Discussion Overview
The discussion centers on whether the product of a maximal left ideal RE with an invertible matrix Q, denoted REQ, remains a maximal left ideal within the ring of n x n matrices over a field F. The scope includes theoretical considerations of ring theory and properties of ideals.
Discussion Character
- Debate/contested
- Technical explanation
Main Points Raised
- One participant asserts that RE is a maximal left ideal in the ring of n x n matrices over a field F.
- Another participant argues that since RE equals R when E is the identity element, REQ would also equal R, which cannot be a maximal ideal as it is not a proper ideal.
- A subsequent post clarifies that E is not the identity element due to its structure, which may affect the previous claims.
- A different participant introduces a general result about maximal left ideals in associative rings, suggesting that if L is a maximal left ideal, then the transformed ideal q^{-1}Lq is also maximal when q is invertible.
- This participant applies the general result to the specific case of RE and Q, proposing that REQ should also be a maximal left ideal based on the earlier assertion about RE.
Areas of Agreement / Disagreement
Participants express disagreement regarding the status of REQ as a maximal left ideal. Some assert that it cannot be maximal if it equals R, while others propose that it remains maximal based on the properties of maximal left ideals under ring automorphisms.
Contextual Notes
There are unresolved assumptions regarding the structure of E and its implications for the properties of RE and REQ. The discussion also highlights the dependence on definitions of maximal ideals and the conditions under which the transformations apply.