definition of module as direct sum of finite number of submodules
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 sum of finite number of subsets of module.
Target Context
- The reader will have a definition of module as direct sum of finite number of submodules.
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:
\( R\): \(\in \{\text{ the rings }\}\)
\(*M\): \(\in \{\text{ the } R \text{ modules }\}\)
\( \{M_1, ..., M_n\}\): \(\subseteq \{\text{ the submodules of } M\}\)
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Conditions:
\(\forall M_j \in \{M_1, ..., M_n\} (M_j \cap (M_1 + ... + M_{j - 1} + \widehat{M_j} + M_{j + 1} + ... + M_n) = \{0\})\)
\(\land\)
\(M = M_1 + ... + M_n\)
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2: Note
The name is "module as direct sum of finite number of submodules" instead of just "direct sum of finite number of submodules" because the direct sum does not create any new module but it is about judging an existing module as the direct sum.
Do not confuse this definition with 'direct sum of modules' (although they are most prevalently confused), which is about creating a new module from the constituent modules. For the submodules, the direct sum of the submodules is not exactly the direct sum of the constituent modules by this definition: any element of \(M_1 \oplus M_2\) is of the form, \((m_1, m_2)\), while any element of \(M_1\), \(m_1\), is not of the form, so, is not any element of \(M_1 \oplus M_2\), so, \(M_1\) is not any submodule of \(M_1 \oplus M_2\); in fact, \(M_1 \oplus M_2\) is the direct sum of \(M_1 \times \{0\}\) and \(\{0\} \times M_2\) by this definition.
Certainly, \(M_1 \oplus ... \oplus M_n\) is 'modules - linear morphisms' isomorphic to \(M\), but still, they are not the same entity.
Inevitably, the decomposition of each \(m \in M\), \(m = m_1 + ... + m_n\) where \(m_j \in M_j\), is unique, because supposing that \(m = m_1 + ... + m_n = m'_1 + ... + m'_n\), \(m_j - m'_j = (- m_1 + m'_1) + ... + (- m_{j - 1} + m'_{j - 1}) + \widehat{(- m_j + m'_j)} + (- m_{j + 1} + m'_{j + 1}) + ... + (- m_n + m'_n)\), but as \(m_j - m'_j \in M_j\) and \((- m_1 + m'_1) + ... + (- m_{j - 1} + m'_{j - 1}) + \widehat{(- m_j + m'_j)} + (- m_{j + 1} + m'_{j + 1}) + ... + (- m_n + m'_n) \in (M_1 + ... + M_{j - 1} + \widehat{M_j} + M_{j + 1} + ... + M_n)\), \(m_j - m'_j \in \{0\}\), so, \(m_j = m'_j\).
