definition of associates of element 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.
- The reader knows a definition of units of ring.
Target Context
- The reader will have a definition of associates of element 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:
\( R\): \(\in \{\text{ the commutative rings }\}\)
\( U\): \(= \{\text{ the units of } R\}\)
\( p\): \(\in R\)
\(*Asc (p)\): \(= \{u p \vert u \in U\}\)
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Conditions:
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2: Natural Language Description
For any commutative ring, \(R\), the set of the units of \(R\), \(U\), and any element, \(p \in R\), \(Asc (p) := \{u p \vert u \in U\}\)
3: Note
For each elements, \(p, p' \in R\), \(p' \in Asc (p)\), is an equivalence relation: 1) \(p \in Asc (p)\), because \(p = 1 p\); 2) \(p_1 \in Asc (p_2) \implies p_2 \in Asc (p_1)\), because if \(p_1 = u p_2\), \(p_2 = u^{-1} p_1\); 3) \((p_1 \in Asc (p_2) \land p_2 \in Asc (p_3))\implies p_1 \in Asc (p_3)\), because if \(p_1 = u_2 p_2\) and \(p_2 = u_3 p_3\), \(p_1 = u_2 u_3 p_3\) where \(u_2 u_3\) is a unit, because \(u_3^{-1} u_2^{-1} u_2 u_3 = u_2 u_3 u_3^{-1} u_2^{-1} = 1\).
So, \(R / Asc\) is a quotient set.
