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Solutions of ๐‘“(๐‘ฅ)=๐‘“(๐‘Ž)+(๐‘…๐ฟ)โˆซโ‚^{๐‘ฅ}(๐‘“๐ป+๐‘“๐บ) for rings

Solutions of ๐‘“(๐‘ฅ)=๐‘“(๐‘Ž)+(๐‘…๐ฟ)โˆซโ‚^{๐‘ฅ}(๐‘“๐ป+๐‘“๐บ) for rings

We show that there is a solution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the equation <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x right-parenthesis equals f left-parenthesis a right-parenthesis plus left-parenthesis upper R upper L right-parenthesis integral Subscript a Superscript x Baseline left-parenthesis โ€ฆ