How Does Condition (2) Imply Condition (3) in Bland's Proposition 3.2.7?

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Discussion Overview

The discussion centers around Proposition 3.2.7 from Paul E. Bland's book "Rings and Their Modules," specifically examining the proof that condition (2) implies condition (3) within the context of exact sequences in module theory. Participants are exploring the implications of these conditions and referencing related works, including counterexamples from Rotman's "An Introduction to Homological Algebra."

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Peter questions how Bland deduces that \( M / \text{Ker } g \cong N \) from the direct sum decomposition \( M = \text{Ker } g \oplus N \).
  • Another participant notes that in general, for a direct sum \( A = B \oplus C \), it follows that \( A / B \cong C \).
  • One participant mentions a counterexample provided by Rotman that challenges the validity of Bland's proposition, suggesting that it is not universally applicable.
  • Peter expresses skepticism about the correctness of the proof for the implication \( (3) \Rightarrow (1) \) in Bland's proposition, indicating a need for further scrutiny.
  • A participant elaborates on the distinction between exact sequences and split exact sequences, using a specific example involving \( \mathbb{Z}_2 \) and \( \mathbb{Z}_4 \) to illustrate how certain conditions can be violated.

Areas of Agreement / Disagreement

Participants express differing views on the implications of Bland's proposition, with some supporting its validity while others reference counterexamples that suggest limitations. There is no consensus on the correctness of the proofs or the implications of the conditions involved.

Contextual Notes

Participants reference specific propositions and examples from both Bland and Rotman, indicating that the discussion is deeply rooted in the nuances of module theory and exact sequences. The limitations of the propositions and the conditions under which they hold are acknowledged but not resolved.

Who May Find This Useful

This discussion may be useful for students and researchers in algebra, particularly those studying module theory, exact sequences, and related counterexamples in homological algebra.

  • #31
steenis said:
The proof of (1)=>(3) may be looking correct, but I want to know if it is correct. In contrary, the proof confuses me.

Ok, sorry, here I was wrong which might be clear, because I was quoting you.

So we agree that (3)=>(1) is wrong. What must we do to finish this thread? I think a proof of the non-equivalence of S1 and S0 in post #27. And, if possible, a direct and easy counterexample of (3)=>(1).
 
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  • #32
I already gave a direct counterexample to $(3)\implies (1)$, which is actually the same as the the one in the MSE link you provided.
 
  • #33
Where is your counterexample, I cannot find it?
 
  • #34
It's at the end of post #28, but it's the same as the k.stm's example in your link.
 
  • #35
That specific example is a direct counterexample of "(3) => the ses splits" and therefore an indirect counterexample of (3) => (1). A direct counterexample of (3) => (1) is example in which (3) is true and (1) is not true, that is how I have learned it in my university years. Of course that all in the context of Bland's proposition and if possible very easy. It is not necessary, not mandatory, because the job is already done. It is just meant to enlighten that (3)=>(1) is faulty. I do not know if it is possible or it is too difficult.

I want to stop this discussion now, the only thing that is open is the non-equivalence of S1 and S0 in post #27.
 
  • #36
That does not make sense, since any counterexample to $(3)\implies (1)$ is a a counterexample to the implication that $(3)$ implies the ses splits; for a split exact sequence necessarily satisfies $(1), (2)$, and $(3)$. Seeing that you want to stop the discussion, I'll leave it to other users address your concerns.
 
  • #37
Euge said:
That does not make sense, since any counterexample to $(3)\implies (1)$ is a a counterexample to the implication that $(3)$ implies the ses splits; for a split exact sequence necessarily satisfies $(1), (2)$, and $(3)$. Seeing that you want to stop the discussion, I'll leave it to other users address your concerns.
Thanks to Euge and Steenis for clarifying matters ...

There are some further helpful and informative posts on the Physics Forums in the sub-forum Linear and Abstract Algebra ... here

https://www.physicsforums.com/threa...-bland-proposition-3-2-7.881174/#post-5538854

Hope that helps ...

Peter
 

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