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InquilineKea
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So I'm kind of confused about the definition:
a-b\in I
Why a - b instead of a + b?
a-b\in I
Why a - b instead of a + b?
Why Use a - b Instead of a + b in Equivalence Classes of Rings?
- Context: Graduate
- Thread starter InquilineKea
- Start date
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- Tags
- Classes Equivalence Rings
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Discussion Overview
The discussion revolves around the use of the expression a - b in the context of equivalence classes of rings, specifically questioning why this form is preferred over a + b. The scope includes theoretical aspects of ring theory and equivalence relations.
Discussion Character
- Technical explanation
- Conceptual clarification
- Debate/contested
Main Points Raised
- One participant expresses confusion regarding the definition involving a - b and questions why it is used instead of a + b.
- Another participant asks for clarification on whether the goal is to prove that a - b is an equivalence relation.
- A different participant argues that using a - b is natural because it leads to the conclusion that a - b = 0 when considering the quotient by I, which implies a = b.
- Another point raised suggests that one would want a to be equivalent to itself, which is supported by the expression a - a in I, rather than a + a.
Areas of Agreement / Disagreement
Participants do not appear to reach a consensus, as there are differing views on the appropriateness of using a - b versus a + b in this context.
Contextual Notes
The discussion may involve assumptions about the properties of equivalence relations and the nature of the ideal I, which are not fully articulated.
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