### 数学代写|抽象代数作业代写abstract algebra代考|History of Ring Theory

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## 数学代写|抽象代数作业代写abstract algebra代考|Noncommutative ring theory

Algebra textbooks usually give the definition of a ring first and follow it with examples. Of course the examples came first, and the abstract definition later-much later. So we begin with examples.

Among the most important examples of rings are the integers, polynomials, and matrices. “Simple” extensions of these examples are at the roots of ring theory. Specifically, we have in mind the following three examples:
(a) The integers $Z$ can be thought of as the appropriate subdomain of the field $Q$ of rationals in which to do number theory. (The rationals themselves are unsuitable for that purpose: every rational is divisible by every other (nonzero) rational.) Take a simple extension field $Q(\alpha)$ of the rationals, where $\alpha$ is an algebraic number, that is, a root of a polynomial with integer coefficients. $Q(\alpha)$ is called an algebraic number field; it consists of polynomials in $\alpha$ with rational coefficients. For example, $Q(\sqrt{3})={a+b \sqrt{3}: a, b \in Q}$.

The appropriate subdomain of $Q(\alpha)$ in which to do number theory -the “integers” of $Q(\alpha)$-consists of those elements that are roots of monic polynomials with integer coefficients, polynomials $p(x)$ in which the coefficient of the highest power of $x$ is 1 . For example, the integers of $Q(\sqrt{3})$ are ${a+b \sqrt{3}: a, b \in Z}$ (this is not obvious). This is our first example.
(b) The polynomial rings $\mathbf{R}[x]$ and $\mathbf{R}[x, y]$ in one and in two variables, respectively, share important properties but also differ in significant ways ( $\mathbf{R}$ denotes the real numbers). In particular, while the roots of a polynomial in one variable constitute a discrete set of real numbers, the roots of a polynomial in two variables constitute a curve in the plane-a so-called algebraic curve. Our second example, then, is the ring of polynomials in two (or more) variables.
(c) Square $m \times m$ matrices (for example, over the reals) can be viewed as $m^{2}$ tuples of real numbers with coordinate-wise addition and appropriate multiplication obeying the axioms of a ring. Our third example consists, more generally, of $n$-tuples $\mathbf{R}^{n}$ of real numbers with coordinate-wise addition and appropriate multiplication, so that the resulting system is a (not necessarily commutative) ring. Such systems, often extensions of the complex numbers, were called in the nineteenth and early twentieth centuries hypercomplex number systems.

In what contexts did these examples arise? What was their importance? The answers will lead us to the genesis of ring theory.

Rings fall into two broad categories: commutative and noncommutative. The abstract theories of these two categories came from distinct sources and developed in different directions. Commutative ring theory originated in algebraic number theory, algebraic geometry, and invariant theory. Central to the development of these subjects were the rings of integers in algebraic number fields and algebraic function fields, and the rings of polynomials in two or more variables.

Noncommutative ring theory began with attempts to extend the complex numbers to various hypercomplex number systems. The genesis of the theories of commutative and noncommutative rings dates back to the early nineteenth century, while their maturity was achieved only in the third decade of the twentieth century.
The following is a diagrammatic sketch of the above remarks.

## 数学代写|抽象代数作业代写abstract algebra代考|Examples of Hypercomplex Number Systems

Hamilton’s invention of the quaternions was conceptually groundbreaking-_”a revolution in arithmetic which is entirely similar to the one which Lobachevsky effected

in geometry,” according to Poincaré. Indeed, both achievements were radical violations of prevailing conceptions. Like all revolutions, however, the invention of the quaternions was initially received with less than universal approbation: “I have not yet any clear view as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties,” declared Hamilton’s mathematician friend John Graves.

Most mathematicians, however, including Graves, soon came around to Hamilton’s point of view. The quaternions acted as a catalyst for the exploration of diverse “number systems,” with properties which departed in various ways from those of the real and complex numbers. Among the examples of such hypercomplex number systems are the following:
(i) Octonions
These are 8-tuples of reals which contain the quaternions and form a division algebra in which multiplication is nonassociative. They were introduced in 1844 by Cayley and independently by the very John Graves who questioned Hamilton’s “imaginaries.”
(ii) Exterior algebras
These are $n$-tuples of reals, added componentwise and multiplied via the “exterior product.” They were introduced by Grassmann in 1844 as part of a brilliant attempt to construct a vector algehra in $n$-dimensional space. Grassmann’s style was far from simple and his approach was ahead of its time.
(iii) Group algebras
In 1854 Cayley published a paper on (finite) abstract groups, at the end of which he gave a definition of a group algebra (over the real or complex numbers). He called it a system of “complex quantities” and observed that it is analogous to Hamilton’s quaternions-it is associative, noncommutative, but in general not a division algebra.
(iv) Matrices
In two papers of 1855 and 1858 Cayley introduced square matrices. He noted that they can be treated as “single quantities,” added and multiplied like “ordinary algebraic quantities,” but that “as regards their multiplication, there is the peculiarity that matrices are not in general convertible [commutative].” See Chapter 5.1.3.

## 数学代写|抽象代数作业代写abstract algebra代考|Classification

Over a thirty-year period (c. $1840-1870$ ) a stock of examples of noncommutative number systems had been established. One could now begin to construct a theory. The general concept of a hypercomplex number system (in current terminology, a finite-dimensional algebra) emerged, and work began on classifying certain types

of these structures. We focus on three such developments, dealing with associative algebras. Note that such algebras are, of course, rings.
(i) Low-dimensional algebras
Of fundamental importance here was the work of Benjamin Peirce of Harvard-the first important contribution to algebra in the U.S. We are referring to his groundbreaking paper “Linear Associative Algebra” of 1870. In the last 100 pages of this 150-page paper Peirce classified algebras (i.e., hypercomplex number systems) of dimension $<6$ by giving their multiplication tables. There are, he showed, over 150 such algebras! What is important in this paper, though, is not the classification but the means used to obtain it. For here Peirce introduced concepts, and derived results, which proved fundamental for subsequent developments. Among these conceptual advances were:

(a) An “abstract” definition of a finite-dimensional algebra. Peirce defined such an algebra-he called it a “linear associative algebra”-as the totality of formal expressions of the form $\sum_{i=1}^{n} a_{i} e_{i}$, where the $e_{i}$ are “basis elements.” Addition was defined componentwise and multiplication by means of “structural constants” $c_{i j k}$, namely $e_{i} e_{j}=\sum_{k=1}^{n} c_{i j k} e_{k}$. Associativity under multiplication and distributivity were postulated, but not commutativity. This is probably the earliest explicit definition of an associative algebra.
(b) The use of complex coefficients. Peirce took the coefficients $a_{i}$ in the expressions $\sum a_{i} e_{i}$ to be complex numbers. This conscious broadening of the field of coefficients from $\mathbf{R}$ to $\mathbf{C}$ was an important conceptual advance on the road to coefficients taken from an arbitrary field.
(c) Relaxation of the requirement that an algebra have an identity. This, too, was a departure from past practice and gave an indication of Peirce’s general, abstract approach.

## 数学代写|抽象代数作业代写abstract algebra代考|Noncommutative ring theory

(a) 整数从可以认为是该领域的适当子域问做数论的有理数。（有理数本身不适合这个目的：每个有理数都可以被其他（非零）有理数整除。） 取一个简单的外延域问(一种)的有理数，其中一种是一个代数数，即具有整数系数的多项式的根。问(一种)称为代数数域；它由多项式组成一种有理系数。例如，问(3)=一个+b3:一个,b∈问.

(b) 多项式环R[X]和R[X,是]在一个和两个变量中，分别具有重要的属性，但也存在显着差异（R表示实数）。特别是，多项式在一个变量中的根构成一组离散的实数，而多项式在两个变量中的根构成平面中的一条曲线——所谓的代数曲线。那么，我们的第二个例子是两个（或更多）变量的多项式环。
(c) 广场米×米矩阵（例如，在实数上）可以被视为米2符合环公理的坐标加法和适当乘法的实数元组。我们的第三个例子更一般地包括n-元组Rn具有坐标加法和适当乘法的实数，因此得到的系统是（不一定是可交换的）环。这种系统，通常是复数的扩展，在 19 世纪和 20 世纪初被称为超复数系统。

## 数学代写|抽象代数作业代写abstract algebra代考|Examples of Hypercomplex Number Systems

(i) Octonions

(ii) 外代数

(iii) 群代数
1854 年，Cayley 发表了一篇关于（有限）抽象群的论文，最后他给出了群代数（在实数或复数上）的定义。他称其为“复量”系统，并观察到它类似于汉密尔顿的四元数——它是结合的、不可交换的，但通常不是除法代数。
(iv) 矩阵

## 数学代写|抽象代数作业代写abstract algebra代考|Classification

(i) 低维代数

(a) 有限维代数的“抽象”定义。Peirce 将这样一个代数——他称之为“线性结合代数”——定义为以下形式的形式表达式的总和∑一世=1n一个一世和一世, 其中和一世是“基本要素”。加法是通过“结构常数”定义的组件和乘法C一世jķ，即和一世和j=∑ķ=1nC一世jķ和ķ. 假设乘法和分配下的关联性，但不是交换性。这可能是关联代数最早的明确定义。
(b) 复系数的使用。皮尔斯取系数一个一世在表达式中∑一个一世和一世是复数。这种有意识地扩大系数领域R至C是从任意领域获取系数的重要概念进步。
(c) 放宽代数有恒等式的要求。这也与过去的做法背道而驰，并表明了皮尔士的一般抽象方法。

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