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.

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