Discussion Overview
The discussion revolves around the question of whether the element $1-y$ lacks a right inverse when $y$ is a non-unit in a right ideal. Participants explore this concept within the context of various rings and ideals, including specific examples and counterexamples.
Discussion Character
- Debate/contested
- Mathematical reasoning
Main Points Raised
- Some participants propose that if $I$ is a right ideal and $y \in I$ with $y \neq 1$, then $1-y$ has no right inverse.
- One participant provides an example using $R=\mathbb{Z}$ and $I=2\mathbb{Z}$, questioning whether $1-y$ has a right inverse in this case.
- Another participant suggests that $(1-y) \cdot -1 = 1$, indicating that $1-y$ can have a right inverse in certain instances.
- A participant introduces the unique maximal right ideal in the localization of $\mathbb{Z}$ at the ideal $(3)$, arguing that $1-y$ can have a right inverse in this context.
- One participant attempts to establish a general claim about the lack of right inverses for $1-ir$ when $I$ is a unique maximal right ideal, but this is challenged by another participant.
- Another participant points out that the assumption made in the previous argument is incorrect, referencing the unique maximal ideal being the set of non-units in $\mathbb{Z}_{(3)}$.
Areas of Agreement / Disagreement
Participants express differing views on whether $1-y$ can have a right inverse, with some examples supporting the idea that it can, while others argue against it. The discussion remains unresolved, with multiple competing perspectives presented.
Contextual Notes
Participants highlight the importance of specific ring structures and the nature of ideals in determining the properties of $1-y$. There are unresolved assumptions regarding the conditions under which $1-y$ may or may not have a right inverse.
