<abstract xmlns="http://www.w3.org/1999/xhtml"><p>In this paper we introduce <italic>β</italic>** relation on the lattice of submodules of a module <italic>M</italic>. We say that submodules <italic>X, Y</italic> of <italic>M</italic> are <italic>β</italic>** equivalent, <italic>X β** Y</italic>, if and only if 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/auom-2014-0059_i1.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mfrac><mrow><mi>X</mi><mo>+</mo><mi>Y</mi></mrow><mi>X</mi></mfrac><mo>⊆</mo><mfrac><mrow><mi mathvariant="italic">Rad</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo>+</mo><mi>X</mi></mrow><mi>X</mi></mfrac></math><tex-math>${{X + Y} \over X} \subseteq {{Rad(M) + X} \over X}$</tex-math></alternatives></inline-formula>
 and 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/auom-2014-0059_i2.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mfrac><mrow><mi>X</mi><mo>+</mo><mi>Y</mi></mrow><mi>Y</mi></mfrac><mo>⊆</mo><mfrac><mrow><mi mathvariant="italic">Rad</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo>+</mo><mi>Y</mi></mrow><mi>Y</mi></mfrac></math><tex-math>${{X + Y} \over Y} \subseteq {{Rad(M) + Y} \over Y}$</tex-math></alternatives></inline-formula>
. We show that the <italic>β**</italic> relation is an equivalence relation. We also investigate some general properties of this relation. This relation is used to define and study classes of Goldie-<italic>Rad</italic>-supplemented and <italic>Rad-H</italic>-supplemented modules. We prove <italic>M</italic> = <italic>A</italic> ⊕ <italic>B</italic> is Goldie-<italic>Rad-</italic>supplemented if and only if <italic>A</italic> and <italic>B</italic> are Goldie-<italic>Rad</italic>-supplemented.</p></abstract>

In this paper we introduce β** relation on the lattice of submodules of a module M. We say that submodules X, Y of M are β** equivalent, X β** Y, if and only if 
X+YX⊆Rad(M)+XX${{X + Y} \over X} \subseteq {{Rad(M) + X} \over X}$
 and 
X+YY⊆Rad(M)+YY${{X + Y} \over Y} \subseteq {{Rad(M) + Y} \over Y}$
. We show that the β** relation is an equivalence relation. We also investigate some general properties of this relation. This relation is used to define and study classes of Goldie-Rad-supplemented and Rad-H-supplemented modules. We prove M = A ⊕ B is Goldie-Rad-supplemented if and only if A and B are Goldie-Rad-supplemented.

Goldie-<italic>Rad</italic>-Supplemented Modules

Department of Mathematics, University of Mazandaran, Babolsar, Iran.

Department of Mathematics, Hacettepe University, Beytepe Campus, 06532 Ankara, Turkey.

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

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

{"article-title":{"italic":"Rad","content":["Goldie-","-Supplemented Modules"]}}

In this paper we introduce β** relation on the lattice of submodules of a module M. We say that submodules X, Y of M are β** equivalent, X β** Y, if and only...

Goldie-Rad-Supplemented Modules

Published Online: Dec 22, 2015

Page range: 205 - 218

Received: Jun 01, 2013

Accepted: Oct 01, 2013

DOI: https://doi.org/10.2478/auom-2014-0059

Keywords
-supplemented module, -supplemented module, Goldie*-(lifting)supplemented module, Goldie--supplemented module

© 2014 Yahya Talebi et al., published by De Gruyter Open

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

Goldie-Rad-Supplemented Modules

Published Online: Dec 22, 2015

Page range: 205 - 218

Received: Jun 01, 2013

Accepted: Oct 01, 2013

DOI: https://doi.org/10.2478/auom-2014-0059

Keywords-supplemented module, -supplemented module, Goldie*-(lifting)supplemented module, Goldie--supplemented module

© 2014 Yahya Talebi et al., published by De Gruyter Open

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

Keywords
-supplemented module, -supplemented module, Goldie*-(lifting)supplemented module, Goldie--supplemented module