@article{Ansari2024, 
author = {Abu Zaid Ansari and Suad Alrehaili and Faiza Shujat},
title = {An extension of Herstein's theorem on Banach algebra},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {2},
pages = {4109-4117},
keywords = {semiprime ring, generalized left derivation, algebraic identities, Banach algebra},
url = {https://www.sciopen.com/article/10.3934/math.2024201},
doi = {10.3934/math.2024201},
abstract = {Let        A   be a    (  p  +  q  )  !-torsion free semiprime ring. We proved that if        H    ,      D    :      A    →      A   are two additive mappings fulfilling the algebraic identity    2      H    (      a          p      +      q        )  =      H    (      a    p    )      a    q    +      a    p        D    (      a    q    )  +      H    (      a    q    )      a    p    +      a    q        D    (      a    p    ) for all    a  ∈      A  , then        H   is a generalized derivation with        D   as an associated derivation on        A  . In addition to that, it is also proved in this article that              H        1   is a generalized left derivation associated with a left derivation    δ on        A   if they fulfilled the algebraic identity    2            H        1    (      a          p      +      q        )  =      a    p              H        1    (      a    q    )  +      a    q    δ  (      a    p    )  +      a    q              H        1    (      a    p    )  +      a    p    δ  (      a    q    ) for all    a  ∈      A  . Further, the legitimacy of these hypotheses is eventually demonstrated by examples and at last, an application of Banach algebra is presented.}
}