definition of least common multiples of subset of commutative ring
Topics
About: ring
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.
Target Context
- The reader will have a definition of least common multiples of subset of commutative ring.
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:
\( R\): \(\in \{\text{ the rings }\}\)
\( S\): \(\subseteq R\)
\( S'\): \(= \{m' \in R \vert \forall p \in S (\exists q \in R (m = q p))\}\)
\(*lcm (S)\): \(= \{m \in S' \vert \forall m' \in S' (\exists q \in R (m' = q m))\}\)
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Conditions:
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\(S'\) is called "set of common multiples of \(S\)".
2: Natural Language Description
For any ring, \(R\), any subset of \(R\), \(S\), and the set of the common multiples of \(S\), \(S' := \{m' \in R \vert \forall p \in S (\exists q \in R (m = q p))\}\), \(lcm (S) := \{m \in S' \vert \forall m' \in S' (\exists q \in R (m' = q m))\}\)
3: Note
This definition does not require \(R\) to have any order: "least" is not according to any order.
\(S = \emptyset\) is not excluded, although it is not particularly assumed to be useful.
When \(S = \emptyset\), vacuously, \(S' = R\), and \(1 \in lcm (S)\), because \(d' = d' 1\) for each \(d' \in S'\); if \(0 \in lcm (S)\), \(lcm (S) = \{0\}\), because \(d' = q 0 = 0\), which does not contradict \(1 \in lcm (S)\), because that means that \(1 = 0\), which means that \(R = \{0 = 1\}\); what others \(lcm (S)\) has depends on \(R\): when \(d\) is a unit, \(d' = d' d^{-1} d\), so, \(d \in lcm (S)\); but otherwise, whether there is a \(q \in R\) such that \(d' = q d\) for each \(d' \in R'\) is not clear.
Hereafter, let us suppose that \(S \neq \emptyset\).
Always \(0 \in S'\): \(0 = 0 p\) for each \(p \in S\).
But when \(0 \in lcm (S)\), \(lcm (S) = \{0\}\): \(m' = 0 = q 0\) for each \(m' \in S'\), so, \(S' = \{0\}\).
When \(S\) is finite, \(\prod_{p_j \in S} p_j \in S'\): \(\forall p_k \in S (\prod_{p_j \in S} p_j = (\prod_{p_j \in S \setminus \{p_k\}} p_j) p k)\).
\(lcm (S)\) may be empty or have multiple elements.
If \(0 \in S\), \(S' = \{0\}\): \(0 = q 0\) for each \(q \in R\). The reverse is not necessarily true: as a counterexample, let \(R = \mathbb{Z} / (6 \mathbb{Z})\) and \(S = \{[2], [3]\}\), then, \(S' = \{[0]\}\): the multiples of \([2]\) are \([2 * 0] = [0], [2 * 1] = [2], [2 * 2] = [4], [2 * 3] = [6] = [0], [2 * 4] = [8] = [2], [2 * 5] = [10] = [4]\) and the multiples of \([3]\) are \([3 * 0] = [0], [3 * 1] = [3], [3 * 2] = [6] = [0], [3 * 3] = [9] = [3], [3 * 4] = [12] = [0], [3 * 5] = [15] = [3]\).
If and only if \(S' = \{0\}\), \(lcm (S) = \{0\}\): if \(S' = \{0\}\), \(0 = 0 0\), so, \(0 \in lcm (S)\), while \(lcm (S)\) is a subset of \(S'\); if \(lcm (S) = \{0\}\), \(\{0\} \subseteq S'\), but as \(0 = q 0\) for each \(q \in R\), only \(0\) can be in \(S'\).
This definition does not exactly specialize to 'least common multiple of subset of integers': the least common multiples of \(\{2, 3\}\) of \(\mathbb{Z}\) by this definition are \(\{-6, 6\}\) (\(-6 = (-3) 2\); \(-6 = (-2) 3\); \(6 = (-1) (-6)\)), while the least common multiple of \(\{2, 3\}\) by 'least common multiple of subset of integers' is \(6\).
