##### Abstract

In Commutative Algebra, there have been numerous signiﬁcant contributions for understanding growth of ideals. Research in this direction is mostly based on the concept of Hilbert-Samuel polynomials. In fact, given an m-primary ideal Q in a local ring (R,m), the length of the R–module R/Qn is equal to an integer-valued polynomial for suﬃciently large n. This polynomial has rational coeﬃcients which provide important parametersforstudyingthepair R and Q.Forinstance,ifthedegreeofthispolynomial is d, then dimR = d and d! times the leading coeﬃcient (called the multiplicity of Q on R) is a signiﬁcant invariant that carry much information about Q. In this thesis, we investigate reductions of ideals as a technique which has proved to have many applications in analytic theory of ideals and which enables us to study ideals by eliminating some superﬂuous elements. This thesis consists of four chapters. In the introductory chapter we give some important notions and results on commutative rings and their modules which will be used in the sequel. In the second chapter, we start with some fundamental properties of polynomials with rational coeﬃcients. Then we study Hilbert polynomials of Z-graded modules as wellastheconceptofmultiplicityarisingfromHilbertpolynomials.Moreover,weprove some important results related to Hilbert-Samuel polynomials and multiplicity which will be crucial for the remaining part of the thesis. In the third chapter, we begin with the deﬁnition and some basic properties of reductions of ideals and prove the existence of minimal reductions. Next we deﬁne the notion of analytic spread and give its relation with the extent of any minimal basis
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of minimal reductions. Finally in this chapter we deﬁne integral closure of ideals and explore its close relation with reductions. In the last chapter, ﬁrstly, in order to set the stage necessary for developping a parallel theory of reductions for modules, we give some important results on Rees valuations.Thenwedeﬁnereductionsofmodulesandprovesomefundamentaltheorems. Among them are two results by which we can explain reductions of modules in terms of symmetric and exterior algebras.