description/proof of that identity map on module is linear
Topics
About: module
The table of contents of this article
Starting Context
- The reader knows a definition of linear map.
Target Context
- The reader will have a description and a proof of the proposition that the identity map on any module is linear.
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:
\(R\): \(\in \{\text{ the rings }\}\)
\(M\): \(\in \{\text{ the } R \text{ modules }\}\)
\(id_M\): \(: M \to M, m \mapsto m\)
//
Statements:
\(id_M \in \{\text{ the linear maps }\}\)
//
2: Proof
Whole Strategy: Step 1: see that \(id_M\) satisfies the conditions to be linear.
Step 1:
Let \(r_1, r_2 \in R\) and \(m_1, m_2 \in M\) be any.
\(id_M (r_1 m_1 + r_2 m_2) = r_1 m_1 + r_2 m_2 = r_1 id_M (m_1) + r_2 id_M (m_2)\).
So, \(id_M\) is linear.
