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|MATH2322

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:
R & \stackrel{\iota}{\longrightarrow} & S^{-1} R \
\varphi \searrow & \curvearrowright & \swarrow \pi \
& S &
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|MATH2322


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

交换代数的根可以在 18 世纪末和 19 世纪上半叶找到。在同一时期,矩阵和行列式的理论仍然步履蹒跚,直到 19 世纪末才成为一个坚实的理论:“矩阵”这个术语在 1850 年由西尔维斯特首次使用。

理想的概念通过其公理化定义归功于 R. Dedekind。根据最好的消息来源,该术语是从 Kummer 为解决代数数环扩展中唯一分解失败的努力中挖掘出来的从.

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

让R代表戒指。Λ子集小号⊂R这样,对于任何一个,b⊂小号还一个bC小号, 称为乘法闭合。为了避免下面要介绍的分数中令人不安的零分母,假设一个乘法闭集不包含 0(因此,也不包含任何幂零)。


套装一个:=⋃在∈小号(0:在)很容易被视为理想的R. 此外,元素小号是模 a 的非零除数。定义关系≡上R×小号通过法令:

(一个,s)≡(b,吨) 当且仅当 一个吨−bs∈一个.
清楚地,≡是自反的和对称的。它也是传递的:如果(一个,s)≡(b,吨)≡(C,在)然后一个吨−bs∈一个和b在−C吨∈一个; 将第一个(或第二个)包含乘以在(分别由s)并添加结果产生吨(一个在−Cs)∈一个, 因此一个在−Cs∈一个由上述
自≡是等价关系,可以考虑商集小号−1R:= (R×小号)/≡这种等价关系。然后一个装备小号−1R具有交换环结构,使得商映射我:R→小号−1R是同态。事实上,要求这个和进一步那个小号−1R由图像生成我(R)和元素的逆l(小号)从高中开始就知道的操作非常自然。

因此,如果l(s)−1l(一个)=一个/s会成立 最好是因为(1/s)(一个/1)= 一个/s,所以这至少表明了如何乘以生成器。因此,经过无害识别后,必须有




  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



命题 2.1.2(普遍财产)。给了一个戒指小号和同态披:R,小号这样的元素ङ披(?)⊂小号是可逆的,则存在唯一的同态圆周率:小号−1R→小号这样披=圆周率⋅我


R⟶我小号−1R 披↷圆周率 小号
为了存在,设置圆周率(一个/s):=披(一个)(披(s))−1. 这是有道理的,因为通过假设披(s)是可逆的小号. 要看到这是一个定义明确的地图,让一个/s=b/吨. 然后,通过施工,一个吨−bs∈一个. 说,一个吨−bs=一个∈一个. 根据定义,一个在=0对于一些在∈小号. 所以,披(一个)披(在)=披(一个在)=0, 因此披(一个)=0再次假设元素披(小号)是可逆的小号. 它遵循披(一个)披(吨)=披(b)披(s),如将要显示的那样。

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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


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