definition of sum of finite number of subsets of module
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 definition of sum of finite number of subsets 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:
\( R\): \(\in \{\text{ the rings }\}\)
\( M\): \(\in \{\text{ the } R \text{ modules }\}\)
\( \{S_1, ..., S_n\}\): \(\subseteq Pow (M)\)
\(*S_1 + ... + S_n\): \(= \{m \in M \vert \exists m_1 \in S_1, ..., \exists m_n \in S_n (m = m_1 + ... + m_n)\}\)
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Conditions:
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2: Note
Typically but not necessarily, \(S_j\) s are some submodules.
Let us see that when \(S_j\) s are some submodules, \(S_1 + ... + S_n\) is a submodule.
1) \(\forall m, m' \in S_1 + ... + S_n (m + m' \in S_1 + ... + S_n)\) (closed-ness under addition): \(m = s_1 + ... + s_n\) and \(m' = s'_1 + ... + s'_n\), and \(m + m' = s_1 + ... + s_n + s'_1 + ... + s'_n = (s_1 + s'_1) + ... + (s_n + s'_n)\), but \(s_j + s'_j \in S_j\), because \(S_j\) is a submodule.
2) \(\forall m, m' \in S_1 + ... + S_n (m + m' = m' + m)\) (commutativity of addition): \(m = s_1 + ... + s_n\) and \(m' = s'_1 + ... + s'_n\), and \(m + m' = s_1 + ... + s_n + s'_1 + ... + s'_n = s'_1 + ... + s'_n + s_1 + ... + s_n = m' + m\).
3) \(\forall m, m', m'' \in S_1 + ... + S_n ((m + m') + m'' = m + (m' + m''))\) (associativity of additions): \(m = s_1 + ... + s_n\), \(m' = s'_1 + ... + s'_n\), and \(m'' = s''_1 + ... + s''_n\), and \((m + m') + m'' = (s_1 + ... + s_n + s'_1 + ... + s'_n) + s''_1 + ... + s''_n = s_1 + ... + s_n + (s'_1 + ... + s'_n + s''_1 + ... + s''_n) = m + (m' + m'')\).
4) \(\exists 0 \in S_1 + ... + S_n (\forall m \in S_1 + ... + S_n (m + 0 = m))\) (existence of 0 element): \(0 = 0 + ... + 0 \in S_1 + ... + S_n\) where \(0 \in S_j\) because \(S_j\) is a submodule, and \(m + 0 = m\).
5) \(\forall m \in S_1 + ... + S_n (\exists m' \in S_1 + ... + S_n (m' + m = 0))\) (existence of inverse vector): \(m = s_1 + ... + s_n\), and \(m' := (- s_1) + ... + (- s_n) \in S_1 + ... + S_n\), because \(S_j\) is a submodule, \(m' + m = (- s_1) + ... + (- s_n) + s_1 + ... + s_n = 0\).
6) \(\forall m \in S_1 + ... + S_n, \forall r \in R (r . m \in S_1 + ... + S_n)\) (closed-ness under scalar multiplication): \(m = s_1 + ... + s_n\), and \(r . m = r (s_1 + ... + s_n) = r s_1 + ... + r s_n \in S_1 + ... + S_n\), because \(S_j\) is a submodule and so, \(r s_j \in S_j\).
7) \(\forall m \in S_1 + ... + S_n, \forall r_1, r_2 \in R ((r_1 + r_2) . m = r_1 . m + r_2 . m)\) (scalar multiplication distributability for scalars addition): \(m = s_1 + ... + s_n\), and \((r_1 + r_2) . m = (r_1 + r_2) . (s_1 + ... + s_n) = r_1 (s_1 + ... + s_n) + r_2 (s_1 + ... + s_n) = r_1 . m + r_2 . m\).
8) \(\forall m, m' \in S_1 + ... + S_n, \forall r \in R (r . (m + m') = r . m + r . m')\) (scalar multiplication distributability for elements addition): \(m = s_1 + ... + s_n\) and \(m' = s'_1 + ... + s'_n\), and \(r . (m + m') = r (s_1 + ... + s_n + s'_1 + ... + s'_n) = r (s_1 + ... + s_n) + r (s'_1 + ... + s'_n) = r . m + r . m'\).
9) \(\forall m \in S_1 + ... + S_n, \forall r_1, r_2 \in R ((r_1 r_2) . m = r_1 . (r_2 . m))\) (associativity of scalar multiplications): \(m = s_1 + ... + s_n\), and \((r_1 r_2) . m = (r_1 r_2) (s_1 + ... + s_n) = r_1 (r_2 (s_1 + ... + s_n)) = r_1 . (r_2 . m)\).
10) \(\forall m \in S_1 + ... + S_n (1 . m = m)\) (identity of 1 multiplication): \(m = s_1 + ... + s_n\), and \(1 . m = 1 (s_1 + ... + s_n) = s_1 + ... + s_n = m\).
