Abu Zaid Ansari

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SYMMETRIC SKEW 4-DERIVATIONS ON PRIME RINGS

2014-08-26

For a ring R with an automorphism α a 4-additive mapping D : R4−→ R is called a skew 4-derivation w.r.t. α if it is a α-derivation of R for … For a ring R with an automorphism α a 4-additive mapping D : R4−→ R is called a skew 4-derivation w.r.t. α if it is a α-derivation of R for each argument. Namely it is always an α-derivation of R for the argument being left once (3) arguments are fixed by (3) elements in R. In the present note, begin with a result of Jung and Park [5], we prove that if a skew 4-derivation D associated with an automorphism α with trace f of a noncommutative prime ring R under suitable torsion condition satisfying [f(x),α(x)] = 0 for all x ∈ I, a nonzero ideal of R, then D = 0.

Lie Ideals and Generalized Derivations in Semiprime Rings

2015-10-10

Vincenzo De Filippis , Nadeem ur Rehman , Abu Zaid Ansari

Let R be a 2-torsion free ring and L a Lie ideal of R. An additive mapping F : R! R is called a generalized derivation on R if there … Let R be a 2-torsion free ring and L a Lie ideal of R. An additive mapping F : R! R is called a generalized derivation on R if there exists a derivation d : R! R such that F (xy) = F (x)y+xd(y) holds for all x;y2 R. In the present paper we describe the action of generalized derivations satisfying several conditions on Lie ideals of semiprime rings.

Generalized Derivations on Lie Ideals and Power Values on Prime Rings

2015-10-01

Giovanni Scudo , Abu Zaid Ansari

Generalized Derivations on Power Values of Lie Ideals in Prime and Semiprime Rings

2014-01-01

Vincenzo De Filippis , Nadeem ur Rehman , Abu Zaid Ansari

Let<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>R</mml:mi></mml:mrow></mml:math>be a 2-torsion free ring and let<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:math>be a noncentral Lie ideal of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi>R</mml:mi></mml:mrow></mml:math>, and let<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mi>F</mml:mi><mml:mo>:</mml:mo><mml:mi>R</mml:mi><mml:mo>→</mml:mo><mml:mi>R</mml:mi></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mi>G</mml:mi><mml:mo>:</mml:mo><mml:mi>R</mml:mi><mml:mo>→</mml:mo><mml:mi>R</mml:mi></mml:math>be two generalized derivations of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mrow><mml:mi>R</mml:mi></mml:mrow></mml:math>. … Let<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>R</mml:mi></mml:mrow></mml:math>be a 2-torsion free ring and let<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:math>be a noncentral Lie ideal of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mi>R</mml:mi></mml:mrow></mml:math>, and let<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mi>F</mml:mi><mml:mo>:</mml:mo><mml:mi>R</mml:mi><mml:mo>→</mml:mo><mml:mi>R</mml:mi></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mi>G</mml:mi><mml:mo>:</mml:mo><mml:mi>R</mml:mi><mml:mo>→</mml:mo><mml:mi>R</mml:mi></mml:math>be two generalized derivations of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mrow><mml:mi>R</mml:mi></mml:mrow></mml:math>. We will analyse the structure of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mrow><mml:mi>R</mml:mi></mml:mrow></mml:math>in the following cases: (a)<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:mrow><mml:mi>R</mml:mi></mml:mrow></mml:math>is prime and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M9"><mml:mi>F</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>for all<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M10"><mml:mi>u</mml:mi><mml:mo>∈</mml:mo><mml:mi>L</mml:mi></mml:math>and fixed positive integers<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M11"><mml:mi>m</mml:mi><mml:mo>≠</mml:mo><mml:mi>n</mml:mi></mml:math>; (b)<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M12"><mml:mrow><mml:mi>R</mml:mi></mml:mrow></mml:math>is prime and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M13"><mml:mi>F</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>m</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>for all<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M14"><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo>∈</mml:mo><mml:mi>L</mml:mi></mml:math>and fixed integers<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M15"><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>,</mml:mo><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>s</mml:mi><mml:mo>≥</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:math>; (c)<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M16"><mml:mrow><mml:mi>R</mml:mi></mml:mrow></mml:math>is semiprime and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M17"><mml:mi>F</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mi>v</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mi>u</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>for all<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M18"><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mi>R</mml:mi><mml:mo>,</mml:mo><mml:mi>R</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:math>and fixed integer<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M19"><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:math>; and (d)<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M20"><mml:mrow><mml:mi>R</mml:mi></mml:mrow></mml:math>is semiprime and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M21"><mml:mi>F</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mi>v</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>v</mml:mi><mml:mi>u</mml:mi><mml:msup><mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>for all<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M22"><mml:mi>u</mml:mi><mml:mo>,</mml:mo><mml:mi>v</mml:mi><mml:mo>∈</mml:mo><mml:mi>R</mml:mi></mml:math>and fixed integer<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M23"><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:math>.

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On Lie ideals of $$*$$ ∗ -prime rings with generalized derivations

2014-07-31

Nadeem ur Rehman , Radwan M. Al-omary , Abu Zaid Ansari

Generalized left derivations acting as homomorphisms and anti-homomorphisms on Lie ideal of rings

2014-02-01

Nadeem ur Rehman , Abu Zaid Ansari

Let R be a prime ring with characteristic different from 2 and L be a Lie ideal of R. In this paper, we characterize generalized left derivation, which acts as … Let R be a prime ring with characteristic different from 2 and L be a Lie ideal of R. In this paper, we characterize generalized left derivation, which acts as a homomorphisms or an anti-homomorphisms on L.

On Identities with Additive Mappings in Rings

2020-04-01

Abu Zaid Ansari

If F, D : R → R are additive mappings which satisfySimilar type of result has been done for the other identity forcing to generalized derivation and at last an … If F, D : R → R are additive mappings which satisfySimilar type of result has been done for the other identity forcing to generalized derivation and at last an example has given in support of the theorems.

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Spacelike Surfaces with a Common Line of Curvature in Lorentz-Minkowski 3-Space

2021-05-04

M. Khalifa Saad , Abu Zaid Ansari , Muhammad Nadeem Akram +1 more

This paper aims to study spacelike surfaces from a given spacelike curve in Minkowski 3–space. Also, we investigate the necessary and sufficient conditions for the given space-like curve to be … This paper aims to study spacelike surfaces from a given spacelike curve in Minkowski 3–space. Also, we investigate the necessary and sufficient conditions for the given space-like curve to be the line of curvature on the space-like surface. Depending on the causal character of the curve, the necessary and sufficient conditions for the given space-like curve to satisfy the line of curvature and the geodesic (resp. asymptotic) requirements are also analyzed. Furthermore, we give with illustration some computational examples in support of our main results.

Jordan *-derivations on standard operator algebras

2023-01-01

Abu Zaid Ansari , Faiza Shujat

LetH be a real or complex Hilbert space with dim(H) > 1, B(H) be algebra of all bounded linear operators on H and A(H) ? B(H) be a standard operator … LetH be a real or complex Hilbert space with dim(H) > 1, B(H) be algebra of all bounded linear operators on H and A(H) ? B(H) be a standard operator algebra on H. If D : A(H) ? B(H) is a linear mapping satisfying D(An+1) = Pn i=0 AiD(A)(A*)n?i for all A ? A(H), then D is a Jordan *-derivation on A(H). Later, we discuss some algebraic identities on semiprime rings.

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Additive mappings satisfying algebraic conditions in rings

2014-04-22

Abu Zaid Ansari , Faiza Shujat

On generalized Jordan ∗-derivation in rings

2013-06-14

Nadeem ur Rehman , Abu Zaid Ansari , Tarannum Bano

Let n ⩾ 1 be a fixed integer and let R be an (n + 1)!-torsion free ∗-ring with identity element e. If F, d:R → R are two additive … Let n ⩾ 1 be a fixed integer and let R be an (n + 1)!-torsion free ∗-ring with identity element e. If F, d:R → R are two additive mappings satisfying F(xn+1) = F(x)(x∗)n + xd(x)(x∗)n−1 + x2d(x)(x∗)n−2+ ⋯ +xnd(x) for all x ∈ R, then d is a Jordan ∗-derivation and F is a generalized Jordan ∗-derivation on R.

An extension of Herstein's theorem on Banach algebra

2024-01-01

Abu Zaid Ansari , Suad Alrehaili , Faiza Shujat

<abstract>Let $ \mathcal{A} $ be a $ (p+q)! $-torsion free semiprime ring. We proved that if $ \mathcal{H}, \mathcal{D} : \mathcal{A}\to \mathcal{A} $ are two additive mappings fulfilling the algebraic … <abstract>Let $ \mathcal{A} $ be a $ (p+q)! $-torsion free semiprime ring. We proved that if $ \mathcal{H}, \mathcal{D} : \mathcal{A}\to \mathcal{A} $ are two additive mappings fulfilling the algebraic identity $ 2\mathcal{H}(a^{p+q}) = \mathcal{H}(a^p) a^q+ a^p \mathcal{D}(a^q)+\mathcal{H}(a^q) a^p+ a^q \mathcal{D}(a^p) $ for all $ a\in \mathcal{A} $, then $ \mathcal{H} $ is a generalized derivation with $ \mathcal{D} $ as an associated derivation on $ \mathcal{A} $. In addition to that, it is also proved in this article that $ \mathcal{H}_1 $ is a generalized left derivation associated with a left derivation $ \delta $ on $ \mathcal{A} $ if they fulfilled the algebraic identity $ 2\mathcal{H}_1(a^{p+q}) = a^p \mathcal{H}_1(a^q)+ a^q \delta(a^p)+a^q \mathcal{H}_1(a^p)+ a^p \delta(a^q) $ for all $ a \in \mathcal{A} $. Further, the legitimacy of these hypotheses is eventually demonstrated by examples and at last, an application of Banach algebra is presented.</abstract>

A Note on Homomorphisms and Anti-Homomorphisms on ∗-Ring

2013-03-22

Nadeem ur Rehman , Abu Zaid Ansari , Claus Haetinger

In this paper we describe generalized left $\ast$-derivation $F:R\to R$ in $\ast$-prime ring and prove that if $F$ acts as homomorphism or anti-homomorphism on $R$, then either $R$ is commutative … In this paper we describe generalized left $\ast$-derivation $F:R\to R$ in $\ast$-prime ring and prove that if $F$ acts as homomorphism or anti-homomorphism on $R$, then either $R$ is commutative or $F$ is a right $\ast$-centralizer on $R$. Analogous results have been proved for generalized left $\ast$-biderivation and Jordan $\ast$-centralizer on $R$.

On Lie Ideals and Generalized Jordan Left Derivations of Prime Rings

2014-01-01

Nadeem ur Rehman , Abu Zaid Ansari

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Strong commutativity preserving biderivations on prime rings

2017-02-26

Faiza Shujat , Abu Zaid Ansari , Shahoor Khan

In the present paper we investigate the commutativity in a prime ring R which admits biderivation D : R×R → R satisfying [D(x,x),D(y,y)] = [x,y] for all x,y ∈ R. … In the present paper we investigate the commutativity in a prime ring R which admits biderivation D : R×R → R satisfying [D(x,x),D(y,y)] = [x,y] for all x,y ∈ R. More precisely, we generalize the result of Bell et.al.[3] on strong commutativity preserving biderivations. Moreover, we obtain that generalized biderivation acts as left bimultiplier, whenever it behaves as right R-homomorphisms.

Identities on additive mappings in semiprime rings

2023-01-16

Abu Zaid Ansari , Nadeem ur Rehman

Consider a ring $R$, which is semiprime and also having $k$-torsion freeness. If $F, d : R\to R$ are two additive maps fulfilling the algebraic identity $$F(x^{n+m})=F(x^m) x^n+ x^m d(x^n)$$ … Consider a ring $R$, which is semiprime and also having $k$-torsion freeness. If $F, d : R\to R$ are two additive maps fulfilling the algebraic identity $$F(x^{n+m})=F(x^m) x^n+ x^m d(x^n)$$ for each $x$ in $R.$ Then $F$ will be a generalized derivation having $d$ as an associated derivation on $R$. On the other hand, in this article, it is also derived that $f$ is a generalized left derivation having a linked left derivation $\delta$ on $R$ if they satisfy the algebraic identity $$f(x^{n+m})=x^n f(x^m)+ x^m \delta(x^n)$$ for each $x$ in $R$ and $k\in \{2, m, n, (n+m-1)!\}$ and at last an application on Banach algebra is presented.

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Generalized differential identities on prime rings and algebras

2023-01-01

Abu Zaid Ansari , Faiza Shujat , Ahlam Fallatah

<abstract>The goal of this study is to bring out the following conclusion. Let $ R $ be a noncommutative prime ring with $ 2(m+n)! $ torsion freeness and let $ … <abstract>The goal of this study is to bring out the following conclusion. Let $ R $ be a noncommutative prime ring with $ 2(m+n)! $ torsion freeness and let $ m $ and $ n $ be fixed, non-negative integers and $ d, g $ be Jordan derivations on $ R $. If $ x^{m+n}d(x)+x^mg(x)x^n\in Z(R) $ or $ d(x)x^{m+n}+x^mg(x)x^n\in Z(R) $ or $ x^{n}d(x)x^{m}+x^mg(x)x^n\in Z(R) $ then $ d = g = 0 $ follows for every $ x\in R $.</abstract>

On Additive Mappings in a ∗-Ring with an Identity Element

2015-08-13

Nadeem ur Rehman , Abu Zaid Ansari

On identities with derivations in rings

2012-01-01

Abu Zaid Ansari

Additive mappings act as a generalized left $$(\alpha , \beta )$$ ( α , β ) -derivation in rings

2018-07-18

Faiza Shujat , Abu Zaid Ansari , F. Salama

A NOTE ON GENERALIZED JORDAN m-DERIVATIONS IN RINGS

2021-01-20

Faiza Shujat , Abu Zaid Ansari

COMMUTING SYMMETRIC BI-SEMIDERIVATIONS ON RINGS

2021-09-22

Faiza Shujat , Abu Zaid Ansari , K. Kiran Kumar

Additive mappings covering generalized (alpha_1, alpha_2)-derivations in semiprime rings

2021-09-12

Faiza Shujat , Abu Zaid Ansari

Let R be a semiprime ring, (α1,α2) be automorphisms on R and Δ, δ be additive maps from R to R. If Δ and δ satisfying any one of the … Let R be a semiprime ring, (α1,α2) be automorphisms on R and Δ, δ be additive maps from R to R. If Δ and δ satisfying any one of the following identities:i) Δ(xnyn)=Δ(xn)α1(yn)+α2(xn)δ(yn)ii) Δ(xn)=Δ(xn-m)α1(xm)+α2(xn-m)δ(xm), for all x,y∈Rthen Δ will be generalized (α1,α2)-derivation with associated (α1,α2)-derivation on R.

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Additive mappings satisfying algebraic identities in semiprime rings

2023-01-01

Abu Zaid Ansari

Let R be a k-torsion free semiprime ring.Suppose that F, d : R → R be two additive mappings which satisfy the algebraic identitywhere α and β are automorphisms on … Let R be a k-torsion free semiprime ring.Suppose that F, d : R → R be two additive mappings which satisfy the algebraic identitywhere α and β are automorphisms on R. Then F is a generalized (α, β)-derivation with associated (α, β)derivation d on R, where k ∈ {2, n, 2n -1}.On the other hand, it is proved that f is a generalized Jordan left (α, β)-derivation associated with Jordan left (α, β)-derivation δ on R if they satisfy the algebraic identity f) for all x ∈ R together with some restrictions on R.

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Characterization of additive mappings on semiprime rings

2023-10-17

Abu Zaid Ansari

Classification of additive mappings on certain rings and algebras

2023-11-14

Abu Zaid Ansari

Abstract The objective of this research is to prove that an additive mapping $$\Delta :{\mathcal {A}}\rightarrow {\mathcal {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mo>:</mml:mo> <mml:mi>A</mml:mi> <mml:mo>→</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> will be … Abstract The objective of this research is to prove that an additive mapping $$\Delta :{\mathcal {A}}\rightarrow {\mathcal {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mo>:</mml:mo> <mml:mi>A</mml:mi> <mml:mo>→</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> will be a generalized derivation associated with a derivation $$\partial :{\mathcal {A}}\rightarrow {\mathcal {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mo>:</mml:mo> <mml:mi>A</mml:mi> <mml:mo>→</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> if it satisfies the following identity $$\Delta (r^{m+n+p})=\Delta (r^m)r^{n+p}+r^m\partial (r^{n})r^p+r^{m+n}\partial (r^p)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>r</mml:mi> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>Δ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>r</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:msup> <mml:mi>r</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>r</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:mi>∂</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>r</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:msup> <mml:mi>r</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>r</mml:mi> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mi>∂</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>r</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for all $$r\in {\mathcal {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> , where $$m, n\ge 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and $$p\ge 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> are fixed integers and $${\mathcal {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> is a semiprime ring. Another analogous has been done where an additive mapping behaves like a generalized left derivation associated with a left derivation on $${\mathcal {A}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> satisfying certain algebraic identity. The proofs of these advancements are derived employing algebraic concepts. These theorems have been validated by offering an example that shows they are not insignificant. Furthermore, we provide an application in the framework of Banach algebra.

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Results on Generalized Skew Bi-Semiderivation in Prime Rings

2024-01-01

Faiza Shujat , Abu Zaid Ansari

Algebraic Identities on Prime and Semiprime Rings

2024-01-01

Abu Zaid Ansari , Faiza Shujat

A characterization of σ-prime rings involving generalized derivations

2024-06-24

Md Arshad Madni , Muzibur Rahman Mozumder , Wasim Ahmed +1 more

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

Two Iterates of Symmetric Generalized Skew 3-Derivation

2024-10-21

Abu Zaid Ansari , Suad Alrehaili , Faiza Shujat

Our goal in this study is to validate the following finding: Assume that a prime ring R having , D is a symmetric skew 3-derivation on R with automorphisms α. … Our goal in this study is to validate the following finding: Assume that a prime ring R having , D is a symmetric skew 3-derivation on R with automorphisms α. If ∇1, ∇2 : R3 → R are symmetric generalized skew 3-derivations with α and associated skew 3-derivations D1, D2 respectively such that for every , then either ∇1 = 0 or ∇2 = 0 on R, where ∂1 and ∂2 stands for the traces of ∇1 and ∇2 respectively.

Structure of (σ, ρ)-n-Derivations on Rings

2024-11-29

Abu Zaid Ansari , Faiza Shujat , Ahlam Fallatah

The goal of this research is to describe the structure of Jordan (σ, ρ)-n-derivations on a prime ring. By (σ, ρ)- n-derivations, we mean n-additive maps ℑ : Rn →R … The goal of this research is to describe the structure of Jordan (σ, ρ)-n-derivations on a prime ring. By (σ, ρ)- n-derivations, we mean n-additive maps ℑ : Rn →R satisfying the following property in each n-slot: ℑ(pq, ϖ1, ··· , ϖn−1) = ℑ(p, ϖ1, ··· , ϖn−1)σ(q) +ρ(p)ℑ(q, ϖ1, ··· , ϖn−1), for every p, q, ϖ1, ··· , ϖn−1∈ R. We find the conditions under which every Jordan (σ, ρ)-n-derivation becomes a (σ, ρ)-n-derivation. Moreover, the concept of ∗-n-centralizers on ∗-ring has given. The ∗-ring is also used for examining some outcomes, where left and right ∗-n-centralizers are significant

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Action of Left Identity on n-Skew Commuting 3-Additive Mappings

2024-11-29

Suad Alrehaili , Faiza Shujat , Abu Zaid Ansari

Jordan $\varphi$-centralizers on Semiprime and Involution Rings

2025-01-31

Abu Zaid Ansari , Faiza Shujat , Alwaleed Kamel +1 more

The intention of the current investigation is to demonstrate that if an additive mapping $\mathcal{H}:R\to R$ fulfills any one of the following identities:\begin{enumerate}\item [$(i)$] $3\mathcal{H}(r^{3p})=\mathcal{H}(r^p)\varphi(r^{2p})+\varphi(r^p) \mathcal{H}(r^{p})\varphi(r^p)+ \varphi(r^{2p})\mathcal{H}(r^p)$\item [$(ii)$] $2\mathcal{H}(r^{2p})=\mathcal{H}(r^p)\varphi(r^{p})+\varphi(r^p)\mathcal{H}(r^{p})$\item [$(ii)$] … The intention of the current investigation is to demonstrate that if an additive mapping $\mathcal{H}:R\to R$ fulfills any one of the following identities:\begin{enumerate}\item [$(i)$] $3\mathcal{H}(r^{3p})=\mathcal{H}(r^p)\varphi(r^{2p})+\varphi(r^p) \mathcal{H}(r^{p})\varphi(r^p)+ \varphi(r^{2p})\mathcal{H}(r^p)$\item [$(ii)$] $2\mathcal{H}(r^{2p})=\mathcal{H}(r^p)\varphi(r^{p})+\varphi(r^p)\mathcal{H}(r^{p})$\item [$(ii)$] $\mathcal{H}(r^{3p})=\varphi(r^p) \mathcal{H}(r^{p}) \varphi(r^p)$ for all $r\in R$,\end{enumerate}then $\mathcal{H}$ is a $\varphi$-centralizer on $R$, where $R$ is any suitable, torsion-free semiprime ring and $p$ is a fixed integer greater than or equal to 1. As a result of the primary theorems, involution $I_v$ related observations are also provided. We will also consider criticism and discussion alongside the proofs of theorems. Suitable examples given in favor of justification.

Weak $ (p, q) $-Jordan centralizer and derivation on rings and algebras

2025-01-01

Faiza Shujat , Fawaz Alharbi , Abu Zaid Ansari

Characterization of Multiplicative Mixed Jordan-Type Derivations on Ring with Involution

2025-05-01

Md Arshad Madni , Muzibur Rahman Mozumder , Abu Zaid Ansari +1 more

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.

Characterization of Multiplicative Mixed Jordan-Type Derivations on Ring with Involution

2025-05-01

Md Arshad Madni , Muzibur Rahman Mozumder , Abu Zaid Ansari +1 more

Jordan $\varphi$-centralizers on Semiprime and Involution Rings

2025-01-31

Abu Zaid Ansari , Faiza Shujat , Alwaleed Kamel +1 more

Weak $ (p, q) $-Jordan centralizer and derivation on rings and algebras

2025-01-01

Faiza Shujat , Fawaz Alharbi , Abu Zaid Ansari

Structure of (σ, ρ)-n-Derivations on Rings

2024-11-29

Abu Zaid Ansari , Faiza Shujat , Ahlam Fallatah

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Action of Left Identity on n-Skew Commuting 3-Additive Mappings

2024-11-29

Suad Alrehaili , Faiza Shujat , Abu Zaid Ansari

Two Iterates of Symmetric Generalized Skew 3-Derivation

2024-10-21

Abu Zaid Ansari , Suad Alrehaili , Faiza Shujat

A characterization of σ-prime rings involving generalized derivations

2024-06-24

Md Arshad Madni , Muzibur Rahman Mozumder , Wasim Ahmed +1 more

An extension of Herstein's theorem on Banach algebra

2024-01-01

Abu Zaid Ansari , Suad Alrehaili , Faiza Shujat

Results on Generalized Skew Bi-Semiderivation in Prime Rings

2024-01-01

Faiza Shujat , Abu Zaid Ansari

Algebraic Identities on Prime and Semiprime Rings

2024-01-01

Abu Zaid Ansari , Faiza Shujat

Classification of additive mappings on certain rings and algebras

2023-11-14

Abu Zaid Ansari

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Characterization of additive mappings on semiprime rings

2023-10-17

Abu Zaid Ansari

Identities on additive mappings in semiprime rings

2023-01-16

Abu Zaid Ansari , Nadeem ur Rehman

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Additive mappings satisfying algebraic identities in semiprime rings

2023-01-01

Abu Zaid Ansari

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Jordan *-derivations on standard operator algebras

2023-01-01

Abu Zaid Ansari , Faiza Shujat

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Generalized differential identities on prime rings and algebras

2023-01-01

Abu Zaid Ansari , Faiza Shujat , Ahlam Fallatah

COMMUTING SYMMETRIC BI-SEMIDERIVATIONS ON RINGS

2021-09-22

Faiza Shujat , Abu Zaid Ansari , K. Kiran Kumar

Additive mappings covering generalized (alpha_1, alpha_2)-derivations in semiprime rings

2021-09-12

Faiza Shujat , Abu Zaid Ansari

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Spacelike Surfaces with a Common Line of Curvature in Lorentz-Minkowski 3-Space

2021-05-04

M. Khalifa Saad , Abu Zaid Ansari , Muhammad Nadeem Akram +1 more

A NOTE ON GENERALIZED JORDAN m-DERIVATIONS IN RINGS

2021-01-20

Faiza Shujat , Abu Zaid Ansari

On Identities with Additive Mappings in Rings

2020-04-01

Abu Zaid Ansari

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Additive mappings act as a generalized left $$(\alpha , \beta )$$ ( α , β ) -derivation in rings

2018-07-18

Faiza Shujat , Abu Zaid Ansari , F. Salama

Strong commutativity preserving biderivations on prime rings

2017-02-26

Faiza Shujat , Abu Zaid Ansari , Shahoor Khan

Lie Ideals and Generalized Derivations in Semiprime Rings

2015-10-10

Vincenzo De Filippis , Nadeem ur Rehman , Abu Zaid Ansari

Generalized Derivations on Lie Ideals and Power Values on Prime Rings

2015-10-01

Giovanni Scudo , Abu Zaid Ansari

On Additive Mappings in a ∗-Ring with an Identity Element

2015-08-13

Nadeem ur Rehman , Abu Zaid Ansari

SYMMETRIC SKEW 4-DERIVATIONS ON PRIME RINGS

2014-08-26

Faiza Shujat , Abu Zaid Ansari

On Lie ideals of $$*$$ ∗ -prime rings with generalized derivations

2014-07-31

Nadeem ur Rehman , Radwan M. Al-omary , Abu Zaid Ansari

Additive mappings satisfying algebraic conditions in rings

2014-04-22

Abu Zaid Ansari , Faiza Shujat

Generalized left derivations acting as homomorphisms and anti-homomorphisms on Lie ideal of rings

2014-02-01

Nadeem ur Rehman , Abu Zaid Ansari

On Lie Ideals and Generalized Jordan Left Derivations of Prime Rings

2014-01-01

Nadeem ur Rehman , Abu Zaid Ansari

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Generalized Derivations on Power Values of Lie Ideals in Prime and Semiprime Rings

2014-01-01

Vincenzo De Filippis , Nadeem ur Rehman , Abu Zaid Ansari

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On generalized Jordan ∗-derivation in rings

2013-06-14

Nadeem ur Rehman , Abu Zaid Ansari , Tarannum Bano

A Note on Homomorphisms and Anti-Homomorphisms on ∗-Ring

2013-03-22

Nadeem ur Rehman , Abu Zaid Ansari , Claus Haetinger

On identities with derivations in rings

2012-01-01

Abu Zaid Ansari

Coauthor	Papers Together
Faiza Shujat	18
Nadeem ur Rehman	9
Ahlam Fallatah	3
Suad Alrehaili	3
Muzibur Rahman Mozumder	2
Vincenzo De Filippis	2
Md Arshad Madni	2
K. Kiran Kumar	1
Muhammad Nadeem Akram	1
Wasim Ahmed	1
F. Salama	1
Giovanni Scudo	1
M. Khalifa Saad	1
Radwan M. Al-omary	1
Tarannum Bano	1
Alwaleed Kamel	1
Shahoor Khan	1
F. Alharbi	1
Claus Haetinger	1
Fawaz Alharbi	1

On generalized left derivations in rings and Banach algebras

2011-04-18

Shakir Ali

ON GENERALIZED JORDAN LEFT DERIVATIONS IN RINGS

2008-05-31

Mohammad Ashraf , Shakir Ali

In this paper, we introduce the notion of generalized left derivation on a ring R and prow that every generalized Jordan left derivation on a 2-torsion free primp ring is … In this paper, we introduce the notion of generalized left derivation on a ring R and prow that every generalized Jordan left derivation on a 2-torsion free primp ring is a generalized left derivation on R. Some related results are also obtained.

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Jordan derivations on rings

1975-12-01

J. M. Cusack

I. N. Herstein has shown that every Jordan derivation on a prime ring not of charactetistic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a … I. N. Herstein has shown that every Jordan derivation on a prime ring not of charactetistic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a derivation. This result is extended in this paper to the case of any ring in which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 x equals 0"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2x = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> implies <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x equals 0"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">x = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and which is semiprime or which has a commutator which is not a zero divisor.

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Derivations in prime rings

1957-12-01

Edward C. Posner

iqaqaq)bq + qaqbqaq = 0 iqaqaq)bq + qaqbqaq = 0

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Jordan derivations of prime rings

1957-12-01

I. N. Herstein

1. Given any associative ring A one can construct from its operations and elements a new ring, the Jordan ring of A, by defining the product in this ring to … 1. Given any associative ring A one can construct from its operations and elements a new ring, the Jordan ring of A, by defining the product in this ring to be a o b=ab+ba for all a, bEA, where the product ab signifies the product of a and b in the associative ring A itself. If R is any ring, associative or otherwise, by a derivation of R we shall mean a function, ', mapping R into itself so that

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On centralizers of semiprime rings

1991-01-01

Borut Zalar

Let $\Cal K$ be a semiprime ring and $T:\Cal K\rightarrow \Cal K$ an additive mapping such that $T(x^2)=T(x)x$ holds for all $x\in \Cal K$. Then $T$ is a left centralizer … Let $\Cal K$ be a semiprime ring and $T:\Cal K\rightarrow \Cal K$ an additive mapping such that $T(x^2)=T(x)x$ holds for all $x\in \Cal K$. Then $T$ is a left centralizer of $\Cal K$. It is also proved that Jordan centralizers and centralizers of $\Cal K$ coincide.

On left Jordan derivations of rings and Banach algebras

2008-06-01

Joso Vukman

Topics In Ring Theory

2010-01-01

I. N. Herstein

On Jordan ideals and left (θ, θ)‐derivations in prime rings

2004-01-01

S. M. A. Zaidi , Mohammad Ashraf , Shakir Ali

Let R be a ring and S a nonempty subset of R . Suppose that θ and ϕ are endomorphisms of R . An additive mapping δ : R → … Let R be a ring and S a nonempty subset of R . Suppose that θ and ϕ are endomorphisms of R . An additive mapping δ : R → R is called a left ( θ , ϕ )‐derivation (resp., Jordan left ( θ , ϕ )‐derivation) on S if δ ( x y ) = θ ( x ) δ ( y ) + ϕ ( y ) δ ( x ) (resp., δ ( x 2 ) = θ ( x ) δ ( x ) + ϕ ( x ) δ ( x )) holds for all x , y ∈ S . Suppose that J is a Jordan ideal and a subring of a 2‐torsion‐free prime ring R . In the present paper, it is shown that if θ is an automorphism of R such that δ ( x 2 ) = 2 θ ( x ) δ ( x ) holds for all x ∈ J , then either J ⫅ Z ( R ) or δ ( J ) = (0). Further, a study of left ( θ , θ )‐derivations of a prime ring R has been made which acts either as a homomorphism or as an antihomomorphism of the ring R .

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Jordan mappings of semiprime rings

1989-11-01

Matej Brešar

Continuity of Derivations and a Problem of Kaplansky

1968-10-01

B. E. Johnson , Allan M. Sinclair

The Image of a Derivation is Contained in the Radical

1988-11-01

Marc P. Thomas

In 1955 I. M. Singer and J. Wermer proved that a bounded derivation on a commutative Banach algebra maps into the (Jacobson) radical; they conjectured that this result holds even … In 1955 I. M. Singer and J. Wermer proved that a bounded derivation on a commutative Banach algebra maps into the (Jacobson) radical; they conjectured that this result holds even if the derivation is unbounded. We give a proof of this conjecture. The central idea in the proof is the introduction of the concept of a recalcitrant system of elements in a commutative radical Banach algebra. Such systems put algebraic constraints upon a derivation which prevent the derivation from mapping outside of the radical.

Prime rings satisfying a generalized polynomial identity

1969-08-01

Wallace S. Martindale

An Additive Mapping Satisfying an Algebraic Condition in Rings with Identity

2013-04-01

Ashraf

On left derivations and related mappings

1990-01-01

Matej Brešar , Joso Vukman

Let fi bca ring and X be a left Z?-module.The purpose of this paper is to investigate additive mappings £>, : R -> X and D2: R -» X that … Let fi bca ring and X be a left Z?-module.The purpose of this paper is to investigate additive mappings £>, : R -> X and D2: R -» X that satisfy Dx(ab) = aDx(b) + bDx(a), a, b e R (left derivation) and D2(a ) = 2aD2(a), a e 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. PreliminariesThroughout, R will represent an associative ring with center Z(R).Recall that R is prime if aRb -0 implies a = 0 or b -0, and R is semiprime if aRa -0 implies a = 0.A module X is said to be zz-torsionfree, where zz is an integer, if zzx = 0, x e X implies x = 0. Let R be a ring and X be an T\-bimodule.An additive mapping D: R -► X is called a derivation (Jordan derivation) if D(ab) = D(a)b + aD(b), a, b e R (D(a2) = D(a)a + aD(a), a e R).Obviously, every derivation is a Jordan derivation.The converse in general is not true.A well-known result of I. N. Herstein [6] states that in case R is a prime ring of characteristic not 2, then every Jordan derivation D: R -► R is a derivation.A brief proof of this result in presented in our recent paper [3].An additive mapping D : R -» X, where R is a ring and X is a left Rmodule will be called a left derivation if D(ab) = aD(b) + bD(a), a , b e R.An additive mapping D: R -> X will be called a Jordan left derivation if D(a ) -2aD(a), a € R. It turns out that the notion of Jordan left derivations is in a close connection with so-called commuting mappings.A mapping 7 of a ring R into itself is said to be commuting on R if F(a)a = aF(a) for all a e

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Jordan Left Derivations on Semiprime Rings

1997-01-01

Joso Vukman

On the distance of the composition of two derivations to the generalized derivations

1991-01-01

Matej Brešar

A well-known theorem of E. Posner [10] states that if the composition d 1 d 2 of derivations d 1 d 2 of a prime ring A of characteristic not … A well-known theorem of E. Posner [10] states that if the composition d 1 d 2 of derivations d 1 d 2 of a prime ring A of characteristic not 2 is a derivation, then either d 1 = 0 or d 2 = 0. A number of authors have generalized this theorem in several ways (see e.g. [1], [2], and [5], where further references can be found). Under stronger assumptions when A is the algebra of all bounded linear operators on a Banach space (resp. Hilbert space), Posner's theorem was reproved in [3] (resp. [12]). Recently, M. Mathieu [8] extended Posner's theorem to arbitrary C * -algebras.

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Generalized derivations of left faithful rings

1999-01-01

Tsiu‐Kwen Lee

Abstract Let R be a left faithful ringU its right Utumi quotient ring and ρ a dense right ideal of R. An additive map g: ρ → U is called … Abstract Let R be a left faithful ringU its right Utumi quotient ring and ρ a dense right ideal of R. An additive map g: ρ → U is called a generalized derivation if there exists a derivation δ of ρ into U such that for all x,y∈ρ. In this note, we prove that there exists an element a∈ U such that for all x ∈ ρ. From this characterization, it is proved that if R is a semiprime ring and if g is a generalized derivation with nilpotent values of bounded index, then g = 0. Analogous results are also obtained for the case of generalized derivations with nilpotent values on Lie ideals or one-sided ideals.

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On Identities with Additive Mappings in Rings

2020-04-01

Abu Zaid Ansari

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A NOTE ON GENERALIZED DERIVATIONS OF SEMIPRIME RINGS

2007-06-01

Joso Vukman

In this paper we prove that generalized Jordan derivations and generalized Jordan triple derivation of 2-torsion free semiprime rings are generalized derivations. In this paper we prove that generalized Jordan derivations and generalized Jordan triple derivation of 2-torsion free semiprime rings are generalized derivations.

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GENERALIZED JORDAN DERIVATIONS ON PRIME RINGS AND STANDARD OPERATOR ALGEBRAS

2003-12-01

Wu Jing , Shijie Lu

In this paper we initiate the study of generalized Jordan derivations and generalized Jordan triple derivations on prime rings and standard operator algebras. In this paper we initiate the study of generalized Jordan derivations and generalized Jordan triple derivations on prime rings and standard operator algebras.

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On Lie ideals and Jordan left derivations of prime rings

2000-01-01

Mohammad Ashraf , Nadeem-ur-Rehman

Let $R$ be a 2-torsion free prime ring and let $U$ be a Lie ideal of $R$ such that $u^{2} \in U$ for all $u \in U$. In the present … Let $R$ be a 2-torsion free prime ring and let $U$ be a Lie ideal of $R$ such that $u^{2} \in U$ for all $u \in U$. In the present paper it is shown that if $d$ is an additive mappings of $R$ into itself satisfying $d(u^{2})=2ud(u)$ for all $u \in U$, then $d(uv)=ud(v)+vd(u)$ for all $u,v \in U$.

Jordan derivations on prime rings

1988-06-01

Matej Brešar , Joso Vukman

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 … 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.

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Mixed -Jordan-type derivations on -algebras

2022-02-23

Bruno Leonardo Macedo Ferreira , Feng Wei

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.

Two results concerning symmetric bi-derivations on prime rings

1990-12-01

Joso Vukman

An equation related to centralizers in semiprime rings

2003-12-15

Joso Vukman

The main result: Let R be a 2-torsion free semiprime ring with extended centroid C and let T : R → R be an additive mapping.Suppose that 3T (xyx) = … The main result: Let R be a 2-torsion free semiprime ring with extended centroid C and let T : R → R be an additive mapping.Suppose that 3T (xyx) = T (x)yx + xT (y)x + xyT (x) holds for all x, y ∈ R. Then there exists an element λ ∈ C such that T (x) = λx for all x ∈ R.

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Generalized Derivations on (Semi-)Prime Rings and Noncommutative Banach Algebras

2009-12-31

Feng Wei , Zhankui Xiao

We first give several polynomial identities of semiprime rings which make the additive mappings appearing in the identities to be generalized derivations. Then we study pairs of generalized Jordan derivations … We first give several polynomial identities of semiprime rings which make the additive mappings appearing in the identities to be generalized derivations. Then we study pairs of generalized Jordan derivations on prime rings. Let m,n be fixed positive integers, \mathcal R be a noncommutative 2(m+n)! -torsion free prime ring with the center \mathcal Z and μ, ν be a pair of generalized Jordan derivations on \mathcal R . If μ(x^m)x^n+x^nν(x^m) \in \mathcal Z for all x \in \mathcal R , then μ and ν are left (or right) multipliers. In particular, if μ , ν are a pair of derivations on \mathcal R satisfying the same assumption, then μ = ν = 0 . Then applying these purely algebraic result we obtain several range inclusion results of pair of derivations on Banach algebras.

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An Engel condition with derivation for left ideals

1997-01-01

Charles Lanski

We generalize a number of results in the literature by proving the following theorem: Let $R$ be a semiprime ring, $D$ a nonzero derivation of $R$, $L$ a nonzero left … We generalize a number of results in the literature by proving the following theorem: Let $R$ be a semiprime ring, $D$ a nonzero derivation of $R$, $L$ a nonzero left ideal of $R$, and let $[x,y]=xy-yx$. If for some positive integers $t_0,t_1,\dots , t_n$, and all $x\in L$, the identity $[[\dots [[D(x^{t_0}),x^{t_1}],x^{t_2}],\dots ],x^{t_n}]=0$ holds, then either $D(L)=0$ or else the ideal of $R$ generated by $D(L)$ and $D(R)L$ is in the center of $R$. In particular, when $R$ is a prime ring, $R$ is commutative.

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On additive mappings in rings with identity element

2016-08-31

V. K. Yadav , R. K. Sharma

Multipliers of Banach Algebras

1956-09-01

S. Helgason

Derivations on commutative normed algebras

1955-12-01

I. M. Singer , John Wermer

Jordan derivations of alternative rings

2019-09-10

Bruno Leonardo Macedo Ferreira , Henrique Guzzo , Ruth Nascimento Ferreira +1 more

Let R be a unital alternative ring with nontrivial idempotent and D:R→R be a Jordan derivation. Then D is of the form d+δ, where d is a derivation of R … Let R be a unital alternative ring with nontrivial idempotent and D:R→R be a Jordan derivation. Then D is of the form d+δ, where d is a derivation of R and δ is a singular Jordan derivation of R. Moreover, d and δ are uniquely determined. This extends the main result of Benkovič and Širovnik's to the case of alternative rings.

Lie ideals and derivations of prime rings

1981-07-01

Jeffrey Bergen , I. N. Herstein , Jeanne Wald Kerr

Left centralizers and isomorphisms of group algebras

1952-06-01

J. G. Wendel

In Part II these results are applied to obtain a generalization of a theorem proved in [5].We showed there that if T is an isometric isomorphism of (real, X We … In Part II these results are applied to obtain a generalization of a theorem proved in [5].We showed there that if T is an isometric isomorphism of (real, X We are using the notation and terminology of [5].In particular, for elements x and y of L (G) the symbol xy denotes the usual convolution-product.

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On generalized Jordan ∗-derivation in rings

2013-06-14

Nadeem ur Rehman , Abu Zaid Ansari , Tarannum Bano

On Prime and Semiprime Rings with Derivations

2006-09-01

Nurcan Argaç

Let R be a ring and S a nonempty subset of R. A mapping f: R → R is called commuting on S if [f(x),x] = 0 for all x … Let R be a ring and S a nonempty subset of R. A mapping f: R → R is called commuting on S if [f(x),x] = 0 for all x ∈ S. In this paper, firstly, we generalize the well-known result of Posner related to commuting derivations on prime rings. Secondly, we show that if R is a semiprime ring and I is a nonzero ideal of R, then a derivation d of R is commuting on I if one of the following conditions holds: (i) For all x, y ∈ I, either d([x,y]) = [x,y] or d([x,y]) = -[x,y]. (ii) For all x, y ∈ I, either d(x ◦ y) = x ◦ y or d(x ◦ y) = -(x ◦ y). (iii) R is 2-torsion free, and for all x, y ∈ I, either [d(x),d(y)] = d([x,y]) or [d(x),d(y)] = d([y,x]). Furthermore, if d(I) ≠ {0}, then R has a nonzero central ideal. Finally, we introduce the notation of generalized biderivation and prove that every generalized biderivation on a noncommutative prime ring is a biderivation.

Generalized derivations on Lie ideals in prime rings

2011-01-01

Öznur Gölbaşı , Emine Koç

Let R be a prime ring with characteristic different from two, U a nonzero Lie ideal of R and f be a generalized derivation associated with d. We prove the … Let R be a prime ring with characteristic different from two, U a nonzero Lie ideal of R and f be a generalized derivation associated with d. We prove the following results: (i) If (u, f (u)) ∈ Z, for all u ∈ U, then U ⊂ Z. (ii) (f,d )a nd (g, h) be two generalized derivations of R such that f (u)v = ug(v), for all u, v ∈ U, then U ⊂ Z. (iii) f ((u, v)) = ±(u, v), for all u, v ∈ U, then U ⊂ Z.

On the Lie structure of an associative ring

1970-04-01

I. N. Herstein

Additive mappings satisfying algebraic conditions in rings

2014-04-22

Abu Zaid Ansari , Faiza Shujat

A NOTE ON JORDAN LEFT DERIVATIONS

1996-01-01

Kil-Woung Jun , Byung-Do Kim

Throughout, R will represent an associative ring with center Z(R). A module X is said to be n-torsionfree, where n is an integer, if nx = 0, $x \in X$ … Throughout, R will represent an associative ring with center Z(R). A module X is said to be n-torsionfree, where n is an integer, if nx = 0, $x \in X$ implies x = 0. An additive mapping $D : R \to X$, where X is a left R-module, will be called a Jordan left derivation if $D(a^2) = 2aD(a), a \in R$. M. Bresar and J. Vukman [1] showed that the existence of a nonzero Jordan left derivation of R into X implies R is commutative if X is a 2-torsionfree and 3-torsionfree left R-module.

An Introduction to the Theory of Centralizers

1964-04-01

B. E. Johnson

Generalized Derivations with Commutativity and Anti-commutativity Conditions

2007-01-01

Howard E. Bell , Nadeem ur Rehman

1. IntroductionLet R be an associative ring with center Z = Z(R). For each x;y 2 Rdenote the commutator xy yx by [x;y] and the anti-commutator xy +yxby xy. Recall … 1. IntroductionLet R be an associative ring with center Z = Z(R). For each x;y 2 Rdenote the commutator xy yx by [x;y] and the anti-commutator xy +yxby xy. Recall that a ring R is prime if for any a;b 2 R, aRb = f0g impliesthat a = 0 or b = 0. An additive mapping d : R ! R is called a derivationif d(xy) = d(x)y +xd(y) for all x;y 2 R. In particular, for xed a 2 R, themapping I

On Jordan Left Derivations of Lie Ideals in Prime Rings

2002-02-01

Mohammad Ashraf , Nadeem-ur-Rehman , Shakir Ali

GPIs having coefficients in Utumi quotient rings

1988-01-01

Chen-Lian Chuang

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a prime ring and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding="application/x-tex">U</mml:annotation> … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a prime ring and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding="application/x-tex">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be its Utumi quotient ring. We prove the following: (1) If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfies a GPI having all its coefficients in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding="application/x-tex">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfies a GPI having all its coefficients in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. (2) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding="application/x-tex">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfy the same GPIs having their coefficients in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U"> <mml:semantics> <mml:mi>U</mml:mi> <mml:annotation encoding="application/x-tex">U</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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Differential identities of prime rings

1978-03-01

V. K. Kharchenko

Multipliers of commutative Banach algebras

1961-09-01

Ju-Kwei Wang

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On (m,n)-Jordan centralizers in rings and algebras

2010-05-17

Joso Vukman

Let m ≥ 0, n ≥ 0 be fixed integers with m + n = 0 and let R be a ring.It is our aim in this paper to investigate … Let m ≥ 0, n ≥ 0 be fixed integers with m + n = 0 and let R be a ring.It is our aim in this paper to investigate additive mapping T : R → R satisfying the relation (m + n)T (x 2 ) = mT (x)x + nxT (x) for all x ∈ R.

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A NOTE ON GENERALIZED DERIVATIONS OF SEMIPRIME RINGS

2007-01-06

Joso Vukman

In this paper we prove that generalized Jordan derivations and generalized Jordan triple derivation of 2−torsion free semiprime rings are generalized derivations. In this paper we prove that generalized Jordan derivations and generalized Jordan triple derivation of 2−torsion free semiprime rings are generalized derivations.

Centralizing Maps in Prime Rings with Involution

1993-11-01

Matej Brešar , Wallace S. Martindale , C. Robert Miers

A note on semiprime rings with derivation

1996-03-03

Motoshi Hongan

Let R be a 2‐torsion free semiprime ring, I a nonzero ideal of R , Z the center of R and D : R → R a derivation. If d … Let R be a 2‐torsion free semiprime ring, I a nonzero ideal of R , Z the center of R and D : R → R a derivation. If d [ x , y ] + [ x , y ] ∈ Z or d [ x , y ] − [ x , y ] ∈ Z for all x , y ∈ I , then R is commutative.

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