definition of greatest common divisors 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 greatest common divisors 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:
\( R\): \(\in \{\text{ the rings }\}\)
\( S\): \(\subseteq R\)
\( S'\): \(= \{d' \in R \vert \forall p \in S (\exists q \in R (p = q d'))\}\)
\(*gcd (S)\): \(= \{d \in S' \vert \forall d' \in S' (\exists q \in R (d = q d'))\}\)
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Conditions:
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\(S'\) is called "set of common divisors of \(S\)".
2: Natural Language Description
For any ring, \(R\), any subset of \(R\), \(S\), and the set of the common divisors of \(S\), \(S' := \{d' \in R \vert \forall p \in S (\exists q \in R (p = q d'))\}\), \(gcd (S) := \{d \in S' \vert \forall d' \in S' (\exists q \in R (d = q d'))\}\)
3: Note
This definition does not require \(R\) to have any order: "greatest" 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 \(gcd (S) = \{0\}\), because \(0 = 0 d'\) for each \(d' \in S'\) and as there is \(d' = 0 \in S'\), \(0 = q 0\) is required.
Hereafter, let us suppose that \(S \neq \emptyset\).
Always \(1 \in S'\): \(\forall p \in S (p = p 1)\).
\(gcd (S)\) may be empty or have multiple elements.
If and only if \(S = \{0\}\), \(0 \in gcd (S)\): if \(S = \{0\}\), \(S' = R\), because \(0 = 0 p\) for each \(p \in R\), and so, \(0 \in S'\) and \(0 = 0 d'\) for each \(d' \in S'\), so, \(0 \in gcd (S)\); if \(0 \in gcd (S)\), \(0 \in S'\), and \(p = q 0 = 0\) for each \(p \in S\).
If and only if \(0 \in gcd (S)\), \(gcd (S) = \{0\}\): if \(0 \in gcd (S)\), as \(0 \in S'\) and \(0 = q 0\) for each \(q \in R\), only \(0\) can be in \(gcd (S)\), so, \(gcd (S) = \{0\}\); if \(gcd (S) = \{0\}\), obviously, \(0 \in gcd (S)\).
This definition does not exactly specialize to 'greatest common divisor of subset of integers': the greatest common divisors of \(\{2, 6\}\) of \(\mathbb{Z}\) by this definition are \(\{-2, 2\}\) (\(2 = (-1) (-2)\); \(6 = (-3) (-2)\); \(-2 = (-1) 2\)), while the greatest common divisors of \(\{2, 6\}\) by 'greatest common divisor of subset of integers' is \(2\).
