• 제목/요약/키워드: Rings with involution

검색결과 20건 처리시간 0.02초

NIL-CLEAN RINGS OF NILPOTENCY INDEX AT MOST TWO WITH APPLICATION TO INVOLUTION-CLEAN RINGS

  • Li, Yu;Quan, Xiaoshan;Xia, Guoli
    • 대한수학회논문집
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    • 제33권3호
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    • pp.751-757
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    • 2018
  • A ring is nil-clean if every element is a sum of a nilpotent and an idempotent, and a ring is involution-clean if every element is a sum of an involution and an idempotent. In this paper, a description of nil-clean rings of nilpotency index at most 2 is obtained, and is applied to improve a known result on involution-clean rings.

SOME STUDIES ON JORDAN (𝛼, 1)* -BIDERIVATION IN RINGS WITH INVOLUTION

  • SK. HASEENA;C. JAYA SUBBA REDDY
    • Journal of Applied and Pure Mathematics
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    • 제6권1_2호
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    • pp.13-20
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    • 2024
  • Let R be a ring with involution. In the present paper, we characterize biadditive mappings which satisfies some functional identities related to symmetric Jordan (𝛼, 1)*-biderivation of prime rings with involution. In particular, we prove that on a 2-torsion free prime ring with involution, every symmetric Jordan triple (𝛼, 1)*-biderivation is a symmetric Jordan (𝛼, 1)*-biderivation.

A NOTE ON LOCAL COMMUTATORS IN DIVISION RINGS WITH INVOLUTION

  • Bien, Mai Hoang
    • 대한수학회보
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    • 제56권3호
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    • pp.659-666
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    • 2019
  • In this paper, we consider a conjecture of I. N. Herstein for local commutators of symmetric elements and unitary elements of division rings. For example, we show that if D is a finite dimensional division ring with involution ${\star}$ and if $a{\in}D^*=D{\setminus}\{0\}$ such that local commutators $axa^{-1}x^{-1}$ at a are radical over the center F of D for every $x{\in}D^*$ with $x^{\star}=x$, then either $a{\in}F$ or ${\dim}_F\;D=4$.

RESULTS OF 3-DERIVATIONS AND COMMUTATIVITY FOR PRIME RINGS WITH INVOLUTION INVOLVING SYMMETRIC AND SKEW SYMMETRIC COMPONENTS

  • Hanane Aharssi;Kamal Charrabi;Abdellah Mamouni
    • 대한수학회논문집
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    • 제39권1호
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    • pp.79-91
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    • 2024
  • This article examines the connection between 3-derivations and the commutativity of a prime ring R with an involution * that fulfills particular algebraic identities for symmetric and skew symmetric elements. In practice, certain well-known problems, such as the Herstein problem, have been studied in the setting of three derivations in involuted rings.

CENTRALIZING AND COMMUTING INVOLUTION IN RINGS WITH DERIVATIONS

  • Khan, Abdul Nadim
    • 대한수학회논문집
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    • 제34권4호
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    • pp.1099-1104
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    • 2019
  • In [1], Ali and Dar proved the ${\ast}$-version of classical theorem due to Posner [15, Theorem] with involution of the second kind. The main objective of this paper is to improve the above mentioned result without the condition of the second kind involution. Moreover, a related result has been discussed.

Posner's First Theorem for *-ideals in Prime Rings with Involution

  • Ashraf, Mohammad;Siddeeque, Mohammad Aslam
    • Kyungpook Mathematical Journal
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    • 제56권2호
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    • pp.343-347
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    • 2016
  • Posner's first theorem states that if R is a prime ring of characteristic different from two, $d_1$ and $d_2$ are derivations on R such that the iterate $d_1d_2$ is also a derivation of R, then at least one of $d_1$, $d_2$ is zero. In the present paper we extend this result to *-prime rings of characteristic different from two.

On n-skew Lie Products on Prime Rings with Involution

  • Ali, Shakir;Mozumder, Muzibur Rahman;Khan, Mohammad Salahuddin;Abbasi, Adnan
    • Kyungpook Mathematical Journal
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    • 제62권1호
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    • pp.43-55
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    • 2022
  • Let R be a *-ring and n ≥ 1 be an integer. The objective of this paper is to introduce the notion of n-skew centralizing maps on *-rings, and investigate the impact of these maps. In particular, we describe the structure of prime rings with involution '*' such that *[x, d(x)]n ∈ Z(R) for all x ∈ R (for n = 1, 2), where d : R → R is a nonzero derivation of R. Among other related results, we also provide two examples to prove that the assumed restrictions on our main results are not superfluous.

ON DOMINATION IN ZERO-DIVISOR GRAPHS OF RINGS WITH INVOLUTION

  • Nazim, Mohd;Nisar, Junaid;Rehman, Nadeem ur
    • 대한수학회보
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    • 제58권6호
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    • pp.1409-1418
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    • 2021
  • In this paper, we study domination in the zero-divisor graph of a *-ring. We first determine the domination number, the total domination number, and the connected domination number for the zero-divisor graph of the product of two *-rings with componentwise involution. Then, we study domination in the zero-divisor graph of a Rickart *-ring and relate it with the clique of the zero-divisor graph of a Rickart *-ring.

ON g(x)-INVO CLEAN RINGS

  • El Maalmi, Mourad;Mouanis, Hakima
    • 대한수학회논문집
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    • 제35권2호
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    • pp.455-468
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    • 2020
  • An element in a ring R with identity is called invo-clean if it is the sum of an idempotent and an involution and R is called invoclean if every element of R is invo-clean. Let C(R) be the center of a ring R and g(x) be a fixed polynomial in C(R)[x]. We introduce the new notion of g(x)-invo clean. R is called g(x)-invo if every element in R is a sum of an involution and a root of g(x). In this paper, we investigate many properties and examples of g(x)-invo clean rings. Moreover, we characterize invo-clean as g(x)-invo clean rings where g(x) = (x-a)(x-b), a, b ∈ C(R) and b - a ∈ Inv(R). Finally, some classes of g(x)-invo clean rings are discussed.