description/proof of that for module, element \(0\)-scalar multiplied is \(0\)
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 \(0\)-scalar multiplied is \(0\).
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:
\(\forall m \in M (0 m = 0)\)
//
2: Proof
Whole Strategy: Step 1: see that \((1 + 0) m = m + 0 m = 1 m = m\).
Step 1:
\((1 + 0) m = 1 m + 0 m = m + 0 m\), but the left hand side is \(1 m\), because \(1 + 0 = 1\), \(= m\), so, \(m + 0 m = m\).
So, \((- m) + m + 0 m = (- m) + m = 0\), but the left hand side is \(((- m) + m) + 0 m = 0 + 0 m = 0 m\), so, \(0 m = 0\).
