description/proof of that range of linear map between modules is submodule of codomain
Topics
About: module
The table of contents of this article
Starting Context
- The reader knows a definition of %ring name% module.
- The reader knows a definition of linear map.
Target Context
- The reader will have a description and a proof of the proposition that the range of any linear map between any modules is a submodule of the codomain.
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:
\(R\): \(\in \{\text{ the rings }\}\)
\(M_1\): \(\in \{\text{ the } R \text{ modules }\}\)
\(M_2\): \(\in \{\text{ the } R \text{ modules }\}\)
\(f\): \(: M_1 \to M_2\), \(\in \{\text{ the linear maps }\}\)
Statements:
\(f (M_1) \in \{\text{ the submodules of } M_2\}\)
//
2: Proof
Whole Strategy: Step 1: see that \(f (M_1)\) satisfies the requirements to be a module.
Step 1:
1) \(\forall f (m), f (m') \in f (M_1) (f (m) + f (m') \in f (M_1))\) (closed-ness under addition): \(f (m) + f (m') = f (m + m') \in f (M_1)\).
2) \(\forall f (m), f (m') \in f (M_1) (f (m) + f (m') = f (m') + f (m))\) (commutativity of addition): it holds on ambient \(M_2\).
3) \(\forall f (m), f (m'), f (m'') \in f (M_1) ((f (m) + f (m')) + f (m'') = f (m) + (f (m') + f (m'')))\) (associativity of additions): it holds on ambient \(M_2\).
4) \(\exists 0 \in f (M_1) (\forall f (m) \in f (M_1) (f (m) + 0 = f (m)))\) (existence of 0 element): \(f (0) = 0 \in f (M_1)\) is the one.
5) \(\forall f (m) \in f (M_1) (\exists f (m') \in f (M_1) (f (m') + f (m) = 0))\) (existence of inverse element): \(f (- m) = - f (m) \in f (V_1)\) is the one.
6) \(\forall f (m) \in f (M_1), \forall r \in R (r . f (m) \in f (M_1))\) (closed-ness under scalar multiplication): \(f (r m) = r . f (m) \in f (M_1)\).
7) \(\forall f (m) \in f (M_1), \forall r_1, r_2 \in R ((r_1 + r_2) . f (m) = r_1 . f (m) + r_2 . f (m))\) (scalar multiplication distributability for scalars addition): it holds on ambient \(M_2\).
8) \(\forall f (m), f (m') \in f (M_1), \forall r \in R (r . (f (m) + f (m')) = r . f (m) + r . f (m'))\) (scalar multiplication distributability for elements addition): it holds on ambient \(M_2\).
9) \(\forall f (m) \in f (M_1), \forall r_1, r_2 \in R ((r_1 r_2) . f (m) = r_1 . (r_2 . f (m)))\) (associativity of scalar multiplications): it holds on ambient \(M_2\).
10) \(\forall f (m) \in f (M_1) (1 . f (m) = f (m))\) (identity of 1 multiplication): it holds on ambient \(M_2\).
