Bounds on the number of generators of a module

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The aim of this work is to present the Forster-Swan bound on the number of generators of a module. It is not our intention to present a novel finding or research discovery. Instead, we will develop the commutative algebra necessary to understand its proof, as well as the significance of the result. As a consequence, we will not present merely a series of prerequisites to the proof of the Forster-Swan bound, but rather everything which is strictly necessary to obtain a genuine understanding of the nature of the theorem. With this goal in mind, we will develop not only many of the fundamental results of commutative algebra, but in addition present results which are in a more profound way related to or even go beyond the Forster-Swan theorem itself. In section one, we describe the basic properties of commutative rings and modules over them. In particular, we will give the standard results on prime and maximal ideals, finitely generated modules, exact sequences, tensor products and flatness. In section two we define the notion of the spectrum of a ring, show that this is a geometric object associated with the ring, and present a few examples of spectra of rings. In section three, the theory of localization, the driving force behind the Forster-Swan theorem, is developed and several local-global principles of commutative algebra are demonstrated. In section four, the theory of Noetherian rings and modules, the dimension of rings, the concept of an algebraic variety, and Hilbert's basis and zero-locus theorems are presented. Section five contains the full proof of the Forster-Swan theorem. We conclude in section 6 with a discussion of topics surrounding Serre's problem on projective modules and the Eisenbud-Evans conjectures which give an improvement on the Forster-Swan bound.

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