### 数学代写|交换代数代写commutative algebra代考|MATH2322

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## 数学代写|交换代数代写commutative algebra代考|Early roots

Roots of commutative algebra can be found throughout the late part of the eighteenth century and first half of the nineteenth century. Along the same period, the theory of matrices and determinants was still stumbling and only became a solid theory toward the end of the nineteenth century: the terminology “matrix” was used for the first time in 1850 by Sylvester.

The notion of ideal through its axiomatic definition is due to R. Dedekind. According to the best sources, the terminology has been dug out of the efforts of Kummer to deal with the failure of unique factorization in algebraic number ring extensions of $\mathbb{Z}$.

Kronecker claimed he already had in the 1850 s the main features of ideal and module theory, including a reasonably definite notion of a prime ideal (cf. the Festchrift in honor of Kummer’s Fünfzigjahr, in Kronecker’s Gesammtwerke, where he says he had long before suggested the concept to others, Dedekind included). In a paper, he introduced the idea of the sum of two ideals and the notion of “decomposable” ideals in the sense of being the ideal product of two others. It seems that he had at the time considered some version of primary or prime decomposition, but it is not clear he had the correct notion.

Kronecker already uses the concepts of a field and of an integral domain (named Rationalitätbericht and Ganzhaliggebericht, resp.). In this respect, he uses the respective notation $\left(\mathfrak{S}{1}, \mathfrak{S}{2}, \ldots\right)$ and $\left[\mathfrak{S}{1}, \mathfrak{S}{2}, \ldots\right]$, our modern notation for field and ring extensions being reminiscent of his. However, because Kronecker considered only finitely generated ideals, he seemed to have completely missed the relevance of the Noetherian assumptions only later clarified by E. Noether.

Kronecker and Dedekind were contemporaneous scientists of enormous mathematical caliber and strong personality, not sharing the same philosophical approach toward mathematics. Both approaches left an enormous legacy to modern algebra, and mathematics as a whole. They developed at length the various questions around the notion of a module, with the difference that Dedeking was more focused on a particular class of modules-what nowadays are called fractional ideals in the field of fractions of an integral domain. In fact, his interest was solely in the case where the domain was the integral closure of the ring of integers in a finite extension of the field of rational numbers. Notation was a flagrant difference in the two mathematicians’ styles. While Kronecker always chose a tautological notation, Dedekind’s preference was a unique letter, mostly the capitalized first letter of a notion name (e.g., $K$ for Körper). In a sense, the philosophy of Kronecker’s approach to notational convention was to become well established throughout the time, no matter how cumbersome it looks from our modern view. Here is a tiny example: the term “Bereich” (domain) used by Kronecker became universal, in fact invariant in the translation to other languages. Dedekind’s “Körper” on the other hand, became “corps” in French, “field” in English, while in Spanish both “cuerpo” and “campo” seem to fight each other.

## 数学代写|交换代数代写commutative algebra代考|Rings of fractions

Let $R$ stand for a ring. $\Lambda$ subset $\mathfrak{S} \subset R$ such that, for any $a, b \subset \mathfrak{S}$ also $a b c \mathfrak{S}$, is called multiplicatively closed. In order to avoid a disturbing zero denominator in fractions to be introduced below, one assumes that a multiplicatively closed set does not contain 0 (consequently, does not contain any nilpotents either).

The outset goal is to define a new ring $S$ and a homomorphism $\iota: R \rightarrow S$ such that the elements of $\mathfrak{S}$ become units in $S$ and $S$ is generated by the image of $R$ and the inverses of these units. As expected, the construction involves a universal property that makes $S$ essentially unique.

The set $a:=\bigcup_{u \in \mathcal{S}}(0: u)$ is easily seen to be an ideal of $R$. Besides, the elements of $\mathfrak{S}$ are nonzero divisors modulo a. Define a relation $\equiv$ on $R \times \mathfrak{S}$ by decreeing:
$$(a, s) \equiv(b, t) \text { if and only if } a t-b s \in a .$$
Clearly, $\equiv$ is reflexive and symmetric. It is also transitive: if $(a, s) \equiv(b, t) \equiv(c, u)$ then $a t-b s \in a$ and $b u-c t \in a$; multiplying the first (resp., the second) inclusion by $u$ (resp., by $s)$ and adding the results yields $t(a u-c s) \in \mathfrak{a}$, hence $a u-c s \in a$ by the above
Since $\equiv$ is an equivalence relation, one can consider the quotient set $\mathcal{S}^{-1} R:=$ $(R \times \mathfrak{S}) / \equiv$ of this equivalence relation. Then one equips $\mathfrak{S}^{-1} R$ with a commutative ring structure such that the quotient map $\iota: R \rightarrow \mathfrak{S}^{-1} R$ is a homomorphism. In fact, requiring this and further that $\mathfrak{S}^{-1} R$ be generated by the image $\iota(R)$ and the inverses of the elements of $l(\mathfrak{S})$ make very natural the operations known since high school.
One briefly explains how this comes about. First, denoting the class of a pair $(a, s)$ by $a / s-a$ well-established notation-these requirements force the equalities
$$l(s)^{-1} l(a)=l(s)^{-1} \frac{a}{1}=\left(\frac{s}{1}\right)^{-1} \frac{a}{1}=\frac{1}{s} \frac{a}{1} .$$
Therefore, if $l(s)^{-1} l(a)=a / s$ is going to hold true it would better be because $(1 / s)(a / 1)=$ $a / s$, so this indicates at least how to multiply out generators. Thus, after harmless identification, one must have
$$s t(a / s+b / t)-l(s) l(t)\left(l(s)^{-1} l(a)+l(t)^{-1} l(b)\right)-l(a) l(t)+l(b) l(s)-a t+b s,$$

which imposes us the general rule of addition. The argument for the general multiplication rule is similar and easier.

It is now routine to verify that the rules of addition and multiplication give welldefined operations on $\mathrm{S}^{-1} R$.
Examples of multiplicatively closed sets are:

1. $\mathfrak{S}=\left{s^{n}: n \geq 0\right}$, where $s$ is a nonnilpotent element of $R$.
2. The set of regular elements of $R$.
3. Given a prime ideal $P \subset R, R \backslash P$ is a multiplicatively closed set. More generally, if $\left{P_{\alpha}\right}_{\alpha}$ is a family of prime ideals, $R \backslash \bigcup_{\alpha} P_{\alpha}$ is multiplicatively closed.

## 数学代写|交换代数代写commutative algebra代考|General properties of fractions

One collects in a few propositions the main operational properties of the present notion.

As many constructions in commutative algebra, fractions also enclose a certain universal property.

Proposition 2.1.2 (Universal property). Given a ring $S$ and a homomorphism $\varphi: R, S$ such that the elements of $\varphi(ङ) \subset S$ are invertible, then there is a unique homomorphism $\pi: \mathcal{S}^{-1} R \rightarrow S$ such that $\varphi=\pi \cdot \iota$

Proof. The following commutative diagram of ring homomorphisms encapsulates pictorially the main contents:
$$\begin{array}{ccc} R & \stackrel{\iota}{\longrightarrow} & S^{-1} R \ \varphi \searrow & \curvearrowright & \swarrow \pi \ & S & \end{array}$$
For the existence, set $\pi(a / s):=\varphi(a)(\varphi(s))^{-1}$. This makes sense since by assumption $\varphi(s)$ is invertible in $S$. To see that this is a well-defined map, let $a / s=b / t$. Then, by construction, $a t-b s \in a$. Say, $a t-b s=a \in a$. By definition, $a u=0$ for some $u \in \mathcal{S}$. Therefore, $\varphi(a) \varphi(u)=\varphi(a u)=0$, hence $\varphi(a)=0$ again by the assumption that the elements of $\varphi(\mathfrak{S})$ are invertible in $S$. It follows that $\varphi(a) \varphi(t)=\varphi(b) \varphi(s)$, as was to be shown.

## 数学代写|交换代数代写commutative algebra代考|Early roots

Kronecker 声称他在 1850 年代已经掌握了理想和模块理论的主要特征，包括一个合理确定的素理想概念（参见 Kronecker 的 Gesammtwerke 中纪念 Kummer 的 Fünfzigjahr 的 Festchrift，他说他很久以前就提出过其他人的概念，包括 Dedekind）。在一篇论文中，他介绍了两个理想之和的概念以及“可分解”理想的概念，即成为另外两个理想的产物。似乎他当时考虑过某种版本的初级或质数分解，但不清楚他的想法是否正确。

Kronecker 已经使用了域和积分域的概念（分别命名为 Rationalitätbericht 和 Ganzhaliggebericht）。在这方面，他使用相应的符号(小号1,小号2,…)和[小号1,小号2,…]，我们对域和环扩展的现代表示法让人想起他的。然而，由于 Kronecker 只考虑了有限生成的理想，他似乎完全忽略了后来由 E. Noether 澄清的诺特假设的相关性。

Kronecker 和 Dedekind 是同时代的科学家，他们具有巨大的数学才能和强烈的个性，他们对数学的哲学方法不同。这两种方法都给现代代数和整个数学留下了巨大的遗产。他们详细讨论了围绕模块概念的各种问题，不同之处在于 Dedeking 更专注于特定类别的模块 – 现在在积分域的分数领域中称为分数理想。事实上，他的兴趣仅仅在于定义域是整数环在有理数域的有限扩展中的积分闭包。符号是两位数学家风格的明显差异。虽然克罗内克总是选择重言式符号，但戴德金的偏好是一个独特的字母，ķ为科尔珀）。从某种意义上说，克罗内克的符号约定方法的哲学在整个时代都得到了很好的确立，无论从我们现代的角度来看它看起来多么繁琐。这是一个小例子：Kronecker 使用的术语“Bereich”（域）变得普遍，实际上在翻译成其他语言时是不变的。另一方面，戴德金的“Körper”在法语中变成了“corps”，在英语中变成了“field”，而在西班牙语中，“cuerpo”和“campo”似乎都在互相争斗。

## 数学代写|交换代数代写commutative algebra代考|Rings of fractions

(一个,s)≡(b,吨) 当且仅当 一个吨−bs∈一个.

l(s)−1l(一个)=l(s)−1一个1=(s1)−1一个1=1s一个1.

s吨(一个/s+b/吨)−l(s)l(吨)(l(s)−1l(一个)+l(吨)−1l(b))−l(一个)l(吨)+l(b)l(s)−一个吨+bs,

1. \mathfrak{S}=\left{s^{n}: n \geq 0\right}\mathfrak{S}=\left{s^{n}: n \geq 0\right}， 在哪里s是一个非幂零元素R.
2. 的规则元素集合R.
3. 给定一个素理想磷⊂R,R∖磷是乘法闭集。更一般地说，如果\left{P_{\alpha}\right}_{\alpha}\left{P_{\alpha}\right}_{\alpha}是一个崇高理想的家庭，R∖⋃一个磷一个是乘法闭的。

## 数学代写|交换代数代写commutative algebra代考|General properties of fractions

R⟶我小号−1R 披↷圆周率 小号

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