description/proof of that for module, inverse of nonzero element is nonzero 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 for any module, the inverse of any nonzero element is a nonzero 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:
\(R\): \(\in \{\text{ the rings }\}\)
\(M\): \(\in \{\text{ the } R \text{ modules }\}\)
\(m\): \(\in M\)
//
Statements:
\(m \neq 0\)
\(\implies\)
\(- m \neq 0\)
//
2: Proof
Whole Strategy: Step 1: suppose that \(- m = 0\) and find a contradiction.
Step 1:
\(- m\) is nothing but the inverse element of \(m\).
Let us suppose that \(- m = 0\).
\(m + (- m) = 0\), but the left hand side would be \(m + 0 = m\), so, \(m = 0\), a contradiction against that \(m \neq 0\).
So, \(- m \neq 0\).
