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 Domainfor $a,b,p \in D$ 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.
一意分解整域(Unique Factorization Domain)
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
$\text{Img }f = f(R) = \{f(x)|x \in R\}$
$\text{Ker }f = \{x|f(x) = 0_{R'}\}$