definition of unique factorization domain
Topics
About: ring
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of integral domain.
- The reader knows a definition of irreducible element of commutative ring.
Target Context
- The reader will have a definition of unique factorization domain.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(*R\): \(\in \{\text{ the integral domains }\}\)
\( U\): \(= \{\text{ the units of } R\}\)
\( I\): \(= \{\text{ the irreducible elements of } R\}\)
//
Conditions:
\(\forall p \in R (\exists u \in U, \exists i_j \in I (p = u i_1 ... i_n))\)
\(\land\)
(
\(\exists u' \in U, \exists i'_j \in U (p = u' i'_1 ... i'_m)\)
\(\implies\)
\(m = n \land \exists f: \{i_1, ..., i_n\} \to \{i'_1, ..., i'_n\} \in \{\text{ the bijections }\}, \exists u_j \in U (i'_j = u_j f (i_j))\)
)
//
2: Natural Language Description
Any integral domain, \(R\), such that for the set of the units of \(R\), \(U\), and the set of the irreducible elements of \(R\), \(I\), any element, \(p \in R\), can be expressed as \(p = u i_1 ... i_n\) for a \(u \in U\) and an \(i_j \in I\), and if \(p = u' i'_1 ... i'_m\) for a \(u' \in U\) and an \(i'_j \in I\), \(m = n\) and there is a bijection, \(f: \{i_1, ..., i_n\} \to \{i'_1, ..., i'_n\}\), such that \(i'_j = u_j f (i_j)\) for a \(u_j \in U\)
3: Note
This categorization is useful because some types of rings are known to be unique factorization domains: especially, any principal integral domain (of which each Euclidean domain (of which \(\mathbb{Z}\) is) is) is a unique factorization domain.
