مباحثی روی بستار راتلیف-راش یک ایده ال

بستار راتلیف-راشِ ایده‌ال نا صفر‎  ‎، در حلقه‌ جابه‌جایی، یکدار و نوتری ‎ ‎، به‌صورت ‎  است. در این مقاله، ویژگی‌های بستار راتلیف-راش یک ایده‌ال را با بستار صحیح آن مقایسه شده است. به‌علاوه ایده‌ال‌هایی، مانند   ‎ که عمق حلقه‌ مدرج وابسته به آن مثبت است، را به‌عنوان ایده‌ال‌هایی که تمام توان‌های آن راتلیف-راش است‎ (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. 

topics on the ratliff-rush closure of an ideal

introductionlet  be a noetherian ring with unity and    be a regular ideal of , that is,  contains a nonzerodivisor. let . then . the :union: of this family, , is an interesting ideal first studied by ratliff and rush in [15]. ‎  the ratliff-rush closure of  ‎ is defined by‎ . ‎ a regular ideal  for which ‎‎ is called ratliff-rush ideal.‎‏‎ ‎the present paper, reviews some of the known properties, and compares properties of ratliff-rush closure of ‎‎‎an ‎ideal ‎with ‎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.‎‎‎for a proper regular ideal , we denote by ‎‎‎‎ the graded ring (or form ring) ‎‎‎ . all powers of ‎ ‎ are ratliff-rush ideals if and only if its positively graded ideal‎‎‎‎contains a nonzerodivisor. ‎an ideal  is called a reduction of ‎‎ if ‎ ‎ for some  a reduction ‎‎‎‎ is called a minimal reduction of ‎‎ if it does not properly contain a reduction of . the least such is called the reduction number of  with respect to ‎, and denoted by . a regular ideal i is always a reduction of its associated ratliff-rush ideal the hilbert-samuel function of ‎ is the numerical function that measures the growth of the length of ‎‎ for all ‎. this function, ‎, is a polynomial in, for all large ‎‎‎. ‎finally, ‎in ‎t‏‎he ‎last ‎section, ‎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 discussionthe present paper compares properties of ratliff-rush closure of ‎‎‎an ‎ideal ‎with ‎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.conclusiont‎he 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../files/site1/files/51/%d9%85%d8%a7%d9%81%db%8c.pdf