The major goal of this paper is to investigate the structure of [Formula: see text]-prime rings with involution, satisfying certain [Formula: see text]-differential identities involving [Formula: see text]-centralizing and some …
The major goal of this paper is to investigate the structure of [Formula: see text]-prime rings with involution, satisfying certain [Formula: see text]-differential identities involving [Formula: see text]-centralizing and some special type of products. We also provide an example which shows that center of [Formula: see text]-prime ring is not free from zero divisors.
Let A be a unital ∗-algebra over the complex fields C. For any H1,H2∈A, a product [H1,H2]•=H1H2−H2H1* is called the skew Lie product. In this article, it is shown that …
Let A be a unital ∗-algebra over the complex fields C. For any H1,H2∈A, a product [H1,H2]•=H1H2−H2H1* is called the skew Lie product. In this article, it is shown that if a map ξ : A→A (not necessarily linear) satisfies ξ(Pn(H1,H2,…,Hn))=∑i=1nPn(H1,…,Hi−1,ξ(Hi),Hi+1,…,Hn)(n≥3) for all H1,H2,…,Hn∈A, then ξ is additive. Moreover, if ξ(ie2) is self-adjoint, then ξ is ∗-derivation. As applications, we apply our main result to some special classes of unital ∗-algebras such as prime ∗-algebra, standard operator algebra, factor von Neumann algebra, and von Neumann algebra with no central summands of type I1.
Let [Formula: see text] be a [Formula: see text]-torsion free unital ∗-ring containing nontrivial symmetric idempotent. In this article, it is shown that if a map [Formula: see text] : …
Let [Formula: see text] be a [Formula: see text]-torsion free unital ∗-ring containing nontrivial symmetric idempotent. In this article, it is shown that if a map [Formula: see text] : [Formula: see text] (not necessarily additive) satisfies [Formula: see text] for all [Formula: see text], then [Formula: see text] is additive. Moreover, if [Formula: see text] is self-adjoint, then [Formula: see text] is a ∗-derivation. As an applications, we apply our main result to some special classes of unital ∗-rings and ∗-algebras such as prime ∗-ring, prime ∗-algebra, standard operator algebra, factor von Neumann algebra and von Neumann algebra with no central summands of type [Formula: see text].
This paper’s major goal is to work on commutativity of σ-prime rings with second kind involution σ, involving generalized derivation satisfy the certain differential identities. Finally, we provide some examples …
This paper’s major goal is to work on commutativity of σ-prime rings with second kind involution σ, involving generalized derivation satisfy the certain differential identities. Finally, we provide some examples to demonstrate that the conditions assumed in our results are not unnecessary
Let $\mathcal{B}$ be a unital $\varnothing$-ring with a $2$-torsion free that contains non-trivial symmetric idempotent. For any $B_1,B_2,B_3,\ldots,B_n \in \mathcal{B}$, a product $B_1 \circ B_2=B_1B_2+B_2B_1$ is called Jordan product and …
Let $\mathcal{B}$ be a unital $\varnothing$-ring with a $2$-torsion free that contains non-trivial symmetric idempotent. For any $B_1,B_2,B_3,\ldots,B_n \in \mathcal{B}$, a product $B_1 \circ B_2=B_1B_2+B_2B_1$ is called Jordan product and $B_1 \bullet B_2=B_1B_2+B_2B_1^\varnothing$ is recognized as a skew Jordan product. Characterize mixed Jordan triple product as $Q_3(B_1,B_2,B_3)=B_1 \circ B_2 \bullet B_3$ and mixed Jordan $n$-product as $Q_n(B_1,B_2,\ldots,B_n)=B_1 \circ B_2 \circ \cdots \bullet B_n$ for all integer $n\geq3$. The present paper deals that a mapping which is called multiplicative mixed Jordan $n$-derivation, $\Psi$: $\mathcal{B} \rightarrow \mathcal{B}$ satisfies $\Psi(Q_n(B_1,B_2,\ldots,B_n))=\sum_{i=1}^{n} Q_n(B_1, \ldots, B_{i-1}, \Psi(B_i), B_{i+1}, \ldots,B_n)$ for all $B_1, B_2,\ldots, B_n \in \mathcal{B}$ if and only if $\Psi$ is an additive $\varnothing$-derivation. Finally, primary outcome is applicable in various specific categories of unital $\varnothing$-rings and $\varnothing$-algebras including prime $\varnothing$-rings, prime $\varnothing$-algebras and factor von Neumann algebras.
Let A be a 2-torsion free unital *-ring containing non-trivial symmetric idempotent. For ?1, ?2 ? A, the product ?1 ? ?2 = ?1 ?2 + ?2 ?*1 is called …
Let A be a 2-torsion free unital *-ring containing non-trivial symmetric idempotent. For ?1, ?2 ? A, the product ?1 ? ?2 = ?1 ?2 + ?2 ?*1 is called the skew Jordan product of elements ?1 and ?2. In this article, it is shown that if a map ?: A ? A (not necessarily additive) fulfills ?(nP (?1, ?2,...,?n)) = ?n,i=1 Pn(?1,...,?i-1, ?(?i), ?i+1,...,?n) for all ?1, ?2,..., ?n ? A, then ? is additive. Moreover, if ?(I) is self- adjoint then ? is a *-derivation. As applications, our main result is applied to several special classes of unital *-rings and unital *-algebras such as prime +-ring, prime +-algebra, factor von Neumann algebra.
Let $\mathcal{B}$ be a unital $\varnothing$-ring with a $2$-torsion free that contains non-trivial symmetric idempotent. For any $B_1,B_2,B_3,\ldots,B_n \in \mathcal{B}$, a product $B_1 \circ B_2=B_1B_2+B_2B_1$ is called Jordan product and …
Let $\mathcal{B}$ be a unital $\varnothing$-ring with a $2$-torsion free that contains non-trivial symmetric idempotent. For any $B_1,B_2,B_3,\ldots,B_n \in \mathcal{B}$, a product $B_1 \circ B_2=B_1B_2+B_2B_1$ is called Jordan product and $B_1 \bullet B_2=B_1B_2+B_2B_1^\varnothing$ is recognized as a skew Jordan product. Characterize mixed Jordan triple product as $Q_3(B_1,B_2,B_3)=B_1 \circ B_2 \bullet B_3$ and mixed Jordan $n$-product as $Q_n(B_1,B_2,\ldots,B_n)=B_1 \circ B_2 \circ \cdots \bullet B_n$ for all integer $n\geq3$. The present paper deals that a mapping which is called multiplicative mixed Jordan $n$-derivation, $\Psi$: $\mathcal{B} \rightarrow \mathcal{B}$ satisfies $\Psi(Q_n(B_1,B_2,\ldots,B_n))=\sum_{i=1}^{n} Q_n(B_1, \ldots, B_{i-1}, \Psi(B_i), B_{i+1}, \ldots,B_n)$ for all $B_1, B_2,\ldots, B_n \in \mathcal{B}$ if and only if $\Psi$ is an additive $\varnothing$-derivation. Finally, primary outcome is applicable in various specific categories of unital $\varnothing$-rings and $\varnothing$-algebras including prime $\varnothing$-rings, prime $\varnothing$-algebras and factor von Neumann algebras.
This paper’s major goal is to work on commutativity of σ-prime rings with second kind involution σ, involving generalized derivation satisfy the certain differential identities. Finally, we provide some examples …
This paper’s major goal is to work on commutativity of σ-prime rings with second kind involution σ, involving generalized derivation satisfy the certain differential identities. Finally, we provide some examples to demonstrate that the conditions assumed in our results are not unnecessary
Let [Formula: see text] be a [Formula: see text]-torsion free unital ∗-ring containing nontrivial symmetric idempotent. In this article, it is shown that if a map [Formula: see text] : …
Let [Formula: see text] be a [Formula: see text]-torsion free unital ∗-ring containing nontrivial symmetric idempotent. In this article, it is shown that if a map [Formula: see text] : [Formula: see text] (not necessarily additive) satisfies [Formula: see text] for all [Formula: see text], then [Formula: see text] is additive. Moreover, if [Formula: see text] is self-adjoint, then [Formula: see text] is a ∗-derivation. As an applications, we apply our main result to some special classes of unital ∗-rings and ∗-algebras such as prime ∗-ring, prime ∗-algebra, standard operator algebra, factor von Neumann algebra and von Neumann algebra with no central summands of type [Formula: see text].
Let A be a 2-torsion free unital *-ring containing non-trivial symmetric idempotent. For ?1, ?2 ? A, the product ?1 ? ?2 = ?1 ?2 + ?2 ?*1 is called …
Let A be a 2-torsion free unital *-ring containing non-trivial symmetric idempotent. For ?1, ?2 ? A, the product ?1 ? ?2 = ?1 ?2 + ?2 ?*1 is called the skew Jordan product of elements ?1 and ?2. In this article, it is shown that if a map ?: A ? A (not necessarily additive) fulfills ?(nP (?1, ?2,...,?n)) = ?n,i=1 Pn(?1,...,?i-1, ?(?i), ?i+1,...,?n) for all ?1, ?2,..., ?n ? A, then ? is additive. Moreover, if ?(I) is self- adjoint then ? is a *-derivation. As applications, our main result is applied to several special classes of unital *-rings and unital *-algebras such as prime +-ring, prime +-algebra, factor von Neumann algebra.
Let A be a unital ∗-algebra over the complex fields C. For any H1,H2∈A, a product [H1,H2]•=H1H2−H2H1* is called the skew Lie product. In this article, it is shown that …
Let A be a unital ∗-algebra over the complex fields C. For any H1,H2∈A, a product [H1,H2]•=H1H2−H2H1* is called the skew Lie product. In this article, it is shown that if a map ξ : A→A (not necessarily linear) satisfies ξ(Pn(H1,H2,…,Hn))=∑i=1nPn(H1,…,Hi−1,ξ(Hi),Hi+1,…,Hn)(n≥3) for all H1,H2,…,Hn∈A, then ξ is additive. Moreover, if ξ(ie2) is self-adjoint, then ξ is ∗-derivation. As applications, we apply our main result to some special classes of unital ∗-algebras such as prime ∗-algebra, standard operator algebra, factor von Neumann algebra, and von Neumann algebra with no central summands of type I1.
The major goal of this paper is to investigate the structure of [Formula: see text]-prime rings with involution, satisfying certain [Formula: see text]-differential identities involving [Formula: see text]-centralizing and some …
The major goal of this paper is to investigate the structure of [Formula: see text]-prime rings with involution, satisfying certain [Formula: see text]-differential identities involving [Formula: see text]-centralizing and some special type of products. We also provide an example which shows that center of [Formula: see text]-prime ring is not free from zero divisors.
Abstract Let R be a prime ring and U be a nonzero ideal or quadratic Jordan ideal of R . If L is a nontrivial automorphism or derivation of R …
Abstract Let R be a prime ring and U be a nonzero ideal or quadratic Jordan ideal of R . If L is a nontrivial automorphism or derivation of R such that uL ( u )— L ( u ) u is in the center of R for every u in U , then the ring R is commutative.
Two well-known results of E. C. Posner state that the composition of two nonzero derivations of a prime ring cannot be a nonzero derivation, and that in a prime ring, …
Two well-known results of E. C. Posner state that the composition of two nonzero derivations of a prime ring cannot be a nonzero derivation, and that in a prime ring, if the commutator of each element and its image under a nonzero derivation is central, then the ring is commutative.Our purpose is to show how the theory of differential identities can be used to obtain these results and their generalizations to Lie ideals and to rings with involution.
In this paper, we investigate commutativity of ring R with involution ∗ which admits a derivation satisfying certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been …
In this paper, we investigate commutativity of ring R with involution ∗ which admits a derivation satisfying certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Finally, we provide examples to show that various restrictions imposed in the hypotheses of our theorems are not superfluous.
Let T be a factor von Neumann algebra acting on complex Hilbert space with dim(T) ? 2. For any T, T1, T2,..., Tn ? T, define q1(T) = T, q2(T1, …
Let T be a factor von Neumann algebra acting on complex Hilbert space with dim(T) ? 2. For any T, T1, T2,..., Tn ? T, define q1(T) = T, q2(T1, T2) = T1 ? T2 = T1T+ 2 + T2T+ 1 and qn(T1,..., Tn) = qn?1(T1,..., Tn?1) ? Tn for all integers n ? 2. In this article, we prove that a map ? : T ? T satisfies ?(qn(T1,..., Tn)) = Pni =1 qn(T1,..., Ti?1, ?(Ti), Ti+1,..., Tn) for all T1,..., Tn ? T if and only if ? is an additive *-derivation.
Let be a factor von Neumann algebra and be the *-Jordan derivation on , that is, for every , where , then is additive *-derivation.
Let be a factor von Neumann algebra and be the *-Jordan derivation on , that is, for every , where , then is additive *-derivation.
We extend a result of Herstein concerning a derivation d on a prime ring R satisfying [ d ( x ), d ( y )] = 0 for all x …
We extend a result of Herstein concerning a derivation d on a prime ring R satisfying [ d ( x ), d ( y )] = 0 for all x , y ∈ R , to the case of semiprime rings. An extension of this result is proved for a two‐sided ideal but is shown to be not true for a one‐sided ideal. Some of our recent results dealing with U * ‐ and U ** ‐ derivations on a prime ring are extended to semiprime rings. Finally, we obtain a result on semiprime rings for which d ( x y ) = d ( y x ) for all x , y in some ideal U .
A classical result of Posner states that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. In this paper we extend Posner's …
A classical result of Posner states that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. In this paper we extend Posner's result to generalized derivations centralizing on Jordan ideals of rings with involution and discuss the related results. Moreover, we provide examples to show that the assumed restriction cannot be relaxed.
AbstractLet be a factor von Neumann algebra. Suppose that is a nonlinear skew Jordan derivable map. Then, is an additive -derivation. In particular, if the von Neumann algebra is infinite …
AbstractLet be a factor von Neumann algebra. Suppose that is a nonlinear skew Jordan derivable map. Then, is an additive -derivation. In particular, if the von Neumann algebra is infinite type I factors, a concrete characterization of is given.Keywords: skew Jordan derivable map-derivationvon Neumann algebraAMS Subject Classifications: 47B4946L10 NotesNo potential conflict of interest was reported by the author.Additional informationFundingThis research was supported by the National Natural Science Foundation of China [grant number 11402199], [grant number 11426176].
Let $R$ be an associative ring. An additive map $x\mapsto x^*$ of $R$ into itself is called an involution if (i) $(xy)^*=y^*x^*$ and (ii) $(x^*)^*=x$ hold for all $x\in R$. …
Let $R$ be an associative ring. An additive map $x\mapsto x^*$ of $R$ into itself is called an involution if (i) $(xy)^*=y^*x^*$ and (ii) $(x^*)^*=x$ hold for all $x\in R$. The main purpose of this paper is to study some additive mappings on prime and semiprime rings with involution. Moreover, some examples are given to demonstrate that the restrictions imposed on the hypothesis of the various results are not superfluous.
AbstractLet be the algebra of all bounded linear operators on a complex Hilbert space and be a von Neumann algebra without central abelian projections. Let be a non-zero scalar. In …
AbstractLet be the algebra of all bounded linear operators on a complex Hilbert space and be a von Neumann algebra without central abelian projections. Let be a non-zero scalar. In this paper, it is proved that a mapping satisfies for all if and only if is an additive *-derivation and for all Keywords: *-derivationsξ-Jordan *-derivationsvon Neumann algebrasAMS Subject Classifications: 47B4946L10 AcknowledgmentsThe authors would like to thank the referee for their valuable comments and suggestions. The second author is supported by the National Natural Science Foundation of China (Grant No. 11171244). The third author is supported by the National Natural Science Foundation of China (Grant No. 11071188).
Let 𝒜 be a unital ∗-algebra with the unit I. Assume that 𝒜 contains a nontrivial projection P which satisfies X𝒜P=0 implies X=0 and X𝒜(I−P)=0 implies X=0. In this paper, …
Let 𝒜 be a unital ∗-algebra with the unit I. Assume that 𝒜 contains a nontrivial projection P which satisfies X𝒜P=0 implies X=0 and X𝒜(I−P)=0 implies X=0. In this paper, it is shown that Φ is a nonlinear ∗-Jordan-type derivation on 𝒜 if and only if Φ is an additive ∗-derivation. As applications, the nonlinear ∗-Jordan-type derivations on prime ∗-algebras, von Neumann algebras with no central summands of type I1, factor von Neumann algebras and standard operator algebras are characterized.
Let A be a factor von Neumann algebra with dimA≥2. For any X1,X2,⋯,Xn∈A, define p1(X1)=X1, p2(X1,X2)=[X1,X2]•=X1X2∗−X2X1∗, and pn(X1,X2,⋯,Xn)=[pn−1(X1,X2,⋯,Xn−1),Xn]• for all integers n≥2. In this article, we prove that a map …
Let A be a factor von Neumann algebra with dimA≥2. For any X1,X2,⋯,Xn∈A, define p1(X1)=X1, p2(X1,X2)=[X1,X2]•=X1X2∗−X2X1∗, and pn(X1,X2,⋯,Xn)=[pn−1(X1,X2,⋯,Xn−1),Xn]• for all integers n≥2. In this article, we prove that a map L:A→A satisfies L(pn(X1,X2,⋯,Xn))=∑i=1npn(X1,X2,⋯,Xi−1,L(Xi),Xi+1,⋯,Xn) for all X1,X2,⋯,Xn∈A if and only if L is an additive *-derivation.
Posner's first theorem states that if R is a prime ring of characteristic different from two, d 1 and d 2 are derivations on R such that the iterate d …
Posner's first theorem states that if R is a prime ring of characteristic different from two, d 1 and d 2 are derivations on R such that the iterate d 1 d 2 is also a derivation of R, then at least one of d 1 , d 2 is zero.In the present paper we extend this result to * -prime rings of characteristic different from two.* -ideal of R if I * = I.Let R be a * -prime ring, a ∈ R and aRa = {0}.This implies that aRaRa * = {0} also.Now * -primeness of R insures that a = 0 or aRa * = {0}.aRa * = {0} together with aRa = {0} gives us a = 0. Thus we conclude that every * -prime ring is a semiprime ring.An additive mapping d : R -→ R is said to be a derivation on R if d(xy) = d(x)y + xd(y) holds for all x, y ∈ R. Let I be a nonzero ideal of R. Then an additive mapping d : I -→ R is called a derivation from I to R if d(xy) = d(x)y +
Let M be a unite prime *-algebra containing a non-trivial projection, and Φ:M→M be a nonlinear mixed Lie triple derivation. Then Φ is an additive *-derivation.
Let M be a unite prime *-algebra containing a non-trivial projection, and Φ:M→M be a nonlinear mixed Lie triple derivation. Then Φ is an additive *-derivation.
Let A be a unital *-algebra containing a non-trivial projection. In this paper, it is shown that a map Γ: A → A such that Γ(U°V•W)=Γ(U)°V•W+U°Γ(V)•W+U°V•Γ(W) for all U,V,W∈A. Then …
Let A be a unital *-algebra containing a non-trivial projection. In this paper, it is shown that a map Γ: A → A such that Γ(U°V•W)=Γ(U)°V•W+U°Γ(V)•W+U°V•Γ(W) for all U,V,W∈A. Then Γ is an additive *-derivation.
In the present paper, we prove some commutativity theorems for a prime ring with involution in which generalized derivations satisfy certain differential identities. Some well known results on commutativity of …
In the present paper, we prove some commutativity theorems for a prime ring with involution in which generalized derivations satisfy certain differential identities. Some well known results on commutativity of prime rings have been obtained. Also, we provide an example to show that the assumed restriction imposed on the involution is not superfluous.
Let 𝒜 be a prime ∗ -algebra and suppose that Φ preserves triple ∗ -Jordan derivation on 𝒜 , that is, for every A , B ∈ 𝒜 , Φ …
Let 𝒜 be a prime ∗ -algebra and suppose that Φ preserves triple ∗ -Jordan derivation on 𝒜 , that is, for every A , B ∈ 𝒜 , Φ ( A ◇ B ◇ C ) = Φ ( A ) ◇ B ◇ C + A ◇ Φ ( B ) ◇ C + A ◇ B ◇ Φ ( C ) , where A ◇ B = A B + B A ∗ . Then Φ is additive. Moreover, if Φ ( α I ) is selfadjoint for α ∈ { 1 , i } , then Φ is a ∗ -derivation.
Let R be a ring with involution.The skew Lie product of a, b ∈ R is defined by [a, b] = abba * .In the present paper we study prime …
Let R be a ring with involution.The skew Lie product of a, b ∈ R is defined by [a, b] = abba * .In the present paper we study prime ring with involution satisfying identities involving skew Lie product and left centralizers.
Abstract Let χ be a Banach space of dimension n > 1 and 𝔘 ⊂ 𝔅 ( χ ) be a standard operator algebra. In the present paper it is …
Abstract Let χ be a Banach space of dimension n > 1 and 𝔘 ⊂ 𝔅 ( χ ) be a standard operator algebra. In the present paper it is shown that if a mapping d : 𝔘 → 𝔘 (not necessarily linear) satisfies <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:mi>d</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mi>U</m:mi> <m:mo>,</m:mo> <m:mi>V</m:mi> </m:mrow> <m:mo>]</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi>W</m:mi> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mi>d</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>U</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi>V</m:mi> </m:mrow> <m:mo>]</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi>W</m:mi> </m:mrow> <m:mo>]</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mi>U</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>V</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi>W</m:mi> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:mrow> <m:mo>]</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mi>U</m:mi> <m:mo>,</m:mo> <m:mi>V</m:mi> </m:mrow> <m:mo>]</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi>d</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>W</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:mrow> </m:math> d\left( {\left[ {\left[ {U,V} \right],W} \right]} \right) = \left[ {\left[ {d\left( U \right),V} \right],W} \right] + \left[ {\left[ {U,d\left( V \right),W} \right]} \right] + \left[ {\left[ {U,V} \right],d\left( W \right)} \right] for all U, V, W ∈ 𝔘, then d = ψ + τ , where ψ is an additive derivation of 𝔘 and τ : 𝔘 → 𝔽 I vanishes at second commutator [[ U, V ], W ] for all U, V, W ∈ 𝔘. Moreover, if d is linear and satisfies the above relation, then there exists an operator S ∈ 𝔘 and a linear mapping τ from 𝔘 into 𝔽 I satisfying τ ([[ U, V ], W ]) = 0 for all U, V, W ∈ 𝔘, such that d ( U ) = SU − US + τ ( U ) for all U ∈ 𝔘.
In a previous note on derivations [1] we determined the structure of a prime ring R which has a derivation d≠0 such that the values of d commute, that is, …
In a previous note on derivations [1] we determined the structure of a prime ring R which has a derivation d≠0 such that the values of d commute, that is, for which d ( x ) d ( y ) = d ( y ) d ( x ) for all x, y∈R . Perhaps even more natural might be the question: what elements in a prime ring commute with all the values of a non-zero derivation? We address ourselves to this question here, and settle it.
Let [Formula: see text] be an *-algebra with identity [Formula: see text] and [Formula: see text] and [Formula: see text] nontrivial symmetric idempotents in [Formula: see text]. In this paper …
Let [Formula: see text] be an *-algebra with identity [Formula: see text] and [Formula: see text] and [Formula: see text] nontrivial symmetric idempotents in [Formula: see text]. In this paper we study the characterization of nonlinear mixed *-Jordan-type derivations. In particular, if [Formula: see text] is a factor von Neumann algebra then every unital nonlinear mixed *-Jordan-type derivations are additive *-derivations.
Let ℛ be a ring with an involution ∗ and k a positive integer. The k-skew Lie product of a,b∈ℛ is defined by ∗[a,b]k=∗[a,∗[a,b]k−1]1, where ∗[a,b]0=b and ∗[a,b]1=ab−ba∗. In this …
Let ℛ be a ring with an involution ∗ and k a positive integer. The k-skew Lie product of a,b∈ℛ is defined by ∗[a,b]k=∗[a,∗[a,b]k−1]1, where ∗[a,b]0=b and ∗[a,b]1=ab−ba∗. In this paper, some useful properties of the k-skew Lie products on prime *-rings are given. Then, as an application of these results, k-skew commuting additive maps on prime *-rings are characterized.