definition of generator of module
Topics
About: module
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of %ring name% module.
Target Context
- The reader will have a definition of generator of module.
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:
\( R\): \(\in \{\text{ the rings }\}\)
\( M\): \(\in \{\text{ the modules over } R\}\)
\(*S\): \(\subseteq M\)
//
Conditions:
\(\forall p \in M (\exists S' \in \{\text{ the finite subsets of } S\}, \exists r^j \in R (p = \sum_{b_j \in S} r^j b_j))\)
//
\(S\) has to be a finite subset of \(B\), because otherwise, \(p = \sum_{b_j \in S} r^j b_j\) would not make sense without \(M\) equipped with any norm: definition of convergence of infinite series requires a norm.
2: Natural Language Description
For any ring, \(R\), and any module, \(M\), over \(R\), any (possibly uncountable) subset, \(S \subseteq M\), such that each element of \(M\) is a linear combination of some (finite) elements of \(S\)
3: Note
Any module has a generator, because the module itself is a generator.
Each basis is a generator, but a generator is not necessarily a basis, because the generator may not be linearly independent.
Furthermore, a generator may not be able to be restricted to any basis: for example, let \(R := \mathbb{Z}\); \(M := \mathbb{Z} / (2 \mathbb{Z})\), with the operations, \([z_1] + [z_2] = [z_1 + z_2]\), well-defined because \([z_1 + 2 k + z_2 + 2 l] = [z_1 + z_2 + 2 (k + l)] = [z_1 + z_2]\), and \(z_1 [z_2] = [z_1 z_2]\), well-defined because \([z_1 (z_2 + 2 k)] = [z_1 z_2 + 2 z_1 k] = [z_1 z_2]\): \(M = \{[0], [1]\}\). \(M\) is indeed a module: 1) \([m_1] + [m_2] = [m_1 + m_2] \in M\); 2) \([m_1] + [m_2] = [m_1 + m_2] = [m_2 + m_1] = [m_2] + [m_1]\); 3) \(([m_1] + [m_2]) + [m_3] = [m_1 + m_2] + [m_3] = [m_1 + m_2 + m_3] = [m_1] + [m_2 + m_3] = [m_1] + ([m_2] + [m_3])\); 4) \([m] + [0] = [m + 0] = [m]\); 5) \([1] + [1] = [1 + 1] = [2] = [0]\) and \([0] + [0] = [0 + 0] = [0]\); 6) \(r [m] = [r m] \in M\); 7) \((r_1 + r_2) [m] = [(r_1 + r_2) m] = [r_1 m + r_2 m] = [r_1 m] + [r_2 m] = r_1 [m] + r_2 [m]\); 8) \(r ([m_1] + [m_2]) = r [m_1 + m_2] = [r (m_1 + m_2)] = [r m_1 + r m_2] = [r m_1] + [r m_2] = r [m_1] + r [m_2]\); 9) \((r_1 r_2) [m] = [(r_1 r_2) m] = [r_1 (r_2 m)] = r_1 [r_2 m] = r_1 (r_2 [m])\); 10) \(1 [m] = [1 m] = [m]\). Then, \(\{[1]\}\) is a generator, because \([0] = 2 [1] = [2] = [0]\) and \([1] = 1 [1]\), but the generator is not linearly independent, because \(2 [1] = [0]\), so, is not any basis, or cannot be restricted to be any basis.
As any vectors space is a module, 'generator of vectors space' is nothing but a generator of module.
