Prime & Irreducible element

2020-2학기, 대학에서 ‘현대대수1’ 수업을 듣고 공부한 바를 정리한 글입니다. 지적은 언제나 환영입니다 :)

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Definition. prime element

Let $R$ be a commutative ring, and $p \in R$.

if

• $p$ is not a zero, and not a unit
• for $a, b \in R$, $p \mid ab$ implies $p \mid a$ or $p \mid b$

then, $p$ is a “prime element”.

• Wikipedia/prime element

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Definition. irreducible element

Let $D$ be an integral domain, and $a \in D$.

if $a$ is not a product of two non-units,

then $a$ is a “irreducible element”.

• Wikipedia/irreducible element
• Mathworld.Wolfram/irreducible element

원래는 Irreducible Polynomial에 입각해 Irreducibility를 생각했는데, 그렇게 Irreducibility를 생각하는 것보단 Ring Theory에서 정의하는 Irreducibility 자체를 받아들이는게 좋을 듯!!

An element $a$ is irreducible

$\iff$ possible decompositions of $a$ into the product of two factors are of the form

\[a = u^{-1} \cdot (ua)\]

즉, 이것을 다시 말하면,

“If $a = pq$, then $p$ or $q$ is an unit.”

왜냐하면, 만약 $p$가 unit element라면,

\[\begin{aligned} a &= pq\\ p^{-1}a &= p^{-1}(pq) \\ p^{-1}a &= q \\ \end{aligned}\]

따라서 $a = pq = p(p^{-1}a) = 1 \cdot a$가 된다!!

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Theorem: Prime - Irreducible

$D$: Integral Domain

Prime element in $D$ $\implies$ Irreducible in $D$

일반적으로 역방향은 성립하지 않는다!!

proof.

($\implies$)

Since $p$ is a Prime element,

If $p \mid ab$ for some $a, b \in D$

then, $p \mid a$ or $p \mid b$.

Supp. $p = ab$, (Check) $a$ or $b$ is an unit.

Sine $p$ is a Prime element,

$p \mid ab$ $\implies$ $p \mid a$ or $p \mid b$

Say $p \mid a$, then $a = p \cdot a’$.

Then,

\[\begin{aligned} p &= ab = (p \cdot a') b \\ 1 &= a' b \end{aligned}\]

따라서 $b$는 $a’$를 multiplicative inverse로 갖는 unit이다.

반대로 $p \mid b$라면, $a$가 unit이다.

따라서 $p$는 irreducible이다. $\blacksquare$

($\impliedby$의 반례)

Integer Domain(또는 Qudratic Integr Ring)과 Algebraic Norm에 대한 개념을 알아야 반례를 쉽게 찾을 수 있다.

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Moreover, while an ideal generated by a prime element is a prime ideal,

it is not true in general that an ideal generated by an irreducible element is an irreducible ideal.