Topics on the Ratliff-Rush Closure of an Ideal

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. ./files/site1/files/51/%D9%85%D8%A7%D9%81%DB%8C.pdf