In this note, we characterize $ b- $generalized derivations which are strong commutative preservi... more In this note, we characterize $ b- $generalized derivations which are strong commutative preserving (SCP) on $ \mathscr{R} $. Moreover, we also discuss and characterize $ b- $generalized derivations involving certain $ \ast- $differential/functional identities on rings possessing involution.
The purpose of this paper is to investigate the behavior of prime rings involving skew derivation... more The purpose of this paper is to investigate the behavior of prime rings involving skew derivations with m-potent commutators on Lie ideals. In addition, we provide an example that shows that we cannot expect the same conclusion in case of semiprime rings. Also, we prove some other related results and present some open problems.
Rendiconti del Circolo Matematico di Palermo Series 2
In this manuscript, we discuss the behaviour and nature of generalized derivation G on a (semi-) ... more In this manuscript, we discuss the behaviour and nature of generalized derivation G on a (semi-) prime ring R satisfying certain differential identities over I, a nonzero ideal of R. Moreover, we extend our purely ring theoretic result to a non-commutative Banach algebras and obtained some range inclusion results of continuous generalized derivations.
Let {\mathfrak{R}} be a ring with center {Z(\mathfrak{R})}. In this paper, we study the higher-or... more Let {\mathfrak{R}} be a ring with center {Z(\mathfrak{R})}. In this paper, we study the higher-order commutators with power central values on rings and algebras involving generalized derivations. Motivated by [A. Alahmadi, S. Ali, A. N. Khan and M. Salahuddin Khan, A characterization of generalized derivations on prime rings, Comm. Algebra 44 2016, 8, 3201–3210], we characterize generalized derivations and related maps that satisfy certain differential identities on prime rings. Precisely, we prove that if a prime ring of characteristic different from two admitting generalized derivation {\mathfrak{F}} such that {([\mathfrak{F}(s^{m})s^{n}+s^{n}\mathfrak{F}(s^{m}),s^{r}]_{k})^{l}\in Z(% \mathfrak{R})} for every {s\in\mathfrak{R}}, then either {\mathfrak{F}(s)=ps} for every {s\in\mathfrak{R}} or {\mathfrak{R}} satisfies {s_{4}} and {\mathfrak{F}(s)=sp} for every {s\in\mathfrak{R}} and {p\in\mathfrak{U}}, the Utumi quotient ring of {\mathfrak{R}}. As an application, we prove that any ...
Let R be a prime ring with U the Utumi quotient ring and Q be the Martindale quotient ring of R, ... more Let R be a prime ring with U the Utumi quotient ring and Q be the Martindale quotient ring of R, respectively. Let d be a derivation of R and m, n be fixed positive integers. In this paper, we study the case when one of the following holds: (i) d(x) •n d(y)=x •m y (ii) d(x) •m d(y)=(d(x • y)) n for all x, y in some appropriate subset of R. We also examine the case where R is a semiprime ring. Finally, as an application we apply our result to the continuous derivations on Banach algebras. 1. Introduction and preliminaries In all that follows, unless specifically stated otherwise, R will be an associative ring, Z(R) the center of R, Q its Martindale quotient ring and U its Utumi quotient ring. The center of U , denoted by C, is called the extended centroid of R (we refer the reader to [3, Chapter 2], for the definitions and related properties of these objects). For any x, y ∈ R, the symbol [x, y] and x • y stands for the commutator xy − yx and anti-commutator xy + yx, respectively. Given
Let {\mathscr{R}} be a prime ring, {\mathscr{Q}_{r}} the right Martindale quotient ring of {\math... more Let {\mathscr{R}} be a prime ring, {\mathscr{Q}_{r}} the right Martindale quotient ring of {\mathscr{R}} and {\mathscr{C}} the extended centroid of {\mathscr{R}}. In this paper, we discuss the relationship between the structure of prime rings and the behavior of skew derivations on multilinear polynomials. More precisely, we investigate the m-potent commutators of skew derivations involving multilinear polynomials, i.e.,\big{(}[\delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})]\big{)}^{m}=[% \delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})],where {1
A map δ : S ⊆ A → A is called a strong commutativity preserving (SCP) on S if [δ(x), δ(y)] = [x, ... more A map δ : S ⊆ A → A is called a strong commutativity preserving (SCP) on S if [δ(x), δ(y)] = [x, y] for all x, y ∈ S, where A is a Banach algebra over C or R and S is a nonempty subset of A. The main goal of this manuscript is to investigate the SCP skew derivations on Banach algebra A and we obtain action of skew derivations on rings and the structure of prime Banach algebras.
In the present paper, we prove that a prime ring R with center Z satisfies s4, the standard ident... more In the present paper, we prove that a prime ring R with center Z satisfies s4, the standard identity in four variables if R admits a non-identity automorphism σ such that (uσ,u]vσ+vσ[uσ,u])n∈Z for all u,v in some non-central Lie ideal L of R whenever either char(R)>n or char(R)=0, where n is a fixed positive integer.
Let R be a prime ring and set OEx; y 1 D OEx; y D xy yx for all x; y 2 R and inductively OEx; y k... more Let R be a prime ring and set OEx; y 1 D OEx; y D xy yx for all x; y 2 R and inductively OEx; y k D OEOEx; y k 1 ; y for k > 1. We apply the theory of generalized polynomial identities with automorphism and skew derivations to obtain the following result: Let R be a prime ring and I a nonzero ideal of R. Suppose that .ı; '/ is a skew derivation of R such that ı.OEx; y/ D OEx; y n for all x; y 2 I , then R is commutative.
In this article we study certain differential identities in prime and semiprime rings. In particu... more In this article we study certain differential identities in prime and semiprime rings. In particular, we prove if a prime rings satisfies certain differential identity on a nonzero ideal of a ring R, then R is commutative. Finally, as an application we obtain some range inclusion results of continuous generalized derivations on non-commutative Banach algebras.
Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive intege... more Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d ( r ○ s)(r ○ s) + ( r ○ s) d ( r ○ s)n - d ( r ○ s))m for all r, s ϵ I, then R is commutative. (ii) If (d ( r ○ s)( r ○ s) + ( r ○ s) d ( r ○ s)n - d (r ○ s))m ϵ Z(R) for all r, s ϵ I, then R satisfies s4, the standard identity in four variables. Moreover, we also examine the case when R is a semiprime ring.
Let $R$ be an $(n+1)!$-torsion free unital $\ast$-ring, $n$ a fixed positive integer. If $F,d:R\t... more Let $R$ be an $(n+1)!$-torsion free unital $\ast$-ring, $n$ a fixed positive integer. If $F,d:R\to R$ are two additive mappings satisfying $F(x^{n+1})=F(x)(\alpha(x^\ast ))^{n}+\beta (x)d(x)(\alpha(x^\ast ))^{n-1}+(\beta(x))^2d(x)(\alpha(x^\ast ))^{n-2}+\cdots+(\beta(x))^nd(x)$ for all $x\in R$, where $\alpha$ and $\beta$ be automorphisms of $R$, then $d$ is a Jordan $(\alpha,\beta)^{\ast}$-derivation and $F$ is a generalized Jordan $(\alpha,\beta)^{\ast}$-derivation on $R$.
Let R be a prime ring and set [x, y](1) = [x, y] = xy - yx for all x, y is an element of R and in... more Let R be a prime ring and set [x, y](1) = [x, y] = xy - yx for all x, y is an element of R and inductively [x, y](k) = [[x, y](k-1) , y] for k > 1. We apply the theory of generalized polynomial identities with automorphism and skew derivations to obtain the following result: Let R be a prime ring and I a nonzero ideal of R. Suppose that (delta, phi) is a skew derivation of R such that delta([x, y]) = [x, y](n) for all x, y is an element of I, then R is commutative.
Journal of the Egyptian Mathematical Society, 2015
Let R be a prime ring, extended centroid C, Utumi quotient ring U, and m; n P 1 are fixed positiv... more Let R be a prime ring, extended centroid C, Utumi quotient ring U, and m; n P 1 are fixed positive integers, F a generalized derivation associated with a nonzero derivation d of R. We study the case when one of the following holds: (i) FðxÞ m dðyÞ ¼ ðx yÞ n and (ii) ðFðxÞ dðyÞÞ m ¼ ðx yÞ n , for all x; y in some appropriate subset of R. We also examine the case where R is a semiprime ring.
Let R be a prime ring of charðRÞ -2; Z the center of R, and L a nonzero 9 Lie ideal of R. If R ad... more Let R be a prime ring of charðRÞ -2; Z the center of R, and L a nonzero 9 Lie ideal of R. If R admits a generalized derivation F associated with a derivation d 10 which acts as a homomorphism or as anti-homomorphism on L, then either d ¼ 0 or 11 L # Z. This result generalizes a theorem of Wang and You. 12 13 Homomorphisms and anti-homomorphisms 15 2010 MSC: 16N60; 16W25 16 17
In this note, we characterize $ b- $generalized derivations which are strong commutative preservi... more In this note, we characterize $ b- $generalized derivations which are strong commutative preserving (SCP) on $ \mathscr{R} $. Moreover, we also discuss and characterize $ b- $generalized derivations involving certain $ \ast- $differential/functional identities on rings possessing involution.
The purpose of this paper is to investigate the behavior of prime rings involving skew derivation... more The purpose of this paper is to investigate the behavior of prime rings involving skew derivations with m-potent commutators on Lie ideals. In addition, we provide an example that shows that we cannot expect the same conclusion in case of semiprime rings. Also, we prove some other related results and present some open problems.
Rendiconti del Circolo Matematico di Palermo Series 2
In this manuscript, we discuss the behaviour and nature of generalized derivation G on a (semi-) ... more In this manuscript, we discuss the behaviour and nature of generalized derivation G on a (semi-) prime ring R satisfying certain differential identities over I, a nonzero ideal of R. Moreover, we extend our purely ring theoretic result to a non-commutative Banach algebras and obtained some range inclusion results of continuous generalized derivations.
Let {\mathfrak{R}} be a ring with center {Z(\mathfrak{R})}. In this paper, we study the higher-or... more Let {\mathfrak{R}} be a ring with center {Z(\mathfrak{R})}. In this paper, we study the higher-order commutators with power central values on rings and algebras involving generalized derivations. Motivated by [A. Alahmadi, S. Ali, A. N. Khan and M. Salahuddin Khan, A characterization of generalized derivations on prime rings, Comm. Algebra 44 2016, 8, 3201–3210], we characterize generalized derivations and related maps that satisfy certain differential identities on prime rings. Precisely, we prove that if a prime ring of characteristic different from two admitting generalized derivation {\mathfrak{F}} such that {([\mathfrak{F}(s^{m})s^{n}+s^{n}\mathfrak{F}(s^{m}),s^{r}]_{k})^{l}\in Z(% \mathfrak{R})} for every {s\in\mathfrak{R}}, then either {\mathfrak{F}(s)=ps} for every {s\in\mathfrak{R}} or {\mathfrak{R}} satisfies {s_{4}} and {\mathfrak{F}(s)=sp} for every {s\in\mathfrak{R}} and {p\in\mathfrak{U}}, the Utumi quotient ring of {\mathfrak{R}}. As an application, we prove that any ...
Let R be a prime ring with U the Utumi quotient ring and Q be the Martindale quotient ring of R, ... more Let R be a prime ring with U the Utumi quotient ring and Q be the Martindale quotient ring of R, respectively. Let d be a derivation of R and m, n be fixed positive integers. In this paper, we study the case when one of the following holds: (i) d(x) •n d(y)=x •m y (ii) d(x) •m d(y)=(d(x • y)) n for all x, y in some appropriate subset of R. We also examine the case where R is a semiprime ring. Finally, as an application we apply our result to the continuous derivations on Banach algebras. 1. Introduction and preliminaries In all that follows, unless specifically stated otherwise, R will be an associative ring, Z(R) the center of R, Q its Martindale quotient ring and U its Utumi quotient ring. The center of U , denoted by C, is called the extended centroid of R (we refer the reader to [3, Chapter 2], for the definitions and related properties of these objects). For any x, y ∈ R, the symbol [x, y] and x • y stands for the commutator xy − yx and anti-commutator xy + yx, respectively. Given
Let {\mathscr{R}} be a prime ring, {\mathscr{Q}_{r}} the right Martindale quotient ring of {\math... more Let {\mathscr{R}} be a prime ring, {\mathscr{Q}_{r}} the right Martindale quotient ring of {\mathscr{R}} and {\mathscr{C}} the extended centroid of {\mathscr{R}}. In this paper, we discuss the relationship between the structure of prime rings and the behavior of skew derivations on multilinear polynomials. More precisely, we investigate the m-potent commutators of skew derivations involving multilinear polynomials, i.e.,\big{(}[\delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})]\big{)}^{m}=[% \delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})],where {1
A map δ : S ⊆ A → A is called a strong commutativity preserving (SCP) on S if [δ(x), δ(y)] = [x, ... more A map δ : S ⊆ A → A is called a strong commutativity preserving (SCP) on S if [δ(x), δ(y)] = [x, y] for all x, y ∈ S, where A is a Banach algebra over C or R and S is a nonempty subset of A. The main goal of this manuscript is to investigate the SCP skew derivations on Banach algebra A and we obtain action of skew derivations on rings and the structure of prime Banach algebras.
In the present paper, we prove that a prime ring R with center Z satisfies s4, the standard ident... more In the present paper, we prove that a prime ring R with center Z satisfies s4, the standard identity in four variables if R admits a non-identity automorphism σ such that (uσ,u]vσ+vσ[uσ,u])n∈Z for all u,v in some non-central Lie ideal L of R whenever either char(R)>n or char(R)=0, where n is a fixed positive integer.
Let R be a prime ring and set OEx; y 1 D OEx; y D xy yx for all x; y 2 R and inductively OEx; y k... more Let R be a prime ring and set OEx; y 1 D OEx; y D xy yx for all x; y 2 R and inductively OEx; y k D OEOEx; y k 1 ; y for k > 1. We apply the theory of generalized polynomial identities with automorphism and skew derivations to obtain the following result: Let R be a prime ring and I a nonzero ideal of R. Suppose that .ı; '/ is a skew derivation of R such that ı.OEx; y/ D OEx; y n for all x; y 2 I , then R is commutative.
In this article we study certain differential identities in prime and semiprime rings. In particu... more In this article we study certain differential identities in prime and semiprime rings. In particular, we prove if a prime rings satisfies certain differential identity on a nonzero ideal of a ring R, then R is commutative. Finally, as an application we obtain some range inclusion results of continuous generalized derivations on non-commutative Banach algebras.
Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive intege... more Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d ( r ○ s)(r ○ s) + ( r ○ s) d ( r ○ s)n - d ( r ○ s))m for all r, s ϵ I, then R is commutative. (ii) If (d ( r ○ s)( r ○ s) + ( r ○ s) d ( r ○ s)n - d (r ○ s))m ϵ Z(R) for all r, s ϵ I, then R satisfies s4, the standard identity in four variables. Moreover, we also examine the case when R is a semiprime ring.
Let $R$ be an $(n+1)!$-torsion free unital $\ast$-ring, $n$ a fixed positive integer. If $F,d:R\t... more Let $R$ be an $(n+1)!$-torsion free unital $\ast$-ring, $n$ a fixed positive integer. If $F,d:R\to R$ are two additive mappings satisfying $F(x^{n+1})=F(x)(\alpha(x^\ast ))^{n}+\beta (x)d(x)(\alpha(x^\ast ))^{n-1}+(\beta(x))^2d(x)(\alpha(x^\ast ))^{n-2}+\cdots+(\beta(x))^nd(x)$ for all $x\in R$, where $\alpha$ and $\beta$ be automorphisms of $R$, then $d$ is a Jordan $(\alpha,\beta)^{\ast}$-derivation and $F$ is a generalized Jordan $(\alpha,\beta)^{\ast}$-derivation on $R$.
Let R be a prime ring and set [x, y](1) = [x, y] = xy - yx for all x, y is an element of R and in... more Let R be a prime ring and set [x, y](1) = [x, y] = xy - yx for all x, y is an element of R and inductively [x, y](k) = [[x, y](k-1) , y] for k > 1. We apply the theory of generalized polynomial identities with automorphism and skew derivations to obtain the following result: Let R be a prime ring and I a nonzero ideal of R. Suppose that (delta, phi) is a skew derivation of R such that delta([x, y]) = [x, y](n) for all x, y is an element of I, then R is commutative.
Journal of the Egyptian Mathematical Society, 2015
Let R be a prime ring, extended centroid C, Utumi quotient ring U, and m; n P 1 are fixed positiv... more Let R be a prime ring, extended centroid C, Utumi quotient ring U, and m; n P 1 are fixed positive integers, F a generalized derivation associated with a nonzero derivation d of R. We study the case when one of the following holds: (i) FðxÞ m dðyÞ ¼ ðx yÞ n and (ii) ðFðxÞ dðyÞÞ m ¼ ðx yÞ n , for all x; y in some appropriate subset of R. We also examine the case where R is a semiprime ring.
Let R be a prime ring of charðRÞ -2; Z the center of R, and L a nonzero 9 Lie ideal of R. If R ad... more Let R be a prime ring of charðRÞ -2; Z the center of R, and L a nonzero 9 Lie ideal of R. If R admits a generalized derivation F associated with a derivation d 10 which acts as a homomorphism or as anti-homomorphism on L, then either d ¼ 0 or 11 L # Z. This result generalizes a theorem of Wang and You. 12 13 Homomorphisms and anti-homomorphisms 15 2010 MSC: 16N60; 16W25 16 17
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Papers by Mohd Arif Raza