08.03 Unique Factorization Domains (U.F.D.s)

Definition.

• $R$,
  • integral domain
• $r \neq 0 \in R$,
  • not a unit
• $p \neq 0 \in R$,

(1) $r$ is said to be irreducible if

\[\exsits a, b \in R, \text{ s.t. } r = ab \Rightarrow a \text{ or } b \text{ is unit.}\]

(2) $p$ is said to be prime if $(p)$ is prime ieal.

(3) $a \in R$ and $b \in R$ are said to be associte if there is unit $u \in R$ such that

\[a = ub .\]

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Definition. U.F.D.

• $R$,
  • integral domain

$R$ is said to be Unique Factorization Domain (U.F.D.) if for every $r \neq 0 \in R$, which is not unit,

(i) there exists irreducibles $p_{1},\ldots, p_{n} \in R$ such that

\[r = p_{1}\cdots p_{n} .\]

(ii) the decomposition in (i) is unique up to associates; namely if $r = q_{1}\cdots q_{m}$ is another factorization, then $m = n$ and

\[\forall i = 1, \ldots, n, \ \exists u_{i} \in R, \text{ s.t. } u_{i}: \text{ unit}, \ p_{1} = u_{i} q_{i} .\]

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