characterize (m,n) -Jordan derivations on {C}^{\ast } -algebras and some non-self-adjoint operator algebras.
... They proved that the existence of a nonzero left derivation of a prime ring of characteristic not 2 implies the commutativity of the ring. After that many new results have been established on Jordan left derivations over different rings and algebras [1,4,8,10,12,13]. Recently, in 2016, Brešar introduced {g, h}-derivation and studied over semiprime algebras and tensor product of algebras [6]. ...
In this article, left {g, h}-derivation and Jordan left {g, h}-derivation on algebras are introduced. It is shown that there is no Jordan left {g, h}-derivation over M n (C) and H R , for g = h. Examples are given which show that every Jordan left {g, h}-derivation over T n (C), M n (C) and H R are not left {g, h}-derivations. Also, the Jordan left {g, h}-derivations over T n (C), M n (C) and H R are right centralizers, where C is a 2-torsionfree commutative ring. Moreover, we prove the result of Jordan left {g, h}-derivation to be a left {g, h}-derivation over tensor products of algebras as well as for algebra of polynomials.
... Let A be a non-commutative Banach algebra and D be a continuous derivation on A. Brešar and Vukman [5] proved that if [D(x), x] ∈ rad(A) for all x ∈ A, then D maps A into rad(A). Vukman [30] proved that the same conclusion holds if [D(x), x] 3 ∈ rad(A) for all x ∈ A. In [17], Kim proved that if D is a continuous linear Jordan derivation in a Banach algebra A, such that [D(x), x]D(x)[D(x), x] ∈ rad(A), for all x ∈ A, then D maps A into rad(A). ...
Let R be a prime ring with extended centroid C, I a non-zero ideal of R and n ≥ 1 a fixed integer. If R admits the generalized derivations H and G such that (H(xy) + G(yx)) n = (xy ± yx) for all x, y ∈ I, then one of the following holds: 1. R is commutative; 2. n = 1 and H(x) = x and G(x) = ±x for all x ∈ R. Moreover, we examine the case where R is a semiprime ring. Finally, we apply the above result to non-commutative Banach algebras.
... It is well known that every derivation on a given ring is a Jordan derivation, however, the converse is not true in general. The converse which is referred in the literature as Jordan derivation problem, was investigated by Herstein [11] and later on improved by M. Bresar and J. Vukman [7] by establishing that a every Jordan derivation on a prime ring is a derivation. In 1996, M. Spanci and A. Nakajima [16] introduced the notions of derivations and Jordan derivations on gamma rings. ...
... The first result in this derivation is due to Posner [15] in 1957. In the year 1990, Bresar and Vukman [1] established that a prime ring must be commutative if it admits a nonzero left derivation. Kim [2,3,4 ] studied right derivation and generalized derivation of incline algebra. ...
Over the last few decades, several authors have investigated the relationship between the commutativity of ring R and the existence of certain specified derivations of R. In this paper, we introduce the concept of right derivation on semigroups and we study some of the properties of right derivation of semigroups. We prove that if d be a non-zero right derivation of a cancellative ordered semigroup M, then M is a commutative ordered semigroup.
... In the year 1990, Bresar and Vukman [2] established that a prime ring must be commutative if it admits a non-zero left derivation. Kim [3], [4] studied right derivation and generalized derivation of incline algebra. ...
In this paper, we introduce the concept of f-derivation and the concept of da derivation of an ordered semiring. We study some of the properties of f and da derivations of ordered semirings. We prove that, if d is a f-derivation of an ordered integral semiring M then kerd is a m − k−ideal of M.
... Over the last few decades, several authors have investigated the relationship between the commutativity of ring R and the existence of certain specified derivations of R. The first result in this direction is due to Posner [9] in 1957. In the year 1990, Bresar and Vukman [2] established that a prime ring must be commutative if it admits a nonzero left derivation. Kim [3,4 ] studied right derivation and generalized derivation of incline algebra. ...
In this paper, we introduce the concept of (f, g)-derivation of an ordered semiring and study its properties. We prove that, if d is a (f, g)-derivation of an ordered integral semiring M then, ker d is a m − k−ideal of M .
... The first result in this direction is due to Posner [17] in 1957. In 1990, Bresar and Vukman [2] established that a prime ring must be commutative if it admits a nonzero left derivation. In this paper our main aim is to study the properties of derivations of fuzzy ideals in an ordered Gsemiring. ...
In this paper, we introduce the notion of derivation of ordered G-semirings, study some properties of derivations of ordered G-semirings, derivations of fuzzy ideals in ordered G-semirings and study the relations between derivation and homomorphism of ordered G-semiring. We prove that if m is a fuzzy prime ideal and d is an onto derivation of idempotent ordered G-semiring M with identity then () d m and () 1 d mare fuzzy prime ideals of M .
... In [11], M. Brešar and J. Vukman introduce the concepts of left derivation and Jordan left derivation. A linear mapping δ from A into M is called a left derivation if δ(ab) = aδ(b) + bδ(a) for each a, b in A; and δ is called a Jordan left derivation if δ(a • b) = 2aδ(b) + 2bδ(a) for each a, b in A. In [7], the authors prove that every Jordan left derivation from a C * -algebra into its Banach left module is identical with zero. ...
Let be a unital -algebra and be a unital --bimodule, we study the local properties of -derivations, -Jordan derivations and -left derivations from into through zero products or zero Jordan products. Moreover, we give some applications on matrix algebras, algebras of locally measurable operators and von Neumann algebras. We also investigate when a linear mapping from a unital -algebra into its unital Banach -bimodule is a -derivation or a -left derivation under certain conditions.
The main purpose of this paper is to provide a simple and short way to obtain Leibniz rule for various types of derivations. As an application of this discussion, the generalized types of higher derivations are also studied.
Let G be a simple graph of order n and A(G) be an adjacency matrix of G, the spectra of G is the set of eigenvalues of A(G). In other side, the resolvent matrix is a matrix with property that all of its eigenvalues are outside the spectra. In this talk, we present an exponential growth to the graph resolvent.
In this paper, we introduced the concepts of semi global domination number in product fuzzy graph, and is denoted by γsg(G) and semi complimentary product fuzzy graph. We determine the semi global domination number γsg(G) for several classes of product fuzzy graph and obtain Nordhaus-Gaddum type results for this parameter. Further, some bounds of γsg(G) are investigated. Moreover, the relations between γsg(G) and other known parameters in Product fuzzy graphs are investigated.
In this part, we will introduce definitions of special functions. Gamma, beta and Mittag- Leffler functions will be given. Then, derivatives and integration of Mittag-Leffler function will be defined before definitions of fractional operators will have been mentioned. Since, this topic are basic to introduce q-Mittag Leffler function and their applications on q-fractional differential operators. Finally, q-gamma, q-beta and q-Mittag Leffler functions will be mentioned before q-fractional differential operators.
In this section, initially we will mention that the definitions of fractional calculus. To give a detail as, Grunwald-Letnikov, Riemann-Liouville & Caputo's definitions will be given because, these definitions are connected with the difference operators. So, after these definitions are introduced, difference equations and linear type of difference equations will be given in this part. Later, definitions of fractional calculus that are our mentioning will be adapted in q-version as Riemann-Liouville and Caputo type on difference operators.
Let be a -algebra and be a --bimodule. We study the local properties of -derivations and -Jordan derivations from into under the following orthogonality conditions on elements in : , and . We characterize the mappings on zero product determined algebras and zero Jordan product determined algebras. Moreover, we give some applications on -algebras, group algebras, matrix algebras, algebras of locally measurable operators and von Neumann algebras.
Let R be a prime ring with center Z(R) and alpha,beta be automorphisms of R. This paper is divided into two parts. The first tackles the notions of (generalized) skew derivations on R, as the subject of the present study, several characterization theorems concerning commutativity of prime rings are obtained and an example proving the necessity of the primeness hypothesis of R is given. The second part of the paper tackles the notions of symmetric Jordan bi (alpha,beta)-derivations. In addition, the researchers illustrated that for a prime ring with char(R) different from 2, every symmetric Jordan bi (alpha,alpha)-derivation D of R is a symmetric bi (alpha,alpha)-derivation.
Let R be a prime ring, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R and F a generalized derivation with associated derivation d of R. Suppose that (F([x, y])) m = [x, y] n for all x, y ∈ I, a nonzero ideal of R, where m ≥ 1 and n ≥ 1 are fixed integers, then one of the following holds: (1) R is commutative; (2) there exists a ∈ C such that F(x) = ax for all x ∈ R with a m = 1. Moreover, in this case if m = n, then either char (R) = 2 or char (R) = 2 |m−n| − 1. We also extend the result to the one sided case for m = n. Finally as an application we obtain a range inclusion result of continuous generalized derivations on Banach algebras.
Contain of JOURNAL OF IMVI, 10(2)(2020), 199--376.
In this article we introduce A*-involution in additively inverse semirings. This involution have potential to extend the striking results of B*-algebras, C*-algebras and involutory rings in the domain of semirings. The remarkable result due to Herstein [XII] states that every Jordan derivation on a 2-torsion free prime ring is a derivation. In the present paper, we shall study the above mentioned result for Jordan left derivations in semirings with A* -Involution.
Let R be a ring with center Z, and S a nonempty subset of R . A mapping F from R to R is called centralizing on S if [ x, F(x) ] ∊ Z for all x ∊ S . We show that a semiprime ring R must have a nontrivial central ideal if it admits an appropriate endomorphism or derivation which is centralizing on some nontrivial one-sided ideal. Under similar hypotheses, we prove commutativity in prime rings.
Let R be a prime ring and T be a nontrivial automorphism of R . If xx T — x T x is in the center of the ring for every x in R , then R is a commutative integral domain.
The purpose of this paper is to present a brief proof of the well known result of Herstein which states that any Jordan derivation on a prime ring with characteristic not two is a derivation.
LetR be a *-ring. We study an additive mappingD: R R satisfyingD(x
2) =D(x)x
* +xD(x) for allx R.It is shown that, in caseR contains the unit element, the element 1/2, and an invertible skew-hermitian element which lies in the center ofR, then there exists ana R such thatD(x) = ax
*
– xa for allx R. IfR is a noncommutative prime real algebra, thenD is linear. In our main result we prove that a noncommutative prime ring of characteristic different from 2 is normal (i.e.xx
* =x
*
x for allx R) if and only if there exists a nonzero additive mappingD: R R satisfyingD(x
2) =D(x)x
* +xD(x) and [D(x), x] = 0 for allx R. This result is in the spirit of the well-known theorem of E. Posner, which states that the existence of a nonzero derivationD on a prime ringR, such that [D(x), x] lies in the center ofR for allx R, forcesR to be commutative.
- B E Johnson
B. E. Johnson, Continuity of derivations on commutative Banach algebras, Amer. J. Math.
91 (1969), 1-10.
Automatic continuity of linear operators
_,
Automatic continuity of linear operators, London Math. Soc. Lecture Note Ser. 21,
Cambridge University Press, Cambridge, London, New York, and Melbourne, 1976.