On Commutative Algebra
On Commutative Algebra
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DOI：
10.1007/978-3-642-03064-2_1
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
M. Chlouveraki
中科院分区：
文献类型：
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作者：
M. Chlouveraki
第一章包含了一些关于交换代数的已知事实和一些新的结果，它们对第三章和第四章的结果的证明至关重要。为了读者的方便，前面的定理在这里不作证明（定理8除外）。在本章的第一部分，我们定义了环的局部化，并给出了环的一些主要性质。第二部分专门讨论整闭环。研究了积分闭环的特殊情况，如估值环、离散估值环和Krull环。我们利用它们的性质得到了整闭环上劳伦多项式环上的一些结果。在1.3节中，我们简要地说明了环补全的一些结果。在第四节中，我们介绍了与单项式相关的态射的概念。它们是态射，允许我们从一个Laurent多项式ringAinm+1不定式传递到一个Laurent多项式ringBinmindeterminates，同时将一个特定的单项式映射到1。此外，我们证明了（命题15）每一个从matb映射每一个不定式到一个单项的满射态射都与一个单项相关联。我们称适应态态为与单项式相关的态态组合。它们在第三章和第四章的主要结果的证明中起着关键作用。最后，在第一章的最后一节中，我们给出了一个多项式在域上带系数的Laurent多项式环上不可约的判据（定理10）。
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 ofmorphisms associated with monomials. They are morphisms which allow us to pass from a Laurent polynomial ringAinm+1 indeterminates to a Laurent polynomial ringBinmindeterminates, while mapping a specific monomial to 1. Moreover, we prove (Proposition 15) that every surjective morphism fromAtoBwhich maps each indeterminate to a monomial is associated with a monomial. We calladapted morphismsthe 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.