description/proof of that for unique factorization domain, if multiple of elements is divisible by irreducible element, at least 1 constituent is divisible by irreducible element
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: Proof
Starting Context
- The reader knows a definition of unique factorization domain.
Target Context
- The reader will have a description and a proof of the proposition that for any unique factorization domain, if the multiple of any elements is divisible by any irreducible element, at least 1 constituent is divisible by the irreducible element.
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:
\(R\): \(\in \{\text{ the unique factorization domains }\}\)
\(I\): \(= \{\text{ the irreducible elements of } R\}\)
\(p_1 ... p_n\): \(p_j \in R\)
\(i\): \(\in I\)
//
Statements:
\(i \vert p_1 ... p_n\)
\(\implies\)
\(\exists p_k \in \{p_1, ..., p_n\} (i \vert p_k)\)
//
2: Natural Language Description
For any unique factorization domain, \(R\), the set of the irreducible elements of \(R\), \(I\), any \(p_1 ... p_n\) where \(p_j \in R\), and any \(i \in I\), if \(i \vert p_1 ... p_n\), there is at least 1 \(p_k\) such that \(i \vert p_k\).
3: Proof
Let \(U\) be the set of the units of \(R\).
\(p_1 ... p_n = q i\) for a \(q \in R\).
\(p_j = u_j i_{j, 1} ... i_{j, l_j}\) where \(u_j \in U\) and \(i_{j, k} \in I\).
\(q = u i_1 ... i_l\) where \(u \in U\) and \(i_j \in I\).
So, \(u_1 i_{1, 1} ... i_{1, l_1} ... u_n i_{n, 1} ... i_{n, l_n} = (u_1 ... u_n) i_{1, 1} ... i_{1, l_1} ... i_{n, 1} ... i_{n, l_n} = u i_1 ... i_l i\) where \(u_1 ... u_n \in U\).
As the factorizations are unique, \(i = u' i_{j, k}\) for a \(i_{j, k}\) and a \(u' \in U\). Then, \(i \vert p_j\), because \(p_j = u_j i_{j, 1} ... i_{j, l_j} = u_j i_{j, 1} ... i_{j, k} ... i_{j, l_j} = u_j i_{j, 1} ... u'^{-1} i ... i_{j, l_j} = u_j u'^{-1} i_{j, 1} ... \hat{i_{j, k}} ... i_{j, l_j} i\) where \(u_j u'^{-1} i_{j, 1} ... \hat{i_{j, k}} ... i_{j, l_j} \in R\).
