description/proof of that for module, inverse of element is element \(- 1\)-scalar multiplied
Topics
About: module
The table of contents of this article
Starting Context
- The reader knows a definition of %ring name% module.
- The reader admits the proposition that for any module, each element \(0\)-scalar multiplied is \(0\).
- The reader admits the proposition that for any module, each element has the unique inverse.
Target Context
- The reader will have a description and a proof of the proposition that for any module, the inverse of each element is the element \(- 1\)-scalar multiplied.
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:
\(\forall m \in M (- m = (- 1) m)\)
//
2: Proof
Whole Strategy: Step 1: see that \(m + (- 1) m = 0\).
Step 1:
\(- 1 \in R\) is the additive inverse of \(1\), the multiplicative identity of \(R\).
\(- m\) is the inverse of \(m\).
\(m + (- 1) m = 1 m + (- 1) m = (1 + (- 1)) m = 0 m = 0\), by the proposition that for any module, each element \(0\)-scalar multiplied is \(0\).
So, \((- 1) m\) is an inverse of \(m\), but in fact, it is the inverse of \(m\), by the proposition that for any module, each element has the unique inverse.
So, \((- 1) m = - m\).
