641: Greatest Common Divisors of Subset of Commutative Ring

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definition of greatest common divisors of subset of commutative ring

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Topics

About: ring

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of ring.

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Target Context

• The reader will have a definition of greatest common divisors of subset of commutative ring.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$R$: $\in \{\text{ the rings }\}$
$S$: $\subseteq R$
$S^{\prime }$: $=\{d^{\prime }\in R|\mathrm{\forall }p\in S(\mathrm{\exists }q\in R(p=q{d}^{\prime }))\}$
$\ast gcd(S)$: $=\{d\in S^{\prime }|\mathrm{\forall }{d}^{\prime }\in S^{\prime }(\mathrm{\exists }q\in R(d=q{d}^{\prime }))\}$
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Conditions:
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$S^{\prime }$ is called "set of common divisors of $S$".

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2: Natural Language Description

For any ring, $R$, any subset of $R$, $S$, and the set of the common divisors of $S$, $S^{\prime }:=\{d^{\prime }\in R|\mathrm{\forall }p\in S(\mathrm{\exists }q\in R(p=q{d}^{\prime }))\}$, $gcd(S):=\{d\in S^{\prime }|\mathrm{\forall }{d}^{\prime }\in S^{\prime }(\mathrm{\exists }q\in R(d=q{d}^{\prime }))\}$

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3: Note

This definition does not require $R$ to have any order: "greatest" is not according to any order.

$S=\mathrm{\varnothing }$ is not excluded, although it is not particularly assumed to be useful.

When $S=\mathrm{\varnothing }$, vacuously, $S^{\prime }=R$, and $gcd(S)=\{0\}$, because $0=0d^{\prime }$ for each $d^{\prime }\in S^{\prime }$ and as there is $d^{\prime }=0\in S^{\prime }$, $0=q0$ is required.

Hereafter, let us suppose that $S\ne \mathrm{\varnothing }$.

Always $1\in S^{\prime }$: $\mathrm{\forall }p\in S(p=p1)$.

$gcd(S)$ may be empty or have multiple elements.

If and only if $S=\{0\}$, $0\in gcd(S)$: if $S=\{0\}$, $S^{\prime }=R$, because $0=0p$ for each $p\in R$, and so, $0\in S^{\prime }$ and $0=0d^{\prime }$ for each $d^{\prime }\in S^{\prime }$, so, $0\in gcd(S)$; if $0\in gcd(S)$, $0\in S^{\prime }$, and $p=q0=0$ for each $p\in S$.

If and only if $0\in gcd(S)$, $gcd(S)=\{0\}$: if $0\in gcd(S)$, as $0\in S^{\prime }$ and $0=q0$ 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 $2$.

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References

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