Mathematical Researches

fa مباحثی روی بستار راتلیف-راش یک ایده ال Topics on the Ratliff-Rush Closure of an Ideal جبر alg علمی پژوهشی بنیادی S بستار راتلیف-راشِ ایده‌ال نا صفر&lrm;  &lrm;، در حلقه‌ جابه‌جایی، یکدار و نوتری &lrm; &lrm;، به‌صورت &lrm;  است. در این مقاله، ویژگی‌های بستار راتلیف-راش یک ایده‌ال را با بستار صحیح آن مقایسه شده است. به‌علاوه ایده‌ال‌هایی، مانند   &lrm; که عمق حلقه‌ مدرج وابسته به آن مثبت است، را به‌عنوان ایده‌ال‌هایی که تمام توان‌های آن راتلیف-راش است&lrm; (ایده‌ال با بستار راتلیف-راشِ خود برابر است)، معرفی شده است. ضمن بیان این‌که هر ایده‌ال منظم یک تقلیل بستار راتلیف-راشِ خودش است، دستوری برای محاسبۀ بستار راتلیف-راش یک ایده‌ال از روی یک تقلیل آن ارائه شده است. این حقیقت که چندجمله‌ای هیلبرت یک ایده‌ال با چند جمله‌ای بستار راتلیف-راش آن یک‌سان است، از دیگر نتایج است.  Introduction Let <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="10" > be a Noetherian ring with unity and  <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="6" >  be a regular ideal of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif" width="10" >, that is, <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="6" > contains a nonzerodivisor. Let <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif" width="219" >. Then <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image008.gif" width="143" >. The :union: of this family, <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif" width="102" >, is an interesting ideal first studied by Ratliff and Rush in [15]. &lrm;  The Ratliff-Rush closure of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="6" > &lrm; is defined by&lrm; <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif" width="127" >. &lrm; A regular ideal <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="6" > for which &lrm;<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image014.gif" width="31" >&lrm; is called Ratliff-Rush ideal.&lrm;&rlm;&lrm; &lrm; The present paper, reviews some of the known properties, and compares properties of Ratliff-Rush closure of &lrm;&lrm;&lrm;an &lrm;ideal &lrm;with &lrm;its integral closure. We discuss some general properties of Ratliff-Rush ideals, consider the behaviour of the Ratliff-Rush property with respect to certain ideal and ring-theoretic operations, and try to indicate how one might determine whether a given ideal is Ratliff-Rush or not. &lrm;&lrm;&lrm;For a proper regular ideal <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="6" >, we denote by &lrm;<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image016.gif" width="28" >&lrm;&lrm;&lrm; the graded ring (or form ring) &lrm;<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image018.gif" width="46" >&lrm;&lrm;<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif" width="158" > . All powers of &lrm;<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif" width="9" > &lrm; are Ratliff-Rush ideals if and only if its positively graded ideal&lrm;<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image024.gif" width="172" >&lrm;&lrm;&lrm;contains a nonzerodivisor. &lrm;An ideal <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image026.gif" width="34" > is called a reduction of &lrm;<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif" width="9" >&lrm; if &lrm; <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image028.gif" width="66" >&lrm; for some <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image030.gif" width="40" > A reduction &lrm;&lrm;<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image032.gif" width="6" >&lrm;&lrm; is called a minimal reduction of &lrm;&lrm;<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="6" > if it does not properly contain a reduction of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="6" >. The least such <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image034.gif" width="12" >is called the reduction number of <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="6" > with respect to <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image032.gif" width="6" >&lrm;, and denoted by <img alt="" height="20" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image036.gif" width="29" >. A regular ideal I is always a reduction of its associated Ratliff-Rush ideal <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image038.gif" width="9" > <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image040.gif" width="3" >The Hilbert-Samuel function of &lrm;<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif" width="6" > is the numerical function that measures the growth of the length of &lrm;<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image042.gif" width="31" >&lrm; for all <img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image044.gif" width="24" >&lrm;. This function, <img alt="" height="21" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image046.gif" width="50" >&lrm;, is a polynomial in, for all large &lrm;&lrm;<img alt="" height="19" src="file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image048.gif" width="9" >&lrm;. &lrm;Finally, &lrm;in &lrm;t&rlm;&lrm;he &lrm;last &lrm;section, &lrm;we review some facts on Hilbert function of the Ratliff-Rush closure of an ideal. Ratliff and Rush [15, (2.4)] prove that every nonzero ideal in a Dedekind domain is concerning a Ratliff-Rush ideal. They also [15, Remark 2.5] express interest in classifying the Noetherian domains in which every nonzero ideal is a Ratliff-Rush ideal. This interest motivated the next sequence of results. A domain with this property has dimension at most one. Results and discussion The present paper compares properties of Ratliff-Rush closure of &lrm;&lrm;&lrm;an &lrm;ideal &lrm;with &lrm;its integral closure. Furthermore, ideals in which their associated graded ring has positive depth, are introduced as ideals for which all its powers are Ratliff-Rush ideals. While stating that each regular ideal is always a reduction of its associated Ratliff-Rush ideal, it expresses the command for calculating the Rutliff-Rush closure of an ideal by its reduction. This fact that Hilbert polynomial of an ideal has the same Hilbert polynomial its Ratliff-Rush closure, is from our other results. Conclusion T&lrm;he Ratliff-Rush closure of ideals is a good operation with respect to many properties, it carries information about associated primes of powers of ideals, about zerodivisors in the associated graded ring, preserves the Hilbert function of zero-dimensional ideals, etc. <a href="./files/site1/files/51/%D9%85%D8%A7%D9%81%DB%8C.pdf">./files/site1/files/51/%D9%85%D8%A7%D9%81%DB%8C.pdf</a> بستار راتلیف راش, بستار صحیح, چند جمله‌ای هیلبرت, عدد تقلیل Ratliff-Rush closure, Integral closure, Hilbert polynomial, Reduction number. 67 78 http://mmr.khu.ac.ir/browse.php?a_code=A-10-304-1&slc_lang=fa&sid=1 Amir Mafi امیر مافی a_mafi@ipm.ir 10031947532846002678 10031947532846002678 Yes دانشگاه کردستان ssh Arkian سید شهاب ارکیان 10031947532846002679 10031947532846002679 No