Mathematical Researches

fa مدول های گلدی مکمل پذیر در ارتباط با پیش رادیکال Goldie supplemented modules with respect to a preradical جبر alg مقاله مستقل Original Manuscript  فرض کنید <m:omath><m:r>τ</m:r></m:omath> یک پیش رادیکال باشد. در این مقاله رابطه <m:omath><m:bar><m:barpr><m:pos m:val="top"><m:ctrlpr></m:ctrlpr></m:pos></m:barpr><m:e><m:ssubsup><m:ssubsuppr><m:ctrlpr></m:ctrlpr></m:ssubsuppr><m:e><m:r>β</m:r></m:e><m:sub><m:r>τ</m:r></m:sub><m:sup><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>*</m:r></m:sup></m:ssubsup></m:e></m:bar></m:omath> روی زیرمدول های یک مدول را تعریف و بررسی می کنیم. نشان می دهیم که رابطه <m:omath><m:bar><m:barpr><m:pos m:val="top"><m:ctrlpr></m:ctrlpr></m:pos></m:barpr><m:e><m:ssubsup><m:ssubsuppr><m:ctrlpr></m:ctrlpr></m:ssubsuppr><m:e><m:r>β</m:r></m:e><m:sub><m:r>τ</m:r></m:sub><m:sup><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>*</m:r></m:sup></m:ssubsup></m:e></m:bar></m:omath> یک رابطه هم ارزی است. این رابطه را برای تعریف مدول های گلدی <m:omath><m:r>τ</m:r></m:omath>-مکمل پذیر و مدول های به طور قوی   <m:omath><m:r>τ</m:r></m:omath> -H&lrm; مکمل پذیر  و  بررسی ویژگی های آنها  بکار می بریم. Introduction Throughout this paper R will denote an associative ring with identity, M a unitary right R-module. A functor <m:omath><m:r>τ</m:r></m:omath>from the category of the right R-modules Mod-R to itself is called a preradicalif it satisfies the following properties: (i) <m:omath><m:r>τ</m:r><m:r>(</m:r><m:r>M</m:r><m:r>)</m:r></m:omath>is a submodule of M, for every R-module M; (ii) If <m:omath><m:r>f</m:r><m:r>:</m:r><m:r>M</m:r><m:r>'→</m:r><m:r>M</m:r></m:omath>is an R-module homomorphism, then <m:omath><m:r>f</m:r><m:r>(</m:r><m:r>τ</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>M</m:r></m:e><m:sup><m:r>'</m:r></m:sup></m:ssup></m:e></m:d><m:r>≤</m:r><m:r>τ</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>M</m:r></m:e></m:d></m:omath> and <m:omath><m:r>τ</m:r><m:r>(</m:r><m:r>f</m:r><m:r>)</m:r></m:omath> is the restriction of <m:omath><m:r>f</m:r></m:omath>to <m:omath><m:r>τ</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>M</m:r></m:e><m:sup><m:r>'</m:r></m:sup></m:ssup></m:e></m:d></m:omath>. For example Rad, Soc, and <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>Z</m:r></m:e><m:sub><m:r>M</m:r></m:sub></m:ssub></m:omath>are preradicals. Note that if K is a summand of M, then <m:omath><m:r>K</m:r><m:r>∩</m:r><m:r>τ</m:r><m:r>(</m:r><m:r>M</m:r><m:r>)=</m:r><m:r>τ</m:r><m:r>(</m:r><m:r>K</m:r><m:r>)</m:r></m:omath>. For a preradical <m:omath><m:r> </m:r><m:r>τ</m:r></m:omath>, Al-Takhman, Lomp and Wisbauer defined and studied the concept of <m:omath><m:r>τ</m:r></m:omath>-lifting and <m:omath><m:r>τ</m:r></m:omath>-supplemented modules. A module M is called <m:omath><m:r>τ</m:r></m:omath>-lifting if every submodule N of M has a decomposition <m:omath><m:r>N</m:r><m:r> =</m:r><m:r>A</m:r><m:r>&oplus; </m:r><m:r>B</m:r></m:omath> such that A is a direct summand of M and <m:omath><m:r>B</m:r><m:r>&sube;</m:r><m:r>τ</m:r><m:r>(</m:r><m:r>M</m:r><m:r>).</m:r></m:omath>A submodule <m:omath><m:r>K</m:r><m:r>&sube; </m:r><m:r>M</m:r></m:omath> is called <m:omath><m:r>τ</m:r><m:r>-</m:r></m:omath>supplement (weak <m:omath><m:r>τ</m:r></m:omath>-supplement) provided there exists some <m:omath><m:r>U</m:r><m:r>&sube; </m:r><m:r>M</m:r></m:omath>such that<m:omath><m:r>M</m:r><m:r>=</m:r><m:r>U</m:r><m:r>+</m:r><m:r>K</m:r></m:omath> and <m:omathpara><m:omath><m:r>U</m:r><m:r>∩ </m:r><m:r>K</m:r><m:r>&sube;</m:r><m:r>τ</m:r><m:r>(</m:r><m:r>K</m:r><m:r>)    (</m:r><m:r>U</m:r><m:r>∩ </m:r><m:r>K</m:r><m:r>&sube;</m:r><m:r>τ</m:r><m:r>(</m:r><m:r>M</m:r><m:r>)).</m:r></m:omath></m:omathpara> M is called <m:omath><m:r>τ</m:r></m:omath>-supplemented (weakly <m:omath><m:r>τ</m:r></m:omath>-supplemented) if each of its submodules <m:omath><m:r>τ</m:r></m:omath>-supplement (weak <m:omath><m:r>τ</m:r></m:omath>-supplement) in M.Talebi, Moniri Hamzekolaei and Keskin-Tütüncü, defined <m:omath><m:r>τ</m:r></m:omath>-H-supplemented modules.  A module M is called<m:omath><m:r>τ</m:r></m:omath>-H-supplemented if for every <m:omath><m:r>N</m:r><m:r>≤ </m:r><m:r>M</m:r></m:omath> there exists a direct summand D of Msuch that <m:omath><m:r>(</m:r><m:r>N</m:r><m:r>+</m:r><m:r>D</m:r><m:r>)/</m:r><m:r>N</m:r><m:r>     &sube;</m:r><m:r>τ</m:r><m:r>(</m:r><m:r>M</m:r><m:r>/</m:r><m:r>N</m:r><m:r>)</m:r></m:omath>and<m:omath><m:r>(</m:r><m:r>N</m:r><m:r>+</m:r><m:r>D</m:r><m:r>)/</m:r><m:r>D</m:r><m:r>&sube;</m:r><m:r>τ</m:r><m:r>(</m:r><m:r>M</m:r><m:r>/</m:r><m:r>D</m:r><m:r>).</m:r></m:omath> The <m:omath><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>β</m:r></m:e><m:sup><m:r>*</m:r></m:sup></m:ssup></m:omath> relation is introduced and investigated by Birkenmeier, Takil Mutlu, Nebiyev, Sokmez and Tercan.  Let X and Y be submodules of M. X and Yare <m:omath><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>β</m:r></m:e><m:sup><m:r>*</m:r></m:sup></m:ssup></m:omath> equivalent, X<m:omath><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>β</m:r></m:e><m:sup><m:r>*</m:r></m:sup></m:ssup></m:omath>Y, provided <m:omath><m:f><m:fpr><m:ctrlpr></m:ctrlpr></m:fpr><m:num><m:r>X</m:r><m:r>+</m:r><m:r>Y</m:r></m:num><m:den><m:r>X</m:r></m:den></m:f><m:r>≪</m:r><m:f><m:fpr><m:ctrlpr></m:ctrlpr></m:fpr><m:num><m:r>M</m:r></m:num><m:den><m:r>X</m:r></m:den></m:f><m:r> </m:r><m:r>and</m:r><m:f><m:fpr><m:ctrlpr></m:ctrlpr></m:fpr><m:num><m:r>X</m:r><m:r>+</m:r><m:r>Y</m:r></m:num><m:den><m:r>Y</m:r></m:den></m:f><m:r>≪</m:r><m:f><m:fpr><m:ctrlpr></m:ctrlpr></m:fpr><m:num><m:r>M</m:r></m:num><m:den><m:r>Y</m:r></m:den></m:f><m:r>.</m:r></m:omath> Based on definition of <m:omath><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>β</m:r></m:e><m:sup><m:r>*</m:r></m:sup></m:ssup></m:omath> relation they introduced two new classes of modules namely <m:omath><m:r>Goldi</m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>e</m:r></m:e><m:sup><m:r>*</m:r></m:sup></m:ssup></m:omath>-lifting and <m:omath><m:r>Goldi</m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>e</m:r></m:e><m:sup><m:r>*</m:r></m:sup></m:ssup><m:r>-</m:r></m:omath>supplemented.They showed that two concept of H-supplemented modules and <m:omath><m:r>Goldi</m:r><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>e</m:r></m:e><m:sup><m:r>*</m:r></m:sup></m:ssup><m:r>-</m:r></m:omath>lifting modules coincide. In this paper, we introduce Goldie<m:omath><m:r>-</m:r><m:r>τ</m:r><m:r>-</m:r></m:omath>supplemented and strongly <m:omath><m:r>τ</m:r></m:omath>-H-supplemented modules. We introduce the<m:omath><m:acc><m:accpr><m:chr m:val="̅"><m:ctrlpr></m:ctrlpr></m:chr></m:accpr><m:e><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r> </m:r><m:r>β</m:r></m:e><m:sup><m:r>*</m:r></m:sup></m:ssup></m:e></m:acc></m:omath> relation. We investigate some properties of this relation and prove that this relation is an equivalence relation. We define Goldie<m:omath><m:r>-</m:r><m:r>τ</m:r><m:r>-</m:r></m:omath>supplemented and strongly <m:omath><m:r>τ</m:r></m:omath>-H-supplemented modules. We call a module M, Goldie<m:omath><m:r>-</m:r><a name="_Hlk62422701">τ</a><m:r>-</m:r></m:omath>supplemented (strongly <a name="_Hlk62422787"></a><m:omath><m:r>τ</m:r></m:omath>-H-supplemented) if for any submodule N of M,there exists a <m:omath><m:r>τ</m:r></m:omath>-supplement submodule (a direct summand) D of M such thatN<m:omath><m:acc><m:accpr><m:chr m:val="̅"><m:ctrlpr></m:ctrlpr></m:chr></m:accpr><m:e><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>β</m:r></m:e><m:sup><m:r>*</m:r></m:sup></m:ssup></m:e></m:acc></m:omath>D. Clearly every strongly <m:omath><m:r>τ</m:r></m:omath>-H-supplemented module is Goldie<m:omath><m:r> </m:r><m:r>τ</m:r></m:omath> -supplemented. We will study direct sums of Goldie<m:omath><m:r> </m:r><m:r>τ</m:r></m:omath> -H-supplemented modules. Let <m:omath><m:r>M</m:r><m:r> = </m:r><m:r>A</m:r><m:r>&oplus; </m:r><m:r>B</m:r></m:omath> be a distributive module. Then M is Goldie<m:omath><m:r> </m:r><m:r>τ</m:r></m:omath> -upplemented(strongly <m:omath><m:r>τ</m:r></m:omath> -H-supplemented) if and only if A and B are Goldie<m:omath><m:r> </m:r><m:r>τ</m:r></m:omath> -supplemented(strongly <m:omath><m:r>τ</m:r></m:omath> -H-supplemented. We also define <m:omath><m:r>τ</m:r></m:omath> -H-cofinitely supplemented modules and obtain some conditions which under the factor module of a <m:omath><m:r>τ</m:r></m:omath> -H-cofinitely supplemented module will be <m:omath><m:r>τ</m:r></m:omath> -H-cofinitely supplemented. Material and methods In this paper, first we define and investigate the<m:omath><m:bar><m:barpr><m:pos m:val="top"><m:ctrlpr></m:ctrlpr></m:pos></m:barpr><m:e><m:ssubsup><m:ssubsuppr><m:ctrlpr></m:ctrlpr></m:ssubsuppr><m:e><m:r> </m:r><m:r>β</m:r></m:e><m:sub><m:r>τ</m:r></m:sub><m:sup><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>*</m:r></m:sup></m:ssubsup></m:e></m:bar></m:omath>  relation on submodules of a module.  We show that the<m:omath><m:bar><m:barpr><m:pos m:val="top"><m:ctrlpr></m:ctrlpr></m:pos></m:barpr><m:e><m:ssubsup><m:ssubsuppr><m:ctrlpr></m:ctrlpr></m:ssubsuppr><m:e><m:r> </m:r><m:r>β</m:r></m:e><m:sub><m:r>τ</m:r></m:sub><m:sup><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>*</m:r></m:sup></m:ssubsup></m:e></m:bar></m:omath>relation is an equivalence relation. We apply this relation to define and investigate the classes of Goldie-<m:omath><m:r>τ</m:r></m:omath> -supplemented modules and strongly<m:omath><m:r>τ</m:r></m:omath>-H-supplemented modules. Results and discussion We investigate some properties of this relation and prove that this relation is an equivalence relation. We define Goldie<m:omath><m:r>-</m:r><m:r>τ</m:r><m:r>-</m:r></m:omath>supplemented and strongly <m:omath><m:r>τ</m:r></m:omath>-H- supplemented modules. We call a module M, Goldie<m:omath><m:r>-</m:r><m:r>τ</m:r><m:r>-</m:r></m:omath>supplemented (strongly <m:omath><m:r>τ</m:r></m:omath> -H-supplemented) if for any submodule N of M, there exists a <m:omath><m:r>τ</m:r></m:omath>-supplement submodule (a direct summand) D of M such that N<m:omath><m:acc><m:accpr><m:chr m:val="̅"><m:ctrlpr></m:ctrlpr></m:chr></m:accpr><m:e><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r>β</m:r></m:e><m:sup><m:r>*</m:r></m:sup></m:ssup></m:e></m:acc></m:omath> D. Clearly every strongly <m:omath><m:r>τ</m:r></m:omath> -H-supplemented module is Goldie<m:omath><m:r> </m:r><m:r>τ</m:r></m:omath> -supplemented. We will study direct sums of Goldie<m:omath><m:r> </m:r><m:r>τ</m:r></m:omath> -H-supplemented modules. Let <m:omath><m:r>M</m:r><m:r> = </m:r><m:r>A</m:r><m:r>&oplus; </m:r><m:r>B</m:r></m:omath> be a distributive module. Then M is Goldie<m:omath><m:r> </m:r><m:r>τ</m:r></m:omath> -upplemented (strongly <m:omath><m:r>τ</m:r></m:omath> -H-supplemented) if and only if A and B are Goldie<m:omath><m:r> </m:r><m:r>τ</m:r></m:omath> -supplemented (strongly <m:omath><m:r>τ</m:r></m:omath> -H-supplemented). We also define <m:omath><m:r>τ</m:r></m:omath> -H-cofinitely supplemented modules and obtain some conditions which under the factor module of a <m:omath><m:r>τ</m:r></m:omath> -H-cofinitely supplemented module will be <m:omath><m:r>τ</m:r></m:omath> -H-cofinitely supplemented. Conclusion The following conclusions were drawn from this research. <ul> <li class="CxSpMiddle">Let <m:omath><m:r>M</m:r><m:r> = </m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>M</m:r></m:e><m:sub><m:r>1</m:r></m:sub></m:ssub><m:r>&oplus; </m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>M</m:r></m:e><m:sub><m:r>2</m:r></m:sub></m:ssub></m:omath>, where <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>M</m:r></m:e><m:sub><m:r>1</m:r></m:sub></m:ssub></m:omath> is a fully invariant submodule of M. Assume that <m:omath><m:r>τ</m:r></m:omath> is a cohereditary preradical. If M is strongly <m:omath><m:r>τ</m:r></m:omath>-H-supplemented, then <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>M</m:r></m:e><m:sub><m:r>1</m:r></m:sub></m:ssub><m:r> </m:r><m:r>and</m:r><m:r> </m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>M</m:r></m:e><m:sub><m:r>2</m:r></m:sub></m:ssub></m:omath> are strongly <m:omath><m:r>τ</m:r><m:r>-</m:r></m:omath>H-supplemented.</li> </ul> <ul> <li class="CxSpMiddle">Let M be an <m:omath><m:r>τ</m:r></m:omath>-H-cofinitely supplemented module and let <m:omath><m:r>N</m:r><m:r>≤ </m:r><m:r>M</m:r></m:omath> be a submodule. Suppose that for every direct summand K of M, there exists a submodule L of M such that <m:omath><m:r>N</m:r><m:r>&sube; </m:r><m:r>L</m:r><m:r>&sube; </m:r><m:r>K</m:r><m:r>+</m:r><m:r>N</m:r></m:omath>, L/N is a direct summand of M/N and<m:omath><m:f><m:fpr><m:ctrlpr></m:ctrlpr></m:fpr><m:num><m:f><m:fpr><m:ctrlpr></m:ctrlpr></m:fpr><m:num><m:r>K</m:r><m:r>+</m:r><m:r>N</m:r></m:num><m:den><m:r>N</m:r></m:den></m:f></m:num><m:den><m:r>L</m:r><m:r>/</m:r><m:r>N</m:r></m:den></m:f><m:r>&sube;</m:r><m:f><m:fpr><m:ctrlpr></m:ctrlpr></m:fpr><m:num><m:r>τ</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:f><m:fpr><m:ctrlpr></m:ctrlpr></m:fpr><m:num><m:r>M</m:r></m:num><m:den><m:r>N</m:r></m:den></m:f></m:e></m:d><m:r>+</m:r><m:f><m:fpr><m:ctrlpr></m:ctrlpr></m:fpr><m:num><m:r>L</m:r></m:num><m:den><m:r>N</m:r></m:den></m:f></m:num><m:den><m:r>L</m:r><m:r>/</m:r><m:r>N</m:r></m:den></m:f><m:r>.</m:r></m:omath> Then M/N is <m:omath><m:r>τ</m:r></m:omath>-H-cofinitelysupplemented.</li> </ul> <ul> <li style="margin-top:8px; margin-left:8px">Let M be a module and let <m:omath><m:r>N</m:r><m:r>≤ </m:r><m:r>M</m:r></m:omath> be a submodule such that for each decomposition <m:omath><m:r>M</m:r><m:r> = </m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>M</m:r></m:e><m:sub><m:r>1</m:r></m:sub></m:ssub><m:r>&oplus; </m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>M</m:r></m:e><m:sub><m:r>2</m:r></m:sub></m:ssub></m:omath> we have <m:omath><m:r>N</m:r><m:r> = </m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>N</m:r><m:r>∩ </m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>M</m:r></m:e><m:sub><m:r>1</m:r></m:sub></m:ssub></m:e></m:d><m:r>&oplus; (</m:r><m:r>N</m:r><m:r>∩ </m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>M</m:r></m:e><m:sub><m:r>2</m:r></m:sub></m:ssub><m:r>).</m:r></m:omath> If M is <m:omath><m:r>τ</m:r></m:omath>-H-cofinitely supplemented, then M/N is <m:omath><m:r>τ</m:r></m:omath>-H-cofinitely supplemented.</li> </ul> مدول های H -مکمل پذیر, مدول های گلدی τ -مکمل پذیر, مدول های به طور قوی ‌ τ ‎- H- ‎مکمل پذیر‌. H-supplemented module, strongly τ -H-supplemented module, Goldie- τ -supplemented module. 1 14 http://mmr.khu.ac.ir/browse.php?a_code=A-10-1144-1&slc_lang=fa&sid=1 Tayyebeh Amouzegar طیبه آموزگار t.amoozegar@yahoo.com 10031947532846005675 10031947532846005675 Yes