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

Faiza Shujat; Faarie Alharbi; Abu Zaid Ansari; Faiza Shujat; Faarie Alharbi; Abu Zaid Ansari

doi:10.3934/math.2025383

AIMS Mathematics

2025, Volume 10, Issue 4: 8322-8330. doi: 10.3934/math.2025383

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

1.
Department of Mathematics, Faculty of Science, Taibah University, Madinah, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Saudi Arabia

Received: 12 February 2025 Revised: 19 March 2025 Accepted: 26 March 2025 Published: 11 April 2025
MSC : 16B99, 16N60, 16W25

In the present paper, the authors discuss two new concepts that will be known as a weak $ (p, q) $-Jordan centralizer and a weak $ (p, q) $-Jordan derivation on an arbitrary ring $ R $ and they prove that every weak $ (p, q) $-Jordan derivation is a derivation on any prime ring $ R $. Furthermore, every weak $ (p, q) $-Jordan centralizer is a centralizer on a semiprime ring $ R $. Later, they discuss the continuity of weak $ (p, q) $-Jordan centralizer on a semisimple Banach algebra and prove that every weak $ (p, q) $ -Jordan centralizer on a semisimple Banach algebra is a linear continuous operator. Moreover, these results are validated with actual examples.
Citation: Faiza Shujat, Faarie Alharbi, Abu Zaid Ansari. Weak $ (p, q) $-Jordan centralizer and derivation on rings and algebras[J]. AIMS Mathematics, 2025, 10(4): 8322-8330. doi: 10.3934/math.2025383

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Abstract

In the present paper, the authors discuss two new concepts that will be known as a weak $ (p, q) $-Jordan centralizer and a weak $ (p, q) $-Jordan derivation on an arbitrary ring $ R $ and they prove that every weak $ (p, q) $-Jordan derivation is a derivation on any prime ring $ R $. Furthermore, every weak $ (p, q) $-Jordan centralizer is a centralizer on a semiprime ring $ R $. Later, they discuss the continuity of weak $ (p, q) $-Jordan centralizer on a semisimple Banach algebra and prove that every weak $ (p, q) $ -Jordan centralizer on a semisimple Banach algebra is a linear continuous operator. Moreover, these results are validated with actual examples.

References

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