description/proof of that for module, each element has unique inverse
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, each element has the unique inverse.
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:
\(\forall m \in M (\exists m', m'' \in M (m' + m = 0 \land m'' + m = 0) \implies m' = m'')\)
//
2: Proof
Whole Strategy: Step 1: see that \(m'' + m + m' = m'' = m'\).
Step 1:
\(m + m' = m' + m = 0\).
So, \(m'' + m + m' = m'' + 0 = m''\), but the left hand side is \((m'' + m) + m' = 0 + m' = m' + 0 = m'\).
So, \(m' = m''\).
