Ring - 2

원소가 하나 뿐인 Ring.

Let $R$ is a ring, THEN

• Unity: a multiplicative identity.
• unit: 곱에 대한 역원이 존재하는 원소

If the multiplicative identity exist, it is unique.

Let $R$ be a ring with unity,

$R$ is a division ring, IF every non-zero element has a mutliplicative inverse.

즉, $R\setminus \{0\}$ is a group w.r.t. multiplication.

Division Ring이 곱에 대해 가환인지 여부에 따라

• Commutative division ring
  • Field
  • 곱셈에 대해 군을 이루면서, 그것이 가환군
• non-Commutative division ring
  • skew Field
  • 곱셈에 대해 군을 이루지만, 가환이 아님

• 덧셈은 component-wise addition으로 정의

• 곱셈은 quaterion units $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$에 대한 규칙에 따라 시행

• 곱셈에 대한 역원: ${\displaystyle \frac{w-x\mathbf{i}-y\mathbf{j}-z\mathbf{k}}{w^2+x^2+y^2+z^2}}$
• 곱셈에 대한 항등원: $1=(1,0,0,0)$
• $q\cdot \overline{q}=w^2+x^2+y^2+z^2$

Let $R$ is a ring, for $a,b\in R$,

IF $ab=0$, THEN $a$ and $b$ is called a zero-divisor.

In ${\mathbb{Z}}_n$, the zero-divisors are precisely the elements which are not relatively prime to $n$.

IF $p$ is a prime, THEN ${\mathbb{Z}}_p$ has no zero-divisor.

IF $p$ is a prime, THEN ${\mathbb{Z}}_p$ is a field.

즉, 모든 non-zero element가 multiplicative inverse를 가진다는 말!!

An Integral Domain is a commutative ring with Unitiy, and without zero-divisors.

For $n,m\in \mathbb{Z}$,

1. if $(n,m)=1$, then $\mathrm{\exists }x,y\in \mathbb{Z}$ s.t. $nx+my=1$.

2. if $(n,m)=d$, then $\mathrm{\exists }x,y\in \mathbb{Z}$ s.t. $nx+my=d$.

If there exist $n\in \mathbb{N}$ s.t. $na\equiv 0$ ($a\in R$),

then call the smallest $n$ as a “Characteristic of a Ring”.

※ If there is no such $n$, then we say Char- of a Ring = 0.

Let $R$ be a ring with unity 1,

then $n\cdot 1=0⟺n\cdot a=0\mathrm{\forall }a\in R$.