On Commutative Algebra
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The first chapter contains some known facts and some novel results on Commutative Algebra which are crucial for the proofs of the results of Chapters 3 and 4. The former are presented here without their proofs (with the exception of Theorem 8) for the convenience of the reader. In the first section of this chapter, we define the localization of a ring and give some main properties. The second section is dedicated to integrally closed rings. We study particular cases of integrally closed rings, such as valuation rings, discrete valuation rings and Krull rings. We use their properties in order to obtain results on Laurent polynomial rings over integrally closed rings. We state briefly some results on the completions of rings in Section 1.3. In the fourth section, we introduce the notion of morphisms associated with monomials. They are morphisms which allow us to pass from a Laurent polynomial ring A in m+1 indeterminates to a Laurent polynomial ring B in m indeterminates, while mapping a specific monomial to 1. Moreover, we prove (Proposition 15) that every surjective morphism from A to B which maps each indeterminate to a monomial is associated with a monomial. We call adapted morphisms the compositions of morphisms associated with monomials. They play a key role in the proof of the main results of Chapters 3 and 4. Finally, in the last section of the first chapter, we give a criterion (Theorem 10) for a polynomial to be irreducible in a Laurent polynomial ring with coefficients in a field.
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