Volume 5, Issue 1 (Vol. 5, No. 1 2019)                   mmr 2019, 5(1): 67-78 | Back to browse issues page

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Mafi A, Arkian S. Topics on the Ratliff-Rush Closure of an Ideal. mmr 2019; 5 (1) :67-78
URL: http://mmr.khu.ac.ir/article-1-2674-en.html
1- , a_mafi@ipm.ir
Abstract:   (2226 Views)
Introduction
Let  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 discussion
The 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.
Conclusion
T‎he 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.
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