Fundamental Mathematical Knowledge¶

This section will introduce "fundamental mathematical knowledge" — note the quotation marks, as the content is not necessarily all that basic.

Algebraic Systems and Modern Algebra¶

Within a set, if one or more algebraic operations (Algebraic Operations) are defined, we generally refer to it as an algebraic system (Algebraic System), also called an algebraic structure (Algebraic Structure).

As a continuously advancing and refining branch of mathematics, the scope of algebra has gradually expanded. The sets it studies have grown from the classical number sets — integers, rationals, reals, and complex numbers — to objects such as vectors, matrices, and linear operators, with a focus on the algebraic operations defined on them. These topics collectively form what is now known as Modern Algebra, or Abstract Algebra.

The algebraic operations mentioned above are rules defined among elements of a set and are closely related to whether a set can form an algebraic system. They are extensions of familiar arithmetic operations such as addition, subtraction, multiplication, and division. By defining appropriate algebraic operations, a set can form algebraic systems such as groups, rings, fields, lattices, etc., which are what we will introduce below.

Groups¶

Given a non-empty set $G\neq\varnothing$ and a binary algebraic operation " $\circ$ " defined on it, if they satisfy the following properties:

Closure: $\forall v, u \in G, \quad v \circ u \in G;$
Associativity: $\forall v, u, w \in G, \quad (v \circ u) \circ w = v \circ (u \circ w);$
Identity: $\exists e \in G, \forall v \in G, \quad e \circ v = v;$
Inverse: $\forall v \in G, \exists v^{-1} \in G, \quad v^{-1} \circ v = e;$

then the set $G$ is said to form a group (Group) under this algebraic operation, denoted $(G,\circ)$ .

A very common example is the integer addition group $(\mathbb{Z},+)$ . It is easy to verify that it is closed under addition, satisfies associativity, has the integer $0$ as the identity element, and for every integer $m$ has its additive inverse $-m$ as the inverse element. Similarly, one can verify that the set of positive rationals $(\mathbb{Q}_+,\times)$ forms a group under multiplication (the identity element is $1$ , and for every element $a$ , the inverse is $\frac{1}a$ ); the set of all invertible $m \times m$ matrices over the real field $\mathbb{R}$ forms a group under matrix multiplication (the identity element is the $m \times m$ identity matrix $E_m$ , and for every element $A$ , the inverse is its matrix inverse $A^{-1}$ ). In modern algebra, this is called the $m$ -th order general linear group $GL_m(\mathbb{R})$ .

Additionally, here is an even simpler example: the multiplication group defined on the set $\{-1,1\}$ , i.e., $(\{-1,1\},\times)$ , which is also easy to verify as forming a group.

In modern algebra, the branch that studies groups is called Group Theory.

Semigroups and Monoids¶

In modern algebra, some algebraic systems possess partial properties of a ring. Although they are not within our main scope of discussion, they also have broad applications and significant research value:

For a non-empty set that is closed under a binary algebraic operation,

If it only satisfies associativity, then the set is said to form a semigroup (Semigroup) under that operation;
If the set, in addition to being closed, satisfies associativity and has an identity element, then it is said to form a monoid (Monoid) under that operation.

Therefore, we can consider:

A monoid is a semigroup with an identity element;
A group is a monoid in which every element has an inverse.

For example, the positive integers form a semigroup under integer addition, while the non-negative integers form a monoid under integer addition, since zero can be regarded as the identity element of integer addition.

Abelian Groups¶

Given a group $(G,\circ)$ , if it satisfies commutativity (Commutativity), i.e. $\forall v, u \in G,$ $ v \circ u = u \circ v, $ then this group is called an abelian group or Abel group (Abelian Group).

It is easy to see that among the examples mentioned above, the integer addition group $(\mathbb{Z},+)$ is an abelian group, but the $m$ -th order general linear group $GL_m(\mathbb{R})$ is not.

Rings and Fields¶

Given a non-empty set $R\neq\varnothing$ and two binary algebraic operations " $+$ " and " $\circ$ " defined on it, if they satisfy the following properties:

$(R,+)$ forms an abelian group;
$R$ satisfies associativity under operation " $\circ$ ": $\forall v, u, w \in R,$ $(v \circ w) \circ u = v \circ (w \circ u);$
Distributivity: $\forall v, u, w \in R,$ $w \circ (v + u) = w \circ v + w \circ u$ and $(v + u) \circ w = v \circ w + u \circ w$ both hold;

then the set $R$ is said to form a ring (Ring) under these two algebraic operations, denoted $(R,+,\circ)$ , where operations " $+$ " and " $\circ$ " are commonly called addition and multiplication, respectively.

If the multiplication on ring $R$ has an identity element, i.e. $\exists e \in G, \forall v \in G,$ $e \circ v = v,$ then ring $R$ is called a ring with identity (Ring with identity);
If the multiplication on ring $R$ satisfies commutativity, then it is called a commutative ring (Commutative Ring);
If for every element $a \neq 0$ in ring $R$ (other than the additive identity) there exists a multiplicative inverse $a^{-1}$ , then $R$ is called a division ring (Division Ring);
If ring $R$ is both a commutative ring and a division ring, then ring $R$ is a field (Field).

In some textbooks, a ring is assumed to contain a multiplicative identity by default, and a ring without a multiplicative identity is called a pseudo ring (Pseudo Ring).

In modern algebra, the branches that study rings and fields are called Ring Theory and Field Theory, respectively.

In some Traditional Chinese contexts, "field" and "field theory" are often referred to as "体" (body) and "体论" (body theory) (written as「體」and「體論」in Traditional Chinese).

Order¶

Exponentiation: Analogous to powers of numbers, we define the exponentiation of elements in a group. For $v \in G, m$ a positive integer,

$v^0 = e;$
$v^m = v \circ v \circ \cdots \circ v,$ where there are $m$ copies of $v$ participating in the operation;
$v^{-m} = \left(v^{-1}\right)^m;$

Order of an element: For any given element $v \in G,$ if a positive integer $m$ satisfies $v^m = e,$ then the order of element $v$ is said to be $m$ . If no such positive integer exists, the element is said to have infinite order.

For example, in the group $\left(\{1,-1,+\mathrm{j},-\mathrm{j}\},\times\right)$ , the orders of each element are as follows:

Element	Order
1	1
-1	2
$+\mathrm{j}$	4
$-\mathrm{j}$	4

Homomorphism¶

A homomorphism between algebraic systems refers to a mapping between different algebraic systems that preserves the algebraic operations.

Specifically, for groups $(G,\circ)$ and $(H,\ast)$ , if a mapping $\psi: G \to H$ satisfies $\forall v, u \in G,$

$\psi(v \circ u) = \psi(v) \ast \psi(u),$

then the mapping $\psi$ is called a group homomorphism from $G$ to $H$ .

References¶

Yang Zixu, Modern Algebra (4^th Edition), Higher Education Press
Introduction to Group Theory - OI-Wiki