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# 数学代写|交换代数代考Commutative Algebra代写|MA8202 Overview of module theory

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## 数学代写|交换代数代考Commutative Algebra代写|Overview of module theory

This chapter assumes a prior knowledge of the basics of general module theory, mainly the definition and the notion of sets of generators. Most elementary properties are envisaged by mimicking the case of an ideal or of a vector space. The notable distinction from ideals, just like a vector space, is that a module is not a priori embedded in a simpler structure (such as a free module), thus making the theory a bit more involved. With Noether and even before, the notion spread out into a vast territory, strongly shared by the noncommutative theory. Sticking to commutative rings makes the general theory a lot more pliable, though not at all trivial. Some of these basics which are useful for the treatment in the book will be covered in the exercises at the end of the chapter. One of the hot topics for homology is the notion of a projective module-some of it will be discussed in Section 6.2.2.

## 数学代写|交换代数代考Commutative Algebra代写|Noetherian modules

In spite of their major role, the various ideas about finite structures in module theory did not take too long to stabilize, as compared to the idea of a module itself. The word “Modul” had been used by Dedekind and Kronecker, but only a bit later, with Noether and others, the general meaning became stable.

Let $R$ denote a commutative ring. In full resemblance to Lemma 2.5.1, one defines an $R$-module to be Noetherian provided it satisfies any of the conditions in the following lemma.
Lemma 3.1.1. The following conditions are equivalent for an R-module M:
(a) Every submodule of $M$ is finitely generated;
(b) Every chain of submodules $M_{1} \subset M_{2} \subset \cdots \subset M$ is stationary, i.e., there exists an index $m$ such that $M_{m}=M_{m+1}=\cdots$
(c) Any nonempty family of submodules of $M$ has a maximal element.
The proof is left to the reader as an encore of the ring case.

## 数学代写|交换代数代考 COMMUTATIVE ALGEBRA代 写INOETHERIAN MODULES

$a$ 每个子模块 $M$ 是有限生成的；
$b$ 每个子模块链 $M_{1} \subset M_{2} \subset \cdots \subset M$ 是平稳的，即存在一个索引 $\mid m$ 这样 $M_{m}=M_{m+1}=\cdots$
$c$ 的任何非空子模块族 $M$ 有一个最大元榡。

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