Discussion Overview
The discussion revolves around proving that the set \( R = \{m/n \in \mathbb{Q}: p \text{ does not divide } n \} \) is a sub-ring of the rational numbers \( \mathbb{Q} \), where \( p \) is a prime number. Participants explore the necessary conditions for \( R \) to qualify as a sub-ring and investigate the group of units \( R^{\times} \) within this context.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Debate/contested
Main Points Raised
- Some participants reference the definition of a subring, noting that it must be closed under multiplication and subtraction, and contain the multiplicative identity.
- Participants question whether the product of two elements in \( R \) remains in \( R \), specifically asking if \( \frac{m_1}{n_1} \times \frac{m_2}{n_2} \) is in \( R \) when \( p \) does not divide \( n_1 \) or \( n_2 \).
- There is a discussion about identifying the units of \( R \), with some suggesting that \( 1 \), \( -1 \), and primes not equal to \( p \) are units.
- Another participant proposes that the multiplicative inverse \( \frac{n}{m} \) must also be in \( R \) for \( \frac{m}{n} \) to be a unit, leading to further exploration of conditions under which this holds.
- One participant concludes that the set of units \( R^{\times} \) includes elements where \( p \) does not divide both the numerator and denominator.
Areas of Agreement / Disagreement
Participants express uncertainty about the closure properties of \( R \) under multiplication and the complete characterization of its units. There is no consensus on the final characterization of \( R \) as a subring or the full set of units.
Contextual Notes
Participants have not fully resolved the implications of the closure properties or the conditions required for elements to be units in \( R \). There are also assumptions regarding the definitions of invertibility and the nature of the elements in \( R \) that remain unexamined.