Narrow Results

In this note, derivations on Morita rings (generalized matrix rings) are introduced and generalized derivations on rings are proved to be Morita invariant without involvement of any homology theory.

Let R be a ring with center Z(R) and n be a fixed positive integer. A mapping f : R → R is said to be n-centralizing on a subset S of R if f(x)xⁿ – xⁿ f(x) ∈ Z(R) holds for all x ∈ S. The main result of this paper states that every n-centralizing generalized derivation F on a (n + 1)!-torsion free semiprime ring is n-commuting. Further, we prove that if a generalized derivation F : R → R is n-centralizing on a nonzero left ideal λ, then either R contains a nonzero central ideal or λD(Z) ⊆ Z(R) for some derivation D of R. As an application, n-centralizing generalized derivations of C*-algebras are characterized.

Let $R$ be a prime ring with the extended centroid $C$, $L$ a noncommutative Lie ideal of $R$ and $g$ a nonzero $b$-generalized derivation of $R$. For $x,y\in R$, let $[x,y]=xy-yx$. We prove that if $[[\cdots [[g(x^{n_0}),x^{n_1}],x^{n_2}],\dots ],x^{n_k}]=0$ for all $x\in L$, where $n_0,n_1,\dots ,n_k$ are fixed positive integers, then there exists $\lambda \in C$ such that $g(x)=\lambda x$ for all $x\in R$ except when $R\subseteq M_2(F)$, the $2\times 2$ matrix ring over a field $F$. The analogous result for generalized skew derivations is also described. Our theorems naturally generalize the cases of derivations and skew derivations obtained by Lanski in [C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc.118 (1993), 75–80, Skew derivations and Engel conditions, Comm. Algebra42 (2014), 139–152.]

The aim of this paper is to show that certain subsets, which are defined by commutativity conditions involving derivations, generalized derivations and epimorphisms, coincide with the center in prime or semiprime rings.

In this paper, we investigate commutativity of prime rings $R$ with involution ∗ of the second kind in which generalized derivations satisfy certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Furthermore, we provide an example to show that the restriction imposed on the involution is not superfluous.

Let ${ℛ}$ be a prime ring with $\mathrm{\text{char}}({ℛ})\ne 2$, ${ℒ}$ a non-central Lie ideal of ${ℛ}$, ${𝒬}$ its Martindale quotient ring and ${𝒞}$ its extended centroid. Let $F:{ℛ}\to {ℛ}$ and $G:{ℛ}\to {ℛ}$ be nonzero generalized derivations on ${ℛ}$ such that

Let $R$ be a prime ring with center $Z(R)$, let $L$ be a nonzero left ideal of $R$ and let $g,h$ be two generalized derivations of $R$. In this paper, we completely characterize the structure of $L$ and all possible forms of $g,h$ such that $g(x^m)x^n-x^{n}h(x^m)\in Z(R)$ for all $x\in L$, where $m,n$ are fixed positive integers. With this, several known results can be either deduced or generalized. Moreover, our result can be regarded as the one-sided ideal version of the theorem obtained by Lee and Zhou in [An identity with generalized derivations, J. Algebra Appl. 8 (2009) 307–317].

Let R be a prime algebra over a commutative ring K, Z and C the center and extended centroid of R, respectively, g a generalized derivation of R, and f (X₁, …,X_t) a multilinear polynomial over K. If g(f (X₁, …,X_t))ⁿ ∈ Z for all x₁, …, x_t ∈ R, then either there exists an element λ ∈ C such that g(x)= λx for all x ∈ R or f(x₁, …,x_t) is central-valued on R except when R satisfies s₄, the standard identity in four variables.

Let R be a prime ring with a nontrivial idempotent. In this paper, we prove that if g is an additive map of R into itself such that xg(y)z = 0 for all x, y, z ∈ R with xy = yz = 0, then g is a generalized derivation. As an application of this result, we show that every local generalized derivation in a prime ring with a nontrivial idempotent is a generalized derivation.

Let R be a prime ring with extended centroid C, maximal right ring of quotients U, a nonzero ideal I and a generalized derivation δ. Suppose δ(x)ⁿ =(ax)ⁿ for all x ∈ I, where a ∈ U and n is a fixed positive integer. Then δ(x)=λax for some λ ∈ C. We also prove two generalized versions by replacing I with a nonzero left ideal and a noncommutative Lie ideal L, respectively.

Let R be a prime ring of characteristic different from 2, L a noncentral Lie ideal of R, H and G two nonzero generalized derivations of R. Suppose u^s(H(u)u-uG(u)) u^t=0 for all u ∈ L, where s, t ≥ 0 are fixed integers. Then either (i) there exists p ∈ U such that H(x)=xp for all x ∈ R and G(x)=px for all x ∈ R unless R satisfies S₄, the standard identity in four variables; or (ii) R satisfies S₄ and there exist p, q ∈ U such that H(x)=px+xq for all x ∈ R and G(x)=qx+xp for all x ∈ R.

Let R be a prime ring of characteristic different from 2 with right Utumi quotient ring U and extended centroid C. Let g be a generalized derivation of R, f(x₁,…,x_n) a multilinear polynomial over C, a ∈ R, and I a nonzero right ideal of R. Suppose that a[g(f(r₁,…,r_n)), f(r₁,…,r_n)]=0 for all r_i∈ I and aI ≠ 0. Then either g(x)=a₁x with (a₁-γ)I=0 for some a₁∈ U and γ ∈ C, or there exists an idempotent element e ∈soc(RC) such that IC=eRC and one of the following holds: (i) f(x₁,…,x_n) is central-valued in eRe; (ii) g(x)=bx+xc, where b, c ∈ U with (c-b-α)e=0 for some α ∈ C and f(x₁,…,x_n) is central-valued in eRe.

The main purpose of this paper is to study generalized derivations in rings with involution which behave like strong commutativity preserving mappings. In fact, we prove the following result: Let R be a noncommutative prime ring with involution of the second kind such that char $\left(R\right)\ne 2$. If R admits a generalized derivation $F:R\to R$ associated with a derivation $d:R\to R$ such that $\left[F\right(x),F(x^{*}\left)\right]-[x,x^{*}]=0$ for all $x\in R$, then $F\left(x\right)=x$ for all $x\in R$ or $F\left(x\right)=-x$ for all $x\in R$. Moreover, a related result is also obtained.

Let R be a prime ring of characteristic different from 2, Q be its maximal right ring of quotients, and C be its extended centroid. Suppose that $f(x_1,\dots ,x_n)$ is a non-central multilinear polynomial over C, $0\ne p\in R$, and F, G are two b-generalized derivations of R. In this paper we describe all possible forms of F and G in the case $pG(F(f(r))f(r))=0$ for all $r=(r_1,\dots ,r_n)$ in Rⁿ.

In this paper we investigate commutativity of prime rings with involution ∗ of the second kind in which endomorphisms satisfy certain algebraic identities. Furthermore, we provide examples to show that the various restrictions imposed by the hypotheses of our theorems are not superfluous.

The hermitian part of the Banach-Lie *-algebra of multiplication operators on the W*-algebra A is a unital GM-space, the base of the dual cone in the dual GL-space of which is affine isomorphic and weak*-homeomorphic to the state space of . It is shown that there exists a Lie *-isomorphism ϕ from the quotient (A ⊕_∞ A^op)/K of an enveloping W*-algebra A ⊕_∞ A^op of A by a weak*-closed Lie *-ideal K onto , the restriction to the hermitian part ((A ⊕_∞ A^op)/K)_h of which is a bi-positive real linear isometry, thereby giving a characterization of the state space of . In the special case in which A is a W*-factor this leads to a further identification of the state space of in terms of the state space of A. For any W*-algebra A, the Banach-Lie *-algebra coincides with the set of generalized derivations of A, and, as an application, a formula is obtained for the norm of an element of in terms of a centre-valued 'norm' on A, which is similar to that previously obtained by non-order-theoretic methods.

Let $R$ be a $2$-torsion free prime ring, and $U$ a square closed Lie ideal of $R.$ Further let $G,$ and $F$ be generalized derivations associated with derivations $g$ and $d$, respectively on $R.$ If one of the following conditions holds: (i) $G(uv)+d(u)F(v)+uv\in Z(R),$ (ii) $G(uv)+d(u)F(v)+vu\in Z(R),$ (iii) $G(uv)+d(u)F(v)+uv\in Z(R),$ (iv) $G(vu)+F(u)F(v)+uv\in Z(R),$ (v) $G(uv)+d(v)F(u)+uv\in Z(R),$ for all $u,v\in U,$ then it is proved that either $g=d=0$ or $U\subseteq R.$

Let $A$ be a unital prime Banach algebra over complex field $ℂ$ with unity and $g$ be a nonzero continuous linear generalized derivation associated with a nonzero continuous linear derivation $d$. In this paper, we investigate the commutativity of $A$. In particular, we prove that a unital prime Banach algebra $A$ is commutative if one of the following holds; (i) either $g({(xy)}^n)+\mu d(x^n)d(y^n)+\nu [x^n,y^n]=0$ or $g({(xy)}^n)+\mu d(y^n)d(x^n)+\nu [y^n,x^n]=0$, (ii) either $g({(xy)}^n)+\mu [d(x^n),d(y^n)]+\nu [x^n,y^n]=0$ or $g({(xy)}^n)+\mu [d(y^n),d(x^n)]+\nu [y^n,x^n]=0$, for sufficiently many $x,y$, for any complex numbers $\mu ,\nu$ and an integer $n=n(x,y)>1$.

Let $R$ be a noncommutative prime ring with its Utumi ring of quotients $U$, $C=Z(U)$ the extended centroid of $R$, $G$ a generalized derivation of $R$ and $I$ a nonzero ideal of $R$. If $I$ satisfies any one of the following conditions:

• ⁽ⁱ⁾$[[G({[x,y]}_n)$, ${[x,y]}_n]$, $G({[x,y]}_n)]\in C$,
• ⁽ⁱⁱ⁾$(G(x{\circ }_{n}y)\circ (x{\circ }_{n}y))\circ G(x{\circ }_{n}y)\in C$,

where $n\ge 1$ is a fixed integer, then one of the following holds:

• ⁽¹⁾there exists $\lambda \in C$ such that $G(x)=\lambda x$ for all $x\in R$;
• ⁽²⁾$R$ satisfies $s_4$ and there exist $a\in U$ and $\lambda \in C$ such that $G(x)=ax+xa+\lambda x$ for all $x\in R$;
• ⁽³⁾char $(R)=2$, $R$ satisfies $s_4$ and there exist $\lambda \in C$ and an outer derivation $d$ of $R$ such that $G(x)=\lambda x+d(x)$ for all $x\in R$.

Our purpose in this paper is to investigate commutativity of a ring with involution $(R,\ast )$ which admits a generalized derivation satisfying certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Moreover, we provide examples to show that the assumed restrictions cannot be relaxed.