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- W2029221652 abstract "Çalışıcı and Türkmen called a module <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M2><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:math> generalized <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M3><mml:mrow><mml:mo>⊕</mml:mo></mml:mrow></mml:math>-supplemented if every submodule has a generalized supplement that is a direct summand of <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M4><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:math>. Motivated by this, it is natural to introduce another notion that we called generalized <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M5><mml:mrow><mml:mo>⊕</mml:mo></mml:mrow></mml:math>-radical supplemented modules as a proper generalization of generalized <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M6><mml:mrow><mml:mo>⊕</mml:mo></mml:mrow></mml:math>-supplemented modules. In this paper, we obtain various properties of generalized <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M7><mml:mrow><mml:mo>⊕</mml:mo></mml:mrow></mml:math>-radical supplemented modules. We show that the class of generalized <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M8><mml:mrow><mml:mo>⊕</mml:mo></mml:mrow></mml:math>-radical supplemented modules is closed under finite direct sums. We attain that over a Dedekind domain a module <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M9><mml:mrow><mml:mi>M</mml:mi></mml:mrow></mml:math> is generalized <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M10><mml:mrow><mml:mo>⊕</mml:mo></mml:mrow></mml:math>-radical supplemented if and only if <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M11><mml:mrow><mml:mrow><mml:mi>M</mml:mi><mml:mo>/</mml:mo><mml:mrow><mml:mi>P</mml:mi><mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy=false>)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow></mml:math> is generalized <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M12><mml:mrow><mml:mo>⊕</mml:mo></mml:mrow></mml:math>-radical supplemented. We completely determine the structure of these modules over left <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M13><mml:mrow><mml:mi>V</mml:mi></mml:mrow></mml:math>-rings. Moreover, we characterize semiperfect rings via generalized <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M14><mml:mrow><mml:mo>⊕</mml:mo></mml:mrow></mml:math>-radical supplemented modules." @default.
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- W2029221652 date "2014-01-14" @default.
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- W2029221652 title "Generalized ⊕-Radical Supplemented Modules" @default.
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- W2029221652 doi "https://doi.org/10.1155/2014/603851" @default.
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