description/proof of that module has unique \(0\) element
Topics
About: module
The table of contents of this article
Starting Context
- The reader knows a definition of %ring name% module.
Target Context
- The reader will have a description and a proof of the proposition that any module has the unique \(0\) 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:
\(R\): \(\in \{\text{ the rings }\}\)
\(M\): \(\in \{\text{ the } R \text{ modules }\}\)
//
Statements:
\(\forall m \in \{\text{ the zero elements of } M\} (m = 0)\)
//
2: Proof
Whole Strategy: Step 1: see that \(m = 0\).
Step 1:
The definition of module requires the existence of a \(0 \in M\), but does not directly require that it is the unique zero element.
Let \(m \in M\) be any zero element.
\(m + 0 = m\), because \(0\) is a zero element, but \(m + 0 = 0\), because \(m\) is a zero element.
So, \(m = 0\).
