<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we introduce the concept of <italic>δss</italic>-supplemented modules and provide the various properties of these modules. In particular, we prove that a ring <italic>R</italic> is <italic>δss</italic>-supplemented as a left module if and only if <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2020-0041_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mi>R</mi><mrow><mi>S</mi><mi>o</mi><mi>c</mi><mrow><mo>(</mo><mrow><msub><mrow></mrow><mi>R</mi></msub><mi>R</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math><tex-math>{R \over {Soc\left( {_RR} \right)}}</tex-math></alternatives></inline-formula> is semisimple and idempotents lift to <italic>Soc</italic>(<italic>RR</italic>) if and only if every left <italic>R</italic>-module is <italic>δss</italic>-supplemented. We define projective <italic>δss</italic>-covers and prove the rings with the property that every (simple) module has a projective <italic>δss</italic>-cover are <italic>δss</italic>-supplemented. We also study on <italic>δss</italic>-supplement submodules.</abstract>

In this paper, we introduce the concept of δss-supplemented modules and provide the various properties of these modules. In particular, we prove that a ring R is δss-supplemented as a left module if and only if RSoc(RR){R \over {Soc\left( {_RR} \right)}} is semisimple and idempotents lift to Soc(RR) if and only if every left R-module is δss-supplemented. We define projective δss-covers and prove the rings with the property that every (simple) module has a projective δss-cover are δss-supplemented. We also study on δss-supplement submodules.

<italic>δss</italic>-supplemented modules and rings

Faculty of Art and Science, Department of Mathematics

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":{"italic":{"sub":"ss","content":"δ"},"content":"-supplemented modules and rings"}}

In this paper, we introduce the concept of δss-supplemented modules and provide the various properties of these modules. In particular, we prove that a ring R...

δ_ss-supplemented modules and rings

Published Online: Dec 28, 2020

Page range: 193 - 216

Received: Oct 08, 2019

Accepted: Jan 09, 2020

DOI: https://doi.org/10.2478/auom-2020-0041

Keywords
semisimple module, strongly -local module, -supplemented module, left -perfect ring, projective -cover

© 2020 Burcu Nişancı Türkmen et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

δss-supplemented modules and rings

Published Online: Dec 28, 2020

Page range: 193 - 216

Received: Oct 08, 2019

Accepted: Jan 09, 2020

DOI: https://doi.org/10.2478/auom-2020-0041

Keywordssemisimple module, strongly -local module, -supplemented module, left -perfect ring, projective -cover

© 2020 Burcu Nişancı Türkmen et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

δ_ss-supplemented modules and rings

Keywords
semisimple module, strongly -local module, -supplemented module, left -perfect ring, projective -cover