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Few months ago, I proved that when there is three morphisms of modules over a commutative ring with zero composition, i.e., a sequence $$A \xrightarrow{\alpha} B \xrightarrow{\beta} C \xrightarrow{\gamma} D$$ with $\gamma\beta\alpha=0$, then there is an induced exact sequence $$0 \to \text{ker }\alpha \xrightarrow{i_1} \text{ker }\beta\alpha \xrightarrow{\alpha'} \text{ker }\beta \xrightarrow{i_2} \text{ker }\gamma\beta/\text{im }\alpha \xrightarrow{\beta'} \text{ker }\gamma/\text{im }\beta\alpha \xrightarrow{i_3} \text{coker }\beta \xrightarrow{\gamma'} \text{coker }\gamma\beta \xrightarrow{i_4} \text{coker }\gamma \to 0.$$ in which all maps are canonical. Using this lemma, it was possible to prove for example the statements about exact couples in this article on my blog: https://fractalofideas.wordpress.com/2019/12/10/closed-complex-of-order-3/

The simple and elementary proof is given here, at the middle of the article: https://fractalofideas.wordpress.com/2019/09/23/how-to-prove-snake-lemma-without-diagrams/

As one may notice, this is nothing but the snake lemma, maybe in its simplest form possible. Is this worth to submit this result to some academic journal?

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    $\begingroup$ The standard proof of the snake lemma seems pretty simple and elementary to me... $\endgroup$
    – lambda
    Dec 9 '19 at 18:05
  • $\begingroup$ @lambda I agree with that. That’s my hesitation too. But I could say that this form of the lemma is most easy to apply in many situations. $\endgroup$
    – HyJu
    Dec 9 '19 at 18:11
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    $\begingroup$ A modern point of view on such lemmas in abelian categories makes systematic use of the salamander lemma. See for example sbseminar.wordpress.com/2007/11/13/… (by MO founding member Anton Geraschenko). $\endgroup$
    – Todd Trimble
    Dec 9 '19 at 18:18
  • $\begingroup$ @Todd Trimble Thank you for your comment and reference! I think the salamander lemma could also be proved using this lemma as well, which now I’m trying. $\endgroup$
    – HyJu
    Dec 9 '19 at 18:30
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    $\begingroup$ Sorry, but I have to link to this: youtube.com/watch?v=etbcKWEKnvg $\endgroup$ Dec 9 '19 at 18:33
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The short and easy answer is "no, you should not submit it to a research journal".

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  • $\begingroup$ I understood. Thanks $\endgroup$
    – HyJu
    Dec 10 '19 at 14:09
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    $\begingroup$ Though just because something is not worth sending to a research journal, that does not always mean it is not worth sending anywhere. If you feel your argument is interesting or novel in some way, there are other options such as a short piece in something like the MAA monthly or Math Horizons, especially if you can sell your argument as educational for students in some way. $\endgroup$ Dec 29 '19 at 8:58
  • $\begingroup$ @GregFriedman Thank you a lot for your encouragement. $\endgroup$
    – HyJu
    May 23 '21 at 14:15

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