# 00.2.環の様々な定義

## 標数

$R$:環
において、$1_R$の自然数$n$個の和を$n1_R$と定義する. $\exists n \in \mathbb{N}, n1_R = 0$
のとき、$R$の標数を$n$と定義して、
$\text{char}(R) = n$
とかく.
$\nexists n \in \mathbb{N}, n1_R = 0$
のとき. $\text{char}(R) = 0$

## 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

## 零元(Zero Element)

$R$:Ring
$0$ is called a zero element

## 単位元(Identity Element)

$R$:Ring
$1$ is called an identity element

## 単元(Unit)または可逆元(Invertible Element)

$R$:Ring
$a$ is a unit $\Leftrightarrow \exists a' \in R$ such that $aa' = e$(単位元)

## 同伴(Associate)

$D$:Integral Domain
$a,b \in D$
if $\exists u \in D,u$:単元,$au=b$
then
$a$ Associate $b$
$a \sim b$
($a \sim id \Leftrightarrow a$:単元)

## 素元(Prime Element)

$D$:Integral Domain
for $a,b,p \in D$ and $p$ is not 単元
$p|ab \Rightarrow p|a \text{ or } p|b$
then $p$ is a prime element.

## 既約元(Irreducible Element)

$D$:Integral Domain
$p \in D, p \neq 0, p$:単元ではない
$a,b \in D, p = ab \Rightarrow a$:単元 or $b$:単元
then
$p$: is a irreducible element.

## イデアル(Ideal)

$R$:Ring
$I \subset R$ is a Right ideal if it fills following properties.
* $I$ is subgroup(for "+") of $R$
* $a \in I, r \in R \Rightarrow ar \in I$

## 単項イデアル(Principal Ideal)

$R$:可換環
$a \in R$
then
$(a) = \{xa|x \in R\}$ is called Principal Ideal generated by $a$.
すなわち
$I$:単項イデアル $\Leftrightarrow \exists a \in I, \forall x \in I \exists r \in R, x = ar$

## 単項イデアル整域(Principal Ideal Domain)

$D$:Indegral Domain
$I$:Ideal $\Rightarrow$ $I$ is principal.
then $D$ is called a principal ideal domain.

## 素イデアル(Prime Ideal)

$R$:可換環
$I$:イデアル
$a \notin I,b \notin I \Rightarrow ab \notin I$
($ab \in I \Rightarrow a \in I \text{ or } b \in I$)

## 極大イデアル(Maximal Ideal)

$R$:可換環
$M(\neq R)$:Ideal
$I(\varsupsetneqq M)$:Ideal $\Rightarrow I = R$
then $M$ is called a maximal ideal.

## 多項式環(Polynomial Ring)

$R$:Ring $R[x] = \{\sum_{i=0}^n a_i x^i|a_i \in R\}$
$+*$:通常の多項式の和と積

## 既約多項式(Irreducible Polynomial)

$R$:可換環
$f(x) \in R[x]$ and $\dim f(x) \ge 1$

if and only if $\nexists g(x),h(x) \in R[x]$ such that $\dim g(x) \ge 1, \dim h(x) \ge 1, f(x) = g(x)h(x)$
then
$f(x)$ is 既約多項式(irreducible polynomial)

## 準同型写像

Supporse
$R,R'$:Ring
$f:R \rightarrow R'$
$f:x \mapsto x'$
then
if
$f(x + y) = f(x) + f(y)$
$f(xy) = f(x)f(y)$ then
$f$ is called a homomorphism(準同型写像)
if $\forall x' \in G', \exists x, f(x) = x'$ then
$f$ is surjective(全射)
if $\forall x,y \in G, x \neq y \Rightarrow f(x) \neq f(y)$ then
$f$ is injective(単射)

## 核、像

$R,R'$:Ring
$f:R \mapsto R'$ is a homomorphism.
then
$\text{Img }f = f(R) = \{f(x)|x \in R\}$
$\text{Ker }f = \{x|f(x) = 0_{R'}\}$