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W2068779782 abstract "A (commutative) ring<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$R$><mml:mi>R</mml:mi></mml:math>(with identity) is called<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$m$><mml:mi>m</mml:mi></mml:math>-linear (for an integer<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$m geqslant 2$><mml:mrow><mml:mi>m</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>) if<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$left( {a + b} right)^m = a^m + b^m $><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>a</mml:mi><mml:mo>+</mml:mo><mml:mi>b</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>m</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>a</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>b</mml:mi><mml:mi>m</mml:mi></mml:msup></mml:mrow></mml:math>for all<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$a$><mml:mi>a</mml:mi></mml:math>and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$b$><mml:mi>b</mml:mi></mml:math>in<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$R$><mml:mi>R</mml:mi></mml:math>. The<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$m$><mml:mi>m</mml:mi></mml:math>-linear reduced rings are characterized, with special attention to the finite case. A structure theorem reduces the study of<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$m$><mml:mi>m</mml:mi></mml:math>-linearity to the case of prime characteristic, for which the following result establishes an analogy with finite fields. For each prime<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$p$><mml:mi>p</mml:mi></mml:math>and integer<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$m geqslant 2$><mml:mrow><mml:mi>m</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math>which is not a power of<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$p$><mml:mi>p</mml:mi></mml:math>, there exists an integer<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$s geqslant m$><mml:mrow><mml:mi>s</mml:mi><mml:mo>≥</mml:mo><mml:mi>m</mml:mi></mml:mrow></mml:math>such that, for each ring<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$R$><mml:mi>R</mml:mi></mml:math>of characteristic<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$p$><mml:mi>p</mml:mi></mml:math>,<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$R$><mml:mi>R</mml:mi></mml:math>is<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$m$><mml:mi>m</mml:mi></mml:math>-linear if and only if<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$r^m = r^{p^s } $><mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>r</mml:mi><mml:mrow><mml:msup><mml:mi>p</mml:mi><mml:mi>s</mml:mi></mml:msup></mml:mrow></mml:msup></mml:mrow></mml:math>for each<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$r$><mml:mi>r</mml:mi></mml:math>in<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$R$><mml:mi>R</mml:mi></mml:math>. Additional results and examples are given." @default.
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W2068779782 date "1992-01-01" @default.
W2068779782 modified "2023-09-25" @default.
W2068779782 title "Commutative rings with homomorphic power functions" @default.
W2068779782 doi "https://doi.org/10.1155/s0161171292000103" @default.
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