<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Let M be an R-module and I be an ideal of R. We say that M is I-Rad-⊕-supplemented, provided for every submodule N of M, there exists a direct summand K of M such that M = N + K, N ∩ K ⊆ IK and N ∩ K Rad(K). The aim of this paper is to show new properties of I-Rad-⊕-supplemented modules. Especially, we show that any finite direct sum of I-Rad-⊕-supplemented modules is I-Rad-⊕-supplemented. We also prove that an R-module M is I-Rad-⊕-supplemented if and only if K and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2019-0017_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mi>M</mi><mi>K</mi></mfrac></mrow></math><tex-math>${M \over K}$</tex-math></alternatives></inline-formula> are I-Rad-⊕-supplemented for a fully invariant direct summand K of M. Finally, we determine the structure of I-Rad-⊕-supplemented modules over a discrete valuation ring.</p></abstract>

Let M be an R-module and I be an ideal of R. We say that M is I-Rad-⊕-supplemented, provided for every submodule N of M, there exists a direct summand K of M such that M = N + K, N ∩ K ⊆ IK and N ∩ K Rad(K). The aim of this paper is to show new properties of I-Rad-⊕-supplemented modules. Especially, we show that any finite direct sum of I-Rad-⊕-supplemented modules is I-Rad-⊕-supplemented. We also prove that an R-module M is I-Rad-⊕-supplemented if and only if K and MK${M \over K}$ are I-Rad-⊕-supplemented for a fully invariant direct summand K of M. Finally, we determine the structure of I-Rad-⊕-supplemented modules over a discrete valuation ring.

I-Rad-⊕-supplemented modules

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"I-Rad-⊕-supplemented modules"}

Let M be an R-module and I be an ideal of R. We say that M is I-Rad-⊕-supplemented, provided for every submodule N of M, there exists a direct summand K of M...

I-Rad-⊕-supplemented modules

Published Online: Aug 17, 2019

Page range: 224 - 233

Received: Jul 21, 2018

DOI: https://doi.org/10.2478/ausm-2019-0017

Keywords
supplement, Rad(-⊕-)supplemented module, I-Rad-⊕-supplemented module

© 2019 Burcu Nişancı Türkmen, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

I-Rad-⊕-supplemented modules

Published Online: Aug 17, 2019

Page range: 224 - 233

Received: Jul 21, 2018

DOI: https://doi.org/10.2478/ausm-2019-0017

Keywordssupplement, Rad(-⊕-)supplemented module, I-Rad-⊕-supplemented module

© 2019 Burcu Nişancı Türkmen, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
supplement, Rad(-⊕-)supplemented module, I-Rad-⊕-supplemented module