definition of %ring name% 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.
- The reader knows a definition of set.
Target Context
- The reader will have a definition of %ring name% 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:
\( R\): \(\in \{\text{ the rings }\}\)
\(*M\): \(\in \{\text{ the sets }\}\) with any \(+: M \times M \to M\) (addition) operation and any \(.: R \times M \to M\) (scalar multiplication) operation
//
Conditions:
1) \(\forall m_1, m_2 \in M (m_1 + m_2 \in M)\) (closed-ness under addition)
\(\land\)
2) \(\forall m_1, m_2 \in M (m_1 + m_2 = m_2 + m_1)\) (commutativity of addition)
\(\land\)
3) \(\forall m_1, m_2, m_3 \in M ((m_1 + m_2) + m_3 = m_1 + (m_2 + m_3))\) (associativity of additions)
\(\land\)
4) \(\exists 0 \in M (\forall m \in M (m + 0 = m))\) (existence of 0 element)
\(\land\)
5) \(\forall m \in M (\exists m' \in M (m' + m = 0))\) (existence of inverse vector)
\(\land\)
6) \(\forall m \in M, \forall r \in R (r . m \in M)\) (closed-ness under scalar multiplication)
\(\land\)
7) \(\forall m \in M, \forall r_1, r_2 \in R ((r_1 + r_2) . m = r_1 . m + r_2 . m)\) (scalar multiplication distributability for scalars addition)
\(\land\)
8) \(\forall m_1, m_2 \in M, \forall r \in R (r . (m_1 + m_2) = r . m_1 + r . m_2)\) (scalar multiplication distributability for elements addition)
\(\land\)
9) \(\forall m \in M, \forall r_1, r_2 \in R ((r_1 r_2) . m = r_1 . (r_2 . m))\) (associativity of scalar multiplications)
\(\land\)
10) \(\forall m \in M (1 . m = m)\) (identity of 1 multiplication)
//
2: Natural Language Description
Any set, \(M\), with any \(+: M \times M \to M\) (addition) operation and any \(.: R \times M \to M\) (scaler multiplication) operation with respect to any ring, \(R\), that satisfies these conditions: 1) for any elements, \(m_1, m_2 \in M\), \(m_1 + m_2 \in M\) (closed-ness under addition); 2) for any elements, \(m_1, m_2 \in M\), \(m_1 + m_2 = m_2 + m_1\) (commutativity of addition); 3) for any elements, \(m_1, m_2, m_3 \in M\), \((m_1 + m_2) + m_3 = m_1 + (m_2 + m_3)\) (associativity of additions); 4) there is a 0 element, \(0 \in M\), such that for each \(m \in M\), \(m + 0 = m\) (existence of 0 element); 5) for any element, \(m \in M\), there is an inverse element, \(m' \in M\), such that \(m' + m = 0\) (existence of inverse vector); 6) for any element, \(m \in M\), and any scalar, \(r \in R\), \(r . m \in M\) (closed-ness under scalar multiplication); 7) for any element, \(m \in M\), and any scalars, \(r_1, r_2 \in R\), \((r_1 + r_2) . m = r_1 . m + r_2 . m\) (scalar multiplication distributability for scalars addition); 8) for any elements, \(m_1, m_2 \in M\), and any scalar, \(r \in R\), \(r . (m_1 + m_2) = r . m_1 + r . m_2\) (scalar multiplication distributability for elements addition); 9) for any element, \(m \in M\), and any scalars, \(r_1, r_2 \in R\), \((r_1 r_2) . m = r_1 . (r_2 . m)\) (associativity of scalar multiplications); 10) for any element, \(m \in M\), \(1 . m = m\) (identity of 1 multiplication)
3: Note
\(.\) is often omitted in notations like \(r m\) instead of \(r . m\).
The requirements 1) ~ 10) are in fact parallel to those for vectors space; the difference between vectors space and module is only that the scalar structure is a field or a ring, which makes a significant difference, because for example, for a module, \(r^1 m_1 + r^2 m_2 + r^3 m_3 = 0\) where \(r^1 \neq 0\) does not imply that \(m_1\) is a linear combination of \(m_2\) and \(m_3\), because \({r^1}^{-1}\) is not guaranteed to exist.
