Discussion Overview
The discussion revolves around the existence of an infinite ring that has exactly two non-trivial maximal ideals. Participants explore various examples and constructions related to this concept.
Discussion Character
- Exploratory
- Technical explanation
- Mathematical reasoning
Main Points Raised
- One participant, LiKeMath, inquires about the existence of an infinite ring with exactly two maximal ideals.
- Another participant suggests \(\mathbb{R} \times \mathbb{R}\) as a potential example, but questions how multiplication is defined in this context.
- A follow-up clarifies that multiplication in \(\mathbb{R} \times \mathbb{R}\) is defined pointwise: \((a,b) \cdot (c,d) = (ac, bd)\).
- One participant proposes a construction involving the integers and rational numbers, suggesting that by inverting certain elements, the only maximal ideals remaining would be (2) and (3).
- Another similar construction is mentioned involving continuous functions on the interval [0,1], where inverting functions that do not vanish at the endpoints would lead to maximal ideals that vanish at those points.
- A participant references micromass's example, discussing continuous functions on a two-point set, where the maximal ideals correspond to functions vanishing at either point.
Areas of Agreement / Disagreement
The discussion does not reach a consensus on a definitive example of an infinite ring with exactly two maximal ideals, and multiple competing views and constructions are presented.
Contextual Notes
Participants explore various constructions and examples, but the assumptions and definitions underlying these examples may not be fully resolved, leaving some ambiguity in the proposed approaches.