Mathematical Researches

fa مدول‎های با خاصیت اشتراک هم‌محض Modules with Copure Intersection Property جبر alg مقاله مستقل Original Manuscript در این مقاله، مدول‌هایی را که داری خاصیت اشتراک هم‌محض هستند را بررسی کرده و نتایج جدیدی در مورد این کلاس از مدول‌ها را به‌دست می‌آوریم.   Paper pages (271-276) Introduction &lrm;Throughout this paper&lrm;, <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="17" > will denote a commutative ring with&lrm; &lrm;identity and <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="13" > will denote the ring of integers. Let <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image006.gif" width="24" >be an <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="17" >-module&lrm;. A submodule <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="17" > of<img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="21" > is said to be pure if <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image014.gif" width="89" >for every ideal <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="13" >of <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="17" >. <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="21" > has the copure sum property if the sum of any two copure submodules is again copure&lrm;. <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="21" > is said to be a comultiplication module if for every submodule<img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="17" > of <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="21" > there exists an ideal <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="13" > of <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="17" >such that <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image022.gif" width="79" >. <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image024.gif" width="20" > satisfies the double annihilator conditions if for each ideal <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="13" > of <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="17" >, we have <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image027.gif" width="116" >. <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="21" >is said to be a strong comultiplication module if <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="21" > is a comultiplication R-module which satisfies the double annihilator conditions. A submodule <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image029.gif" width="20" > of <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image024.gif" width="20" > is called fully invariant if for every endomorphism <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image031.gif" width="76" > ,<img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image033.gif" width="70" >. In [5]&lrm;, &lrm;H&lrm;. &lrm;Ansari-Toroghy and F&lrm;. &lrm;Farshadifar introduced the dual notion of pure submodules (that is copure submodules) and investigated the first properties of this class of modules&lrm;. &lrm;A submodule <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image010.gif" width="17" > of <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="21" > is said to be copure if <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image036.gif" width="146" > for every ideal <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="13" >of <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image018.gif" width="17" >. Material and methods We say that an <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="17" >-module<img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image012.gif" width="21" >has the copure intersection property if the intersection of any two copure submodules is again copure&lrm;. In this paper, we investigate the modules with the copure intersection property and obtain some related results. Conclusion The following conclusions were drawn from this research. <ul> <li>Every distributive <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="17" >-module has the copure intersection property.</li> <li>Every strong comultiplication <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="17" >-module has the copure intersection property.</li> <li>An <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="17" >-module <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image006.gif" width="24" > has the copure intersection property if and only if for each ideal <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image016.gif" width="13" > of <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="17" >and copure submodules <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image040.gif" width="39" > of <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image006.gif" width="24" > we have</li> </ul> <img alt="" height="20" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image043.gif" width="301" > <ul> <li>If <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="17" > is a <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image046.gif" width="31" >, then an <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image008.gif" width="17" >-module <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image006.gif" width="24" > has the copure intersection property if and only if <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image049.gif" width="24" > has the copure sum property.</li> <li>Let <img alt="" height="26" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image051.gif" width="70" >, where <img alt="" height="24" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image053.gif" width="26" >is a submodule of <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image006.gif" width="24" >. If <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image006.gif" width="24" > has the copure intersection property, then each <img alt="" height="26" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image056.gif" width="24" > has the has the copure intersection property. The converse is true if each copure submodule of  <img alt="" height="17" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image006.gif" width="24" >is fully invariant.<a href="./files/site1/files/62/12Abstract.pdf">./files/site1/files/62/12Abstract.pdf</a></li> </ul>   زیرمدول محض, زیرمدول هم‌محض. مدول‌ با خاصیت اشتراک هم‌محض. Pure submodule, copure submodule, copure intersection property. 271 276 http://mmr.khu.ac.ir/browse.php?a_code=A-10-391-2&slc_lang=fa&sid=1 فرانک فرشادی فر f.farshadifar@gmail.com 10031947532846003760 10031947532846003760 Yes دانشگاه فرهنگیان، گروه ریاضی، تهران