description/proof of cancellation rule on integral 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: Proof
Starting Context
- The reader knows a definition of integral domain.
Target Context
- The reader will have a description and a proof of the proposition that the cancellation rule holds on any integral 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:
\(R\): \(\in \{\text{ the integral domains }\}\)
\(p_1\): \(\in R\)
\(p_2\): \(\in R\)
\(p_3\): \(\in R\)
//
Statements:
(
\(p_1 \neq 0\)
\(\land\)
\(p_1 p_2 = p_1 p_3\)
)
\(\implies\)
\(p_2 = p_3\)
//
2: Natural Language Description
For any integral domain, \(R\), and any elements, \(p_1, p_2, p_3 \in R\), if \(p_1 \neq 0\) and \(p_1 p_2 = p_1 p_3\), \(p_2 = p_3\).
3: Proof
Let us suppose that \(p_1 \neq 0\) and \(p_1 p_2 = p_1 p_3\).
\(p_1 (p_2 - p_3) = 0\). As \(p_1 \neq 0\), \(p_2 - p_3 = 0\), by the definition of integral domain. So, \(p_2 = p_3\).
