抽象代數研究對象是代數結構(集合+一套運算規則)，以本人的理解方式整理了代數結構基礎內容，旨在弄清群、環、域等代數結構間的關係。(之前很認真整理了這部分內容，後來裝雙系統把草稿給丟了，這會重新整理)

PS：以下內容直接從個人博客Spark & Shine拷貝，圖片不能正常顯示，部分排版混亂。建議直接閱讀原文，鏈接在這裡：http://sparkandshine.net/algebraic-structure-primer-group-ring-field-vector-space/

文章目錄 [隱藏]

• 1. 抽象代數
  • 1.1 相關概念
  • 1.2 代數結構
  • 1.3 初等代數–>抽象代數
• 2. Group-like
  • 2.1 原群
  • 2.2 半群
  • 2.3 幺半群
  • 2.4 群
  • 2.5 阿貝爾群(交換群)
• 3. 環論
  • 3.1 環
  • 3.2 交換環
  • 3.3 整環
• 4. 域
• 5. 向量空間
  • 5.1 8個公理
• 6. 模
• 7. 代數(環論)
• 8. 格
• 9. 總結

1. 抽象代數

1.1 相關概念

數學的發展通常是先從簡單的開始，然後不斷放寬限制，推廣到更一般化。從初等代數到抽象代數很好說明了這一點。

(1)算術(arithmetic)

算術(arithmetic)無疑是數學中最古老、最基礎和最初等的部分。算術研究數的性質及其運算。把數和數的性質、數和數之間的四則運算在應用過程中的經驗累積起來，並加以整理，就形成了最古老的一門數學——算術[1]。值得一提的是，算術運算不僅僅指加減乘除，還可以是百分比、平方根、取冪和對數；演算法的對象包括自然數、整數、有理數和實數(興許還包括複數)；進位不僅僅是十進位，還可以是二進位、十六進位、六十進位。個人認為，算術的最大特點是關注具體數字。

(2)初等代數

用符號(成了變數)代替具體的數字，就可以得到更一般化(generalization)的等式，舉例如下：

圖1 從算術到代數的例子

初等代數(elementary algebra)是古老算術的推廣與發展。在古代，算術積累了大量數量問題的解法，為尋求更系統、更普遍的求解各種數量關係方法，就產生了以解方程為中心的初等代數[2]。從實際問題的數量關係(即代數式：整式、分式、根式)、等量關係(或者不等式)列出列出方程或者方程組。方程(組)包括一元/二元一次方程(linear equations with one/two variable)、一元二次方程(quadratic equations)、指數和對數方程(exponential and logarithmic equations)、無理方程(radical equations)、線性方程組(system of linear equations)[3]。

高等代數相對於初等代數而言，本質上是一個東西，只是更加系統(深度+廣度)。

初等代數再進一步推廣(generalization)，那就是抽象代數了。初等代數與抽象代數的界限在於初等代數只考慮實數和複數代數結構。摘錄維基百科詞條Elementary algebra：

Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers.

(3)抽象代數

抽象代數(abstract algebra)、近世代數、現代代數(modern algebra)指的都是同一個意思(甚至直接稱為代數學)。抽象代數主要研究對象是代數結構，包括群、環、域、向量空間。

伽羅瓦(évariste Galois, 1811-1832)是現代群論的創始人(與阿貝爾獨立發明)，他利用群的概念徹底解決了用根式求解代數方程的可能性問題[4](稱為伽羅瓦理論)，系統闡釋了為何五次以上之方程式沒有公式解，而四次以下有公式解[5]，使代數學從解方程的科學轉變為研究代數結構的科學，即把代數推廣到抽象代數[4]。

(4)線性代數

線性代數是抽象代數特殊的一類，其代數結構為：向量空間(vector spaces，也叫線性空間) + 線性變換(linear mappings)。很容易將線性代數和矩陣理論等同起來，但其實是不一樣的，討論線性變換是基於選定一組基的前提下。摘抄mathoverflow上的一個回答(原文在這裡)：

When you talk about matrices, you』re allowed to talk about things like the entry in the 3rd row and 4th column, and so forth. In this setting, matrices are useful for representing things like transition probabilities in a Markov chain, where each entry indicates the probability of transitioning from one state to another.

In linear algebra, however, you instead talk about linear transformations, which are not a list of numbers, although sometimes it is convenient to use a particular matrix to write down a linear transformation. However, when you』re given a linear transformation, you』re not allowed to ask for things like the entry in its 3rd row and 4th column because questions like these depend on a choice of basis. Instead, you』re only allowed to ask for things that don』t depend on the basis, such as the rank, the trace, the determinant, or the set of eigenvalues. This point of view may seem unnecessarily restrictive, but it is fundamental to a deeper understanding of pure mathematics.

1.2 代數結構

既然抽象代數研究對象是代數結構(algebraic structure)，那什麼是代數結構呢。看了多個不同角度描述代數結構，如百度百科代數：代數是研究數、數量、關係與結構的數學分支。還是覺得《[轉]MIT牛人解說數學體系》中的描述最深入淺出，如下：

代數主要研究的是運算規則。一門代數， 其實都是從某種具體的運算體系中抽象出一些基本規則，建立一個公理體系，然後在這基礎上進行研究。一個集合再加上一套運算規則，就構成一個代數結構[1](想想計算機的數據結構：數據+操作)。

1.3 初等代數–>抽象代數

抽象代數將初等代數的一些概念延伸。

(1)數 –> 集合

集合在樸素集合論(naive set theory)和公理化集合論(axiomatic set theory)的定義是不一樣的，前者指由一些元素組成；後者指具有某種特定性質事物的總體。還是看維基英文詞條吧[7]：

Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of sets: a collection of all objects (called elements) selected by property specific for the set.

(2)+ –> 二元運算

加號+被抽象為二元運算*(binary operation)，對兩個元素作二元運算，得到的新元素仍然屬於該集合，這叫封閉性(closure)。實際上，加減乘除都叫二元運算(二元指的是兩個操作數)。原文如下[7]：

The notion of addition (+) is abstracted to give a binary operation, ? say. The notion of binary operation is meaningless without the set on which the operation is defined. For two elements a and bin a set S, a ? b is another element in the set; this condition is called closure.

Addition (+), subtraction (-), multiplication (×), and division (÷) can be binary operations when defined on different sets, as are addition and multiplication of matrices, vectors, and polynomials.

(3)0/1 –> 單位元

0和1被抽象成單位元(identity elements)，0為加法單位元，1為乘法單位元。單位元是集合的一個特殊元素(跟二元運算有關)，滿足單位元與其他元素相結合時，不改變該元素，即滿足a ? e = a 與 e ? a = a。可見，單位元取決於元素與二元運算，如矩陣的加法單位元是零矩陣，矩陣的乘法單位元是單位矩陣。值得注意的是，有些集合不存在單位元，如正整數集合(the set of positive natural numbers)沒有加法單位元(no identity element for addition)。維基百科原文如下：

The numbers zero and one are abstracted to give the notion of an identity element for an operation. An identity element is a special type of element of a set with respect to a binary operation on that set. For a general binary operator ? the identity element e must satisfy a ? e = a and e ? a = a. Not all sets and operator combinations have an identity element.

(4)負數 –> 逆元素

負數推廣到逆元素(inverse element)，對於加法，a的逆元素是-a；對於乘法，a的逆元素是倒數a?1。直觀地說，逆元可以撤銷操作，如加了一個數a，再加上該數的逆元-a(相當於撤消操作)，結果還是一樣。維基百科原文如下：

The negative numbers give rise to the concept of inverse elements. For addition, the inverse of a is written ?a, and for multiplication the inverse is written a?1. A general two-sided inverse element (left inverse + right inverse) a?1 satisfies the property that a ? a?1 = 1 and a?1 ? a = 1.

The idea of an inverse element generalises concepts of a negation (sign reversal) in relation toaddition, and a reciprocal in relation to multiplication. The intuition is of an element that can 『undo』 the effect of combination with another given element. While the precise definition of an inverse element varies depending on the algebraic structure involved, these definitions coincide in a group.

(5)結合

律

結合律(Associative property)是某些二元運算的性質，有些二元運算沒有結合律(如減法、除法、八元數)。原文如下：

Addition of integers has a property called associativity. That is, the grouping of the numbers to be added does not affect the sum. In general, this becomes (a ? b) ? c = a ? (b ? c). This property is shared by most binary operations, but not subtraction or division or octonion multiplication.

(6)交換律

交換律(Commutative property)，改變二元運算符兩邊的元素不影響結果。並不是所有二次元運算都滿足交換律(如矩陣的乘法)。維基百科原文如下：

A binary operation is commutative if changing the order of the operands does not change the result. Addition and multiplication of real numbers are both commutative. In general, this becomes a ? b =b ? a.

This property does not hold for all binary operations. For example, matrix multiplication andquaternion multiplication are both non-commutative.

2. Group-like

代數結構(R, *)，二元運算根據封閉性、單位元、逆元、結合律、交換律，可以歸納成不同的群。本節介紹的group-like，從最不嚴格到嚴格(依次添加限制條件)，其關係圖如下：

圖1 群之間的關係

維基百科有一張表，給出更詳細的group-like間的關係，如下：

圖2 Group-like structures (source from here)

2.1 原群

原群(magma)是一種基本的代數結構，只要滿足兩元素作二元運算得到新元素仍屬於該集合，即封閉性。維基百科原文如下：

A magma is a basic kind of algebraic structure. Specifically, a magma consists of a setequipped with a single binary operation. The binary operation must be closed by definition but no other properties are imposed.

2.2 半群

半群(Semigroup)，滿足結合律(associative property)的代數結構。V=<S，* >，其中二元運算*是可結合的，即(a*b)*c=a*(b*c)，則稱V是半群。維基百科原文如下：

A semigroup is analgebraic structure consisting of a set together with an associativebinary operation.

A semigroup generalizes a monoid in that a semigroup need not have an identity element. It also (originally) generalized a group (a monoid with all inverses) in that no element had to have an inverse, thus the name semigr

oup.

2.

3 幺

半群

幺半群(monoid)在半群的基礎上，還需要滿足有一個單位元。維基百科原文如下：

A monoid is an algebraic structure with a single associativebinary operation and an identity element. Monoids are studied in semigroup theory as they are semigroups with identity.

2.4 群

群(group)是兩個元素作二元運算得到的一個新元素，需要滿足群公理(group axioms)，即：

• 封閉性：a ? b is another element in the set
• 結合律：(a ? b) ? c = a ? (b ? c)
• 單位元：a ? e = a and e ? a = a
• 逆 元：加法的逆元為-a，乘法的逆元為倒數1/a，… (對於所有元素)

如整數集合，二次元運算為加法就是一個群(封閉性是顯然的，加法滿足結合律，單位元為0，逆元取相反數-a)。維基百科原文如下：

A group is a set of elements together with an operation that combines any two of its elements to form a third element satisfying four conditions called the group axioms, namely closure,associativity, identity and invert

ibilit

y.

One

o

f the most familiar examples of a group is the set of integers together with the addition operation

2.5 阿貝爾群(交換群)

阿貝爾群(Abelian Group)在群的基礎上，還需滿足交換律。如整數集合和加法運算，(Z,+)，是一個阿貝爾群。

• 群公理：見2.4 群。
• 交換律：a + b = b + a

維基百科原文如下：

An abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order (the axiom ofcomm

utativ

ity).

3

. 環

論

環在交換群基礎上，進一步限制條件。環、交換環、域間的關係如下：

圖3 環、交換環、域間的關係

維基百科有一張表從不同角度呈現這三者的關係，如下：

圖4 Ring-like structures (source from here)

3.1 環

環(ring)在阿貝爾群(也叫交換群)的基礎上，添加一種二元運算·(雖叫乘法，但不同於初等代數的乘法)。一個代數結構是環(R, +, ·)，需要滿足環公理(ring axioms)，如(Z,+, ?)。環公理如下：

(1)(R, +)是交換群

• 封閉性：a + b is another element in the set
• 結合律：(a + b) + c = a + (b + c)
• 單位元：加法的單位元為0，a + 0 = a and 0 + a = a
• 逆 元：加法的逆元為-a，a + (?a) = (?a) + a = 0 (對於所有元素)
• 交換律：a + b = b + a

(2)(R, ·)是幺半群

• 結合律：(a ? b) ? c = a ? (b ? c)
• 單位元：乘法的單位元為1，a ? 1 = a and 1 ? a = a

(3)乘法對加法滿足分配律Multiplication distributes over addition

• a ? (b + c) = (a ? b) + (a ? c) for all a, b, c in R (left distributivity)
• (b + c) ? a = (b ? a) + (c ? a) for all a, b, c in R (right distributivity)

維基百科原文如下：

A ring is an abelian group with a second binary operation (The abelian group operation is called 「addition」 and the second binary operation is called 「multiplication」 in analogy with the integers) that is distributive over addition and is associative.

One familiar example of a ring is the set of integers. The integers are a commutative ring, since a timesb is equal to b times a. The set of polynomials also forms a commutative ring. An example of a non-commutative ring is the ring of square matrices of the same size. Finally, a field is a commutative ring in which one can divide by any nonzero element: an example is the field of real numbers.

3.2 交換環

交換環(commutative ring)在環的基礎上，二元運算乘法還滿足交換律。

A commutative ring is a ring in which the multiplication operation is commutative

3.3 整環

整環(integral domain)在交換環的基礎上，並滿足沒有零因子(如此，集合內任意兩個元素乘積均不等於0)。維基百科原文如下：

An integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain the cancellation property holds for multiplication by a nonzero element a, that is, if a ≠ 0, an equalityab = ac implies b = c.

4. 域

域(Field)在交換環的基礎上，還增加了二元運算除法，要求元素(除零以外)可以作除法運算，即每個非零的元素都要有乘法逆元。由此可見，域是一種可以進行加減乘除(除0以外)的代數結構，是數域與四則運算的推廣。整數集合，不存在乘法逆元(1/3不是整數)，所以整數集合不是域。有理數、實數、複數可以形成域，分別叫有理數域、實數域、複數域。域的幾種定義，直接看維基百科英文吧：

a field is a nonzerocommutative ring that contains a multiplicative inverse for every nonzero element, or equivalently a ring whose nonzero elements form an abelian group under multiplication.

As such it is analgebraic structure with notions of addition,subtraction, multiplication, and divisionsatisfying the appropriate abelian group equations and distributive law.

從有限域到交換環一些代數結構的從屬關係如下：

Commutative rings ? integral domains ? integrally closed domains ? unique factorization domains ? principal ideal domains ? Euclidean domains ? fields ? finite fields.

5. 向量空間

向量空間(vector space)是一些向量的集合。最熟悉的例子是幾何向量或矢量(Euclidean vectors, geometric vector, spatial vector)，表示具有大小和方向的對象，如，矢量可以做加法(addition)和乘法(scalar multiplication)運算，舉例如下：

圖5 Vector addition and scalar multiplication (source from here)

其他例子，還包括坐標空間(Coordinate spaces)、複數、函數空間(Function spaces)、線性方程組(linear equations)。詳情可查閱維基百科詞條：Examples of vector spaces.

5.1 8個公理

摘抄維基百科Vector space部分內容如下：

A vector space is a collection of objects called vectors, which may be added together and multiplied(「scaled」) by numbers, called scalars in this context. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below.

給定域F，向量空間V記為F-向量空間。其二元運算：

• 向量加法：+ : V × V → V 記作 v + w, ? v, w ∈ V
• 標量乘法：·: F × V → V 記作 a v, ?a ∈ F 且 v ∈ V

並且滿足如下8條公理[10]：

• 向量加法結合律：u + (v + w) = (u + v) + w
• 向量加法的單位元：V存在零向量的0，? v ∈ V , v + 0 = v
• 向量加法的逆元素：?v∈V, ?w∈V，使得 v + w = 0
• 向量加法交換律：v + w = w + v
• 標量乘法與域乘法兼容性(compatibility): a(b v) = (ab)v
• 標量乘法有單位元: 1 v = v, 1指域F的乘法單位元
• 標量乘法對於向量加法滿足分配律：a(v + w) = a v + a w
• 標量乘法對於域加法滿足分配律: (a + b)v = a v + b v

另，若F是實數域?，則V稱為實數向量空間；若F是複數域?，則V稱為複數向量空間；若F是有限域，則V稱為有限域向量空間。

6. 模

模(module)是對向量空間的推廣，將標量需為域(向量空間)推廣到任意環(模)。維基百科module部分原文如下：

A module over a ring is a generalization of the notion of vector space over a field, wherein the corresponding scalars are the elements of an arbitrary ring.

7. 代數(環論)

代數(algebra)將algebra over a field中的域推廣到交換環。維基百科原文如下：

An algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base fieldK is replaced by a commutative ringR.

alAn gebra over a field is a vector space equipped with a bilinearproductalgebra. An such that the product is associative and has an identity is therefore a ring that is also a vector space, and thus equipped with a field of scalars. Such an algebra is called here a unitalassociative algebra for clarity, because there are also nonassociative algebras.

8. 格

格(lattice)是任意兩個元素都有上確界和下確界的偏序集合。

A lattice is a partially ordered set in which every two elements have a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).

An example is given by the natural numbers, partially ordered by divisibility, for which the unique supremum is the least common multiple and the unique infimum is the greatest common divisor.

9. 總結

是時候，祭出這張圖了，圖片來源於這裡。YouTube還有一段小視頻講這張圖的，在這裡。

圖6 Mathematical Structures/Objects

參考資料：

[1]百度百科詞條：算術

[2]百度百科詞條：初等代數

[3]Wikipedia: Elementary algebra

[4]百度百科詞條：抽象代數

[5]百度百科詞條：埃瓦里斯特·伽羅瓦

[6][轉]MIT牛人解說數學體系

[7]Wikipedia: Algebra#Abstract_algebra

[8]博文系列：明日枯荷包:數學閑話

[9]Wikipedia: Vector space

[10]百度百科詞條：向量空間

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