On the characterization of certain additive maps in prime * -rings

Mohammad Ashraf; Mohammad Aslam Siddeeque; Abbas Hussain Shikeh

Czechoslovak Mathematical Journal (2024)

  • Volume: 74, Issue: 2, page 549-565
  • ISSN: 0011-4642

Abstract

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Let 𝒜 be a noncommutative prime ring equipped with an involution ‘ * ’, and let 𝒬 m s ( 𝒜 ) be the maximal symmetric ring of quotients of 𝒜 . Consider the additive maps ℋ and 𝒯 : 𝒜 → 𝒬 m s ( 𝒜 ) . We prove the following under some inevitable torsion restrictions. (a) If m and n are fixed positive integers such that ( m + n ) 𝒯 ( a 2 ) = m 𝒯 ( a ) a * + n a 𝒯 ( a ) for all a ∈ 𝒜 and ( m + n ) ℋ ( a 2 ) = m ℋ ( a ) a * + n a 𝒯 ( a ) for all a ∈ 𝒜 , then ℋ = 0 . (b) If 𝒯 ( a b a ) = a 𝒯 ( b ) a * for all a , b ∈ 𝒜 , then 𝒯 = 0 . Furthermore, we characterize Jordan left τ -centralizers in semiprime rings admitting an anti-automorphism τ . As applications, we find the structure of generalized Jordan * -derivations in prime rings and generalize as well as improve all the results of A. Abbasi, C. Abdioglu, S. Ali, M. R. Mozumder (2022).

How to cite

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Ashraf, Mohammad, Siddeeque, Mohammad Aslam, and Shikeh, Abbas Hussain. "On the characterization of certain additive maps in prime $\ast $-rings." Czechoslovak Mathematical Journal 74.2 (2024): 549-565. <http://eudml.org/doc/299562>.

@article{Ashraf2024,
abstract = {Let $\mathcal \{A\}$ be a noncommutative prime ring equipped with an involution ‘$*$’, and let $\mathcal \{Q\}_\{ms\}(\mathcal \{A\})$ be the maximal symmetric ring of quotients of $\mathcal \{A\}$. Consider the additive maps $\mathcal \{H\}$ and $\mathcal \{T\} \colon \mathcal \{A\}\rightarrow \mathcal \{Q\}_\{ms\}(\mathcal \{A\})$. We prove the following under some inevitable torsion restrictions. (a) If $m$ and $n$ are fixed positive integers such that $(m+n)\mathcal \{T\}(a^2)=m\mathcal \{T\}(a)a^*+na\mathcal \{T\}(a)$ for all $a\in \mathcal \{A\}$ and $(m+n)\mathcal \{H\}(a^2)=m\mathcal \{H\}(a)a^*+na\mathcal \{T\}(a)$ for all $a\in \mathcal \{A\}$, then $\mathcal \{H\}=0$. (b) If $\mathcal \{T\}(aba)=a\mathcal \{T\}(b)a^*$ for all $a, b\in \mathcal \{A\}$, then $\mathcal \{T\}=0$. Furthermore, we characterize Jordan left $\tau $-centralizers in semiprime rings admitting an anti-automorphism $\tau $. As applications, we find the structure of generalized Jordan $*$-derivations in prime rings and generalize as well as improve all the results of A. Abbasi, C. Abdioglu, S. Ali, M. R. Mozumder (2022).},
author = {Ashraf, Mohammad, Siddeeque, Mohammad Aslam, Shikeh, Abbas Hussain},
journal = {Czechoslovak Mathematical Journal},
keywords = {prime ring; involution; generalized $(m, n)$-Jordan $*$-centralizer},
language = {eng},
number = {2},
pages = {549-565},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the characterization of certain additive maps in prime $\ast $-rings},
url = {http://eudml.org/doc/299562},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Ashraf, Mohammad
AU - Siddeeque, Mohammad Aslam
AU - Shikeh, Abbas Hussain
TI - On the characterization of certain additive maps in prime $\ast $-rings
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 2
SP - 549
EP - 565
AB - Let $\mathcal {A}$ be a noncommutative prime ring equipped with an involution ‘$*$’, and let $\mathcal {Q}_{ms}(\mathcal {A})$ be the maximal symmetric ring of quotients of $\mathcal {A}$. Consider the additive maps $\mathcal {H}$ and $\mathcal {T} \colon \mathcal {A}\rightarrow \mathcal {Q}_{ms}(\mathcal {A})$. We prove the following under some inevitable torsion restrictions. (a) If $m$ and $n$ are fixed positive integers such that $(m+n)\mathcal {T}(a^2)=m\mathcal {T}(a)a^*+na\mathcal {T}(a)$ for all $a\in \mathcal {A}$ and $(m+n)\mathcal {H}(a^2)=m\mathcal {H}(a)a^*+na\mathcal {T}(a)$ for all $a\in \mathcal {A}$, then $\mathcal {H}=0$. (b) If $\mathcal {T}(aba)=a\mathcal {T}(b)a^*$ for all $a, b\in \mathcal {A}$, then $\mathcal {T}=0$. Furthermore, we characterize Jordan left $\tau $-centralizers in semiprime rings admitting an anti-automorphism $\tau $. As applications, we find the structure of generalized Jordan $*$-derivations in prime rings and generalize as well as improve all the results of A. Abbasi, C. Abdioglu, S. Ali, M. R. Mozumder (2022).
LA - eng
KW - prime ring; involution; generalized $(m, n)$-Jordan $*$-centralizer
UR - http://eudml.org/doc/299562
ER -

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