Algebra - group, ring, field, vector space, module, algebra
Define.
Given a set $S$,
We say $* : S \times S \to S$ is a binary operation(이항 연산)
if $*$ satisfies for any $x,y,z \in S$,
$(x*y)*z = x*(y*z)$ called the associative law(결합 법칙).
We call $(S, *)$ the semi-group(반군).
e.g) $S= \mathbb{N}$, $*=+$. $(a+b)+c = a+(b+c)$.
Non e.g) $S= \mathbb{N}$, $*=-$. : not binary operation.
e.g) $S=\{ 0, 1 \}$. $*= \times$. ($S$를 잘 선택해도 성립함)
Define.
Given a semi-group, $(M, *)$,
$\text{If }\exists e \in M \ \text{s.t.} \ \forall a \in M, \ a * e = e * a = a$.
We will call $e$ the identity of $M$.
and call $(M, *)$ the monoid.
Define.
Given a monoid $(G, *)$, we say $G$ is a group,
if $G$ satisfies the following : $\forall x \in G , \ \exists y \in G \ \text{s.t.} \ x*y=y*x=e$.
e.g) (1) $( \mathbb{N}, +)$: semi-gp, not monoid.
($\because a+e=e+a=a, \ e= 0 \ne \mathbb{N}$)
(2) $( \mathbb{N}, \times)$ : monoid, but not a group.
($\because a \times e = e \times a = a , \ e=1$.
$a \times x = x \times a = e =1 , \ x = \frac{1}{a}$)
(3) $( \mathbb{Z} , +)$ : group.
[$a+e=e+a=a \Rightarrow e=0 \in \mathbb{Z}$
$a+x=x+a=0 \Rightarrow x=-a \in \mathbb{Z}$]
(4) $\underbrace{(\mathbb{N} , +)}_{\text{semi-gp}} \subset \underbrace{(\mathbb{Z} , +)}_{\text{group}} \subset \underbrace{(\mathbb{Q} , +) \subset (\mathbb{R}, +)}_{\text{gps}}$
(5) $\underbrace{(\mathbb{Z} - \{ 0 \} , \times)}_{\text{semi-group}} \subset \underbrace{ (\mathbb{Q} - \{ 0 \} , \times ) \subset ( \mathbb{R} - \{ 0 \}, \times)}_{\text{gps}}$
(6) $( GL_n(\mathbb{R}) , \times)$ : group. ($\times$ : matrix-multiplication)(General Linear Group(nxn))
$= \{ A \in \text{Mat}_{n \times n} ( \mathbb{R} ) \mid \text{det} A \ne 0 \}$.
$e = \text{Id} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & 1 \end{pmatrix}$
$A^{-1}$ : the inverse of $A$.
(7) Given a set $A$, let $\mathcal{F} = \{ f : A \to A \mid f \ \text{is 1-1 onto} \}$.
Then $(\mathcal{F} , \circ )$ : group. ($\circ$ : composition of fns)
[$f \circ id = id \circ f = f$
$f \circ f^{-1} = f^{-1} \circ f = id$]
Remark.
(1) Given a monoid, such $e$ is unique.
(2) Given a group, for each $a \in (G, *) = G$, there exists unique $x \in G \ \text{s.t.} \ a*x = x*a = e$.
(Exercise) 2개 있다고 가정하고 풀면 2개가 같다는 결론이 나옴으로써 증명가능.
Define.
Given a group $(G, *)$,
We say $(G, *)$ is a commutative group if $\forall x,y \in G$, $x*y=y*x$.
If $*=+$, $(G, +)$ is called the abelian group.
Remark.
There is an operation which does not satisfy the associative law.
e.g) Lie-bracket. $[ \ , \ ]$.
$[[A,B],C] \ne [A, [B,C]]$
Define.
A set $R$ is a ring if $R$ is equipped with the addtion $+$ and multiplication $\cdot$.
$+ : R \times R \to R$
$ \cdot : R \times R \to R$
such that
(1) $(R, +)$ is an abelian gp.
(2) $(R, \cdot)$ is a semi-gp.
(3) $+, \cdot$ is compatible, i.e.,
$\begin{cases} (x+y) \cdot z = x \cdot z + y \cdot z \\ x \cdot (y+z) = x \cdot y + x \cdot z \end{cases} \ \ \ \ \ \ \forall x,y,z \in \mathbb{R}$
Define.
We say a ring $(\mathbb{F}, +, \cdot)$ is a field if $\{ \mathbb{F} - \{ 0 \} , \cdot \}$ is a commutative group.
e.g) $(\mathbb{R} , + , \times )$ is a field.
e.g) $(\mathbb{Q}, +, \times )$ is a field.
Define.
Given a ring $R$, a set $M$,
we say $M$ is a $R$-module if $M$ is equipped with the addition $+$ and the scalar multiplication
$R \times M \to M$ such that
$(r,m) \mapsto r \cdot m $ ($r$ : scalar)
(1) $(M, +)$ : abelian gp
(2) $(r_1 + r_2 ) \cdot m = r_1 m + r_2 m$, $\forall r_1 , r_2 \in R , m \in M$.
(3) $r \cdot (m_1 + m_2 ) = r \cdot m_1 + r \cdot m_2$, $\forall r \in R , \forall m_1 , m_2 \in M$.
(4) $O_R \cdot m = O_M$ $\forall m \in M$. ($O$는 덧셈의 항등원)
(5) $(r_1 r_2 ) \cdot m = r_1 \cdot (r_2 \cdot m )$ $\forall r_1 , r_2 \in R , \forall m \in M$.
Define.
Given a Field $\mathbb{F}$, we say $V$ is a vector space over $\mathbb{F}$ if $V$ is a $\mathbb{F}$-module.
Define.
Given a ring $R$ and a set $A$, we say $A$ is the algebra if $A$ has 3 operations with compatability.
$\begin{cases} + : A \times A \to A \ \ \ \ \text{(abelian)} \\ \times : A \times A \to A \ \ \ \ \text{(ring)}\\ \cdot : R \times A \to A \ \ \ \ \text{(scalar multiplication)} \end{cases}$
