description/proof of that for module, \(0\) 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, \(0\) each-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:
\(\forall r \in R (r 0 = 0)\)
//
2: Proof
Whole Strategy: Step 1: see that \(r (0 + 0) = r 0 = r 0 + r 0\).
Step 1:
For each \(r \in R\), \(r (0 + 0) = r 0\), because \(0 + 0 = 0\), but the left hand side is \(r 0 + r 0\), so, \(r 0 + r 0 = r 0\).
So, \(- (r 0) + r 0 + r 0 = - (r 0) + r 0 = 0\), but the left hand side is \((- (r 0) + r 0) + r 0 = 0 + r 0 = r 0 + 0 = r 0\), so, \(r 0 = 0\).
