# 00.0.群環体の定義

## 群(Group)

$G$ is Set.
Defined a mapping.
$$G \times G \longrightarrow G$$ $$a \in G, b \in G \quad (a,b) \mapsto a + b \in G$$ $G$ is Group. if $G$ fills following conditions.
1. $(a + b) + c = a + (b + c) \quad (a,b,c \in G)$
2. $\exists 0 \in G, \forall x \in G, 0 + x = x + 0 = x$
3. $\forall x \in G, \exists -x \in G, x + (-x) = -x + x = e$

## 可換群(Abelian Group)

$G$ is a Group.
$\forall a,b \in G, a + b = b + a$
then $G$ is 可換群(アーベル群)

## 環(Ring)

$R$:可換群 $a,b,c \in R$
* $(ab)c = a(bc)$
* $a(b + c) = ab + ac, (a + b)c = ac + bc$
* $\exists 1 \text{s.t.} \forall x \in R, x1 = 1x = x$

## 可換環(Ring)

$R$:環
$a,b \in R$
* $ab = ba$

## 零因子

$R$:可換環 $a \in R$
if
$\exists x \in R \text{ s.t. } x \neq 0, ax = 0$
then
$a$ is called 零因子

## 整域(Integral Domain)

$D$:零因子を持たない可換環

## 体(Field)

$F$:可換環 $\forall x \in F - \{0\}, \exists x^{-1} \text{ s.t. } xx^{-1} = 1$

## 部分群

$G$:群
$H \subset G$ and $H$:群

## 部分環

$R$:環
$S \subset R$ and $S$:環

## 部分体

$F$:体
$E \subset F$ and $E$:体
($F$と同じ演算 ($+,\cdot$))