# 1. 抽象代数

## 1.1 相关概念

(1)算术(arithmetic)

(2)初等代数

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

(3)抽象代数

(4)线性代数

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.

## 1.3 初等代数–>抽象代数

(1)数 –> 集合

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)+ –> 二元运算

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 b in a set S, ab 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 –> 单位元

0和1被抽象成单位元(identity elements)，0为加法单位元，1为乘法单位元。单位元是集合的一个特殊元素(跟二元运算有关)，满足单位元与其他元素相结合时，不改变该元素，即满足ae = aea = a。可见，单位元取决于元素与二元运算，如矩阵的加法单位元是零矩阵，矩阵的乘法单位元是单位矩阵。值得注意的是，有些集合不存在单位元，如正整数集合(the set of positive natural numbers)没有加法单位元(no identity element for addition)。维基百科原文如下：

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)负数 –> 逆元素

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 to addition, 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)结合律

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 (ab) ∗ c = a ∗ (bc). This property is shared by most binary operations, but not subtraction or division or octonion multiplication.

(6)交换律

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 ab = ba.

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

# 2. Group-like

## 2.1 原群

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

## 2.2 半群

A semigroup is an algebraic structure consisting of a set together with an associative binary 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 semigroup.

## 2.3 幺半群

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

## 2.4 群

• 封闭性：ab is another element in the set
• 结合律：(a ∗ b) ∗ c = a ∗ (b ∗ c)
• 单位元：a ∗ e = a and e ∗ a = a
• 逆  元：加法的逆元为-a，乘法的逆元为倒数1/a，… (对于所有元素)

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 invertibility.

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

## 2.5 阿贝尔群(交换群)

• 群公理：见2.4 群。
• 交换律：a + b = b + a

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 of commutativity).

# 3. 环论

## 3.1 环

(1)(R, +)是交换群

• 封闭性：a + b is another element in the set
• 结合律：(a + b) + c = a + (b + c)
• 单位元：加法的单位元为0，a + 0 = a and 0 + a = a
• 逆  元：加法的逆元为-a，a + (−a) = (−a) + a = 0 (对于所有元素)
• 交换律：a + b = b + a

(2)(R, ·)是幺半群

• 结合律：(ab) ⋅ c = a ⋅ (bc)
• 单位元：乘法的单位元为1，a ⋅ 1 = a and 1 ⋅ a = a

• 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)

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 times b 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 交换环

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

## 3.3 整环

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

a field is a nonzero commutative 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 an algebraic structure with notions of addition,subtraction, multiplication, and division satisfying the appropriate abelian group equations and distributive law.

# 5. 向量空间

## 5.1 8个公理

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.

• 向量加法：+ : V × V → V 记作 v + w, ∃ v, w ∈ V
• 标量乘法：·: F × V → V 记作 a v, ∃a ∈ F 且 v ∈ V

• 向量加法结合律：u + (v + w) = (u + v) + w
• 向量加法的单位元：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
• 向量加法交换律：v + w = w + v
• 标量乘法与域乘法兼容性(compatibility): a(b v) = (ab)v
• 标量乘法有单位元: 1 v = v, 1指域F的乘法单位元
• 标量乘法对于向量加法满足分配律：a(v + w) = a v + a w
• 标量乘法对于域加法满足分配律: (a + b)v = a v + b v

# 6. 模

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. 代数(环论)

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

An algebra over a field is a vector space equipped with a bilinear product. An algebra 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 unital associative algebra for clarity, because there are also nonassociative algebras.

# 8. 格

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. 总结

[1]百度百科词条：算术
[2]百度百科词条：初等代数
[3]Wikipedia: Elementary algebra
[4]百度百科词条：抽象代数
[5]百度百科词条：埃瓦里斯特·伽罗瓦
[6][转]MIT牛人解说数学体系
[7]Wikipedia: Algebra#Abstract_algebra
[8]博文系列：明日枯荷包:数学闲话
[9]Wikipedia: Vector space
[10]百度百科词条：向量空间