A characterization of additive mappings in rings with involution

Ali, Shakir; Alhazmi, Husain; Dar, Nadeem Ahmad; Khan, Abdul Nadim

doi:10.1515/9783110542400-002

Cited by 3 publications

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“…In fact, these mappings appeared first time in the recent paper due to Ali et al [1]. A Banach algebra is a linear associative algebra which, as a vector space, is a Banach space with norm ∥ • ∥ satisfying the multiplicative inequality; ∥xy∥ ≤ ∥x∥∥y∥, ∀ x and y in A.…”

Section: Results On C * -Algebrasmentioning

confidence: 99%

“…Further, Chaudhary and Thaheem [18] extended the above mentioned results for semiprime rings and showed that if R is a semiprime ring and f, g a pair of derivations of R such that f (x)x + xg(x) ∈ Z(R), ∀ x ∈ R, then f and g are central. Inspired by these work's, Ali et al [1] established the following result. Our next theorem is motivated by the above mentioned result.…”

Section: Proof By the Assumption We Havementioning

confidence: 96%

“…Let R be a ring with involution ′ * ′ . An additive mapping [1] and [4] for details). In [13], Brešar considered a pair of additive mappings (derivations) and proved the following result: If a noncommutative prime ring R admits a pair of derivations d and g such that d…”

Section: Proof By the Assumption We Havementioning

confidence: 99%

“…[1, Theorem 4.4] Let m, n be fixed positive integers, and R be a (m + n)!torsion free noncommutative prime ring with involution ′ * ′ of the second kind having the identity element e. Suppose there exist Jordan * -derivations d, g : R → R such that d(x m )x n ± x n g(x m ) = 0, ∀ x ∈ R. Then d = g = 0.…”

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confidence: 99%

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On ideals of prime rings involving $n$-skew commuting additive mappings with applications

Abdioğlu

Ali

Khan

2022

Hacettepe Journal of Mathematics and Statistics

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Let $n > 1 $ be a fixed positive integer and $S$ be a subset of a ring $R$. A mapping $\zeta$ of a ring $R$ into itself is called $n$-skew-commuting on $S$ if $\zeta(x)x^{n} + x^{n}\zeta(x)=0$, $\forall$ $x\in S.$ The main aim of this paper is to describe $n$-skew-commuting mappings on appropriate subsets of $R$. With this, many known results can be either generalized or deduced. In particular, this solves the conjecture in [M. Nadeem, M. Aslam and M.A. Javed, On $2$-skew commuting additive mappings of prime rings, Gen. Math. Notes, 2015]. The second main result of this paper is concerned with a pair of linear mappings of $C^*$-algebras. We show that here, if $C^*$-Algebra admits a pair of linear mappings $f$ and $g$ such that $f(x)x^* + x^*g(x) \in Z(A)$ for all $x \in A,$ then both $f$ and $g$ must be zero. As the applications of first main result (Theorem $2.1$) and apart from proving some other results, we characterize the linear mappings on primitive $C^*$-algebras. Furthermore, we provide an example to show that the assumed restrictions cannot be relaxed.

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Section: Results On C * -Algebrasmentioning

confidence: 99%

Section: Proof By the Assumption We Havementioning

confidence: 96%

Section: Proof By the Assumption We Havementioning

confidence: 99%

mentioning

confidence: 99%

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On ideals of prime rings involving $n$-skew commuting additive mappings with applications

Abdioğlu

Ali

Khan

2022

Hacettepe Journal of Mathematics and Statistics

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“…In 2020, Alahmadi et al [30] extended the above mentioned result for generalized derivations. Over the last few years, the interest on this topic has been increased and numerous papers concerning these mappings on prime rings have been published (see [4,9,[30][31][32][33][34][35][36][37] and references therein). In [38], Creedon studied the action of derivations of prime ideals and proved that if e is a derivation of a ring A and L is a semiprime ideal of A such that A/L is characteristic-free and e k (L) ⊆ L, then e(L) ⊆ L for some positive integer k. Very recently, Idrissi and Oukhtite [39] investigated the structure of a quotient ring A/L via the action of generalized derivations on the prime ideal of L. For more recent works, see [40][41][42] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Posner’s Theorem and ∗-Centralizing Derivations on Prime Ideals with Applications

Ali,

Alsuraiheed,

Khan

et al. 2023

Mathematics

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A well-known result of Posner’s second theorem states that if the commutator of each element in a prime ring and its image under a nonzero derivation are central, then the ring is commutative. In the present paper, we extended this bluestocking theorem to an arbitrary ring with involution involving prime ideals. Further, apart from proving several other interesting and exciting results, we established the ∗-version of Vukman’s theorem. Precisely, we describe the structure of quotient ring A/L, where A is an arbitrary ring and L is a prime ideal of A. Further, by taking advantage of the ∗-version of Vukman’s theorem, we show that if a 2-torsion free semiprime A with involution admits a nonzero ∗-centralizing derivation, then A contains a nonzero central ideal. This result is in the spirit of the classical result due to Bell and Martindale (Theorem 3). As the applications, we extended and unified several classical theorems. Finally, we conclude our paper with a direction for further research.

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Posner’s Theorem and *-Centralizing Derivations on Prime Ideals with Applications

Ali,

Alsuraiheed,

Khan

et al. 2023

Preprint

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A well-known result of Posner’s second theorem states that if the commutator of each element in a prime ring and its image under a nonzero derivation is central, then the ring is commutative. In the present paper, we extend this bluestocking theorem to an arbitrary ring with involution involving prime ideals. Further, apart from proving several other interesting and exciting results, we establish *-version of Vukman’s theorem [48, Theorem 1]. Precisely, we describe the structure of quotient ring A/L, where A is an arbitrary ring and L is a prime ideal of A. Further, by taking advantage of the *-version of Vukman’s theorem, we show that if a 2-trosion free semiprime ring A with involution admits a nonzero *-centralizing derivation, then A contains a nonzero central ideal. This result is in a spirit of the classical result due to Bell and Martindale [19, Theorem 3]. As the applications, we extends and unify several classical theorems proved in [6],[25,[42], and [48] . Finally, we conclude our paper with a direction for further research.

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A characterization of additive mappings in rings with involution

Cited by 3 publications

References 0 publications

On ideals of prime rings involving $n$-skew commuting additive mappings with applications

On ideals of prime rings involving $n$-skew commuting additive mappings with applications

Posner’s Theorem and ∗-Centralizing Derivations on Prime Ideals with Applications

Posner’s Theorem and *-Centralizing Derivations on Prime Ideals with Applications

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