Mathematical Researches

fa توسیع های لَخت مدولی، زیرمجموعه های ضربی بستۀ حافظ زیرمدول‌های دوری و تجزیه در مدول ها Inert Module Extensions, Multiplicatively Closed Subsets Conserving Cyclic Submodules and Factorization in Modules جبر alg مقاله مستقل Original Manuscript فرض کنید  یک حلقه جابه‌جایی یکدار باشد،  یک -مدول یکانی و  یک زیرمجموعۀ ضربی بسته . گوییم  حافظ زیرمدولهای دوری  است، هرگاه انقباض هر زیرمدول دوری  به  یک زیرمدول دوری باشد. در این مقاله ضمن ارائه یک شرط معادل برای حافظ زیرمدولهای دوری بودن، به بررسی ارتباط بین خواص تجزیهای  و  زمانی که  حافظ زیرمدولهای دوری  است میپردازیم. به‌علاوه مفهوم یک توسیع مدولی لَخت و لَخت ضعیف را معرفی کرده و اگر  یک زیرمجموعۀ ضربی بسته  شامل  باشد،  و  یک توسیع -لَخت ضعیف باشد، تجزیه نسبت به  در  را به تجزیۀ نسبت به  در  ارتباط میدهیم. هم‌چنین نشان میدهیم اگر  فارغ از تاب و  حافظ زیرمدولهای دوری  باشد، آن‌گاه  شکافنده  است و  یک توسیع لخت است.    Introduction Suppose that <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="10" > is a commutative ring with identity, <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="13" > is a unitary <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="10" >-module and <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif" width="8" > is a multiplicatively closed subset of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="10" >.  Factorization theory in commutative rings, which has a long history, still gets the attention of many researchers. Although at first, the focus of this theory was factorization properties of elements in integral domains, in the late nineties the theory was generalized to commutative rings with zero-divisors and to modules. Also recently, the factorization properties of an element of a module with respect to a multiplicatively closed subset of the ring has been investigated. It has been shown that using these general views, one can derive new results and insights on the classic case of factorization theory in integral domains. An important and attractive question in this theory is understanding how factorization properties of a ring or a module behave under localization. In particular, Anderson, et al in 1992 showed that if <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="10" > is an integral domain and every principal ideal of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image008.gif" width="15" > contracts to a principal ideal of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="10" >, then there are strong relations between factorization properties of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="10" > and <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image008.gif" width="15" >. In the same paper and also in another paper by Aḡargün, et al in 2001 the concepts of inert and weakly inert extensions of rings were introduced and the relation of factorization properties of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="10" > and <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image008.gif" width="15" >, under the assumption that <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif" width="44" > is (weakly) inert, is studied. In this paper, we generalize the above concepts to modules and with respect to a multiplicatively closed subset. Then we utilize them to relate the factorization properties of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="13" > and <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif" width="18" >.  Material and methods We first recall the concepts of factorization theory in modules with respect to a multiplicatively closed subset of the ring. Then, we define multiplicatively closed subsets conserving cyclic submodules of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="13" > and say that <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif" width="8" > conserves cyclic submodules of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="13" >, when the contraction of every cyclic submodule of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif" width="18" > to <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="13" > is a cyclic submodule. We present conditions on <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif" width="8" > equivalent to conserving cyclic submodules of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="13" > and study how factorization properties of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="13" > is related to those of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif" width="18" >, when <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif" width="8" > coserves cyclic submodules of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image014.gif" width="16" > Finally we present generalizations of inert and weakly inert extensions of rings to modules and investigate how factorization properties behave under localization with respect to <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif" width="8" >, when <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image016.gif" width="50" > is inert or weakly inert.   Results and discussion We show that if <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="10" > is an integral domain, <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="13" > is torsion-free and <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif" width="8" > conserves cyclic submodules of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="13" >, then <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image018.gif" width="11" > splits <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="13" > (as defined by Nikseresht in 2018) and hence factorization properties of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="13" > and those of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif" width="18" > are strongly related. Also we show that under certain conditions, the converse is also true, that is, if <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif" width="8" > splits <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="13" >, then <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif" width="8" > conserves cyclic submodules of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="13" >. Suppose that <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="9" > is a multiplicatively closed subset of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="10" > containing <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif" width="8" > and <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif" width="64" >. We show that if <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image016.gif" width="50" > is a <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image024.gif" width="39" >-weakly inert extension, then there is a strong relationship between <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="9" >- factorization properties of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="13" > and <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image026.gif" width="12" >-factorization properties of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif" width="18" >. For example, under the above assumptions, if <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="13" > is also torsion-free and has unique (or finite or bounded) factorization with respect to <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="9" >, then <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif" width="18" > has the same property with respect to <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image026.gif" width="12" >. Conclusion In this paper, the concepts of a multiplicatively closed subset conserving cyclic submodules and inert and weakly inert extensions of modules are introduced and utilized to derive relations between factorization properties of a module <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="13" > and those of its localization <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif" width="18" >. It is seen that many properties can be delivered from one to another when <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif" width="8" > conserves cyclic submodules or when <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image016.gif" width="50" > is a weakly inert extension, especially when <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="10" > is an integral domain and <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="13" > is torsion-free. <a href="./files/site1/files/51/%D9%86%DB%8C%DA%A9_%D8%B3%D8%B1%D8%B4%D8%AA.pdf">./files/site1/files/51/%D9%86%DB%8C%DA%A9_%D8%B3%D8%B1%D8%B4%D8%AA.pdf</a> زیرمجموعه های ضربی بسته حافظ زیرمدول های دوری, توسیع لَخت, مدول اتمی, مدول تجزیۀ یکتا. Multiplicatively closed subsets conserving cyclic submodules, Inert extension, Atomic module, Unique factorization module 107 120 http://mmr.khu.ac.ir/browse.php?a_code=A-10-171-1&slc_lang=fa&sid=1 Ashkan Nikseresht اشکان نیک سرشت ashkan_nikseresht@yahoo.com 10031947532846002686 10031947532846002686 Yes Shiraz University دانشگاه شیراز، دانشکدۀ علوم، گروه ریاضی