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It seems to me that in history of commutative algebra, you come from interest in rings and get more attention of modules by the time.

In my understanding this is because rings build an "ugly" category where you can't look at exact sequences and so you can't use homological algebra, but you can do this for modules and you can watch every ring as a module over itself. So you learn about rings by watching moules. Is this understanding of historical development correct?

We pay some price for working in this way. For getting very nice arguments from homological algebra, we must do some "dirty work" to get from modules back to rings.

As an example, localization and factorication behave well for modules because the localization functor is exact and we can do factorication with exact sequences. But to see that it also works for rings, we must realise by "ugly calculation" that we can do the same mappings as for the moduls also for the rings. This step is often skipped because it's ugly and seems obvious.

But is there a simple argument, why we needn't be frightened of the "ugly part" and why it's "obvious" in the relevant cases?

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  • $\begingroup$ It sounds as if you are assuming that homological algebra was around and well understood even before the concept of a ring got studied and that therefore it was used to understand rings. However, I'm not sure if that is true, I would rather think that they developed in parallel (e.g. first understand rings and ideals a little bit, then do some module theory, then try to apply this to rings and ideals, etc.). $\endgroup$ – Dirk Nov 24 '17 at 7:49
  • $\begingroup$ It's not clear what you're asking, I'm not aware of any "ugly" part to doing factorization with rings. I think when authors skip it they do so because it's considered pretty basic, not because it's ugly. And I certainly don't think anyone is frightened by it. So what exactly are you looking for in an answer? $\endgroup$ – Jim Nov 27 '17 at 18:32
  • $\begingroup$ As I understand it, it's nice to go from rings to moduls because we can use tools of homological algebra there. But it isn't clear for me, why we can be sure, that everything works when we go back from moduls to rings. If two rings are isomorphic as moduls, why should they be isomorphic as rings? $\endgroup$ – user302982 Nov 27 '17 at 18:46
  • $\begingroup$ They shouldn't. $k[x]$ and $k[x, y]$ are isomorphic as $k$-modules. And in general what are you considering rings as modules over? As a module over one of the rings? I can also make $k[x, y]$ into a $k[x]$ module that's isomorphic to $k[x]$, because I'm just gonna throw away the multiplication of $k[x, y]$ and use that it's isomorphic to $k[x]$ as a vector space. $\endgroup$ – Jim Nov 30 '17 at 0:36
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In representation theory one is interested in groups and so one studies representations of them, representations lead naturally to modules, and modules lead to more general rings. So in that setting modules come first and rings second, which is opposite of the development you've suggested. Another surprising reversal: characters were defined and studied by Frobenius [Uber Gruppencharaktere, 1896] a full year before he defined representations [Uber die Darstellung der endlichen Gruppen durch lineare Substitutionen, 1897]; which is surprising because most current mathematicians don't even know how to define a character of a non-abelian group without using representations in the definition.

This should suggest to you that trying to discern historical motivations from the current utility of the theorems/objects is fraught with problems and that the modern path to introduce mathematical objects often bears no resemblance to the historical path taken to their discovery/definition.

As for what the actual historical development was I'm no expert. The modern axiomatic notion of a ring and its ideals was formalized by Emmy Noether [Idealtheorie in Ringbereichen, 1921] but those concepts had certainly existed previous to that paper, albeit informally. Noether is also responsible for introducing the notion of a module [Hyperkomplexe Grossen und Darstellungstheorie, 1929]. I don't know if this concept predates her or not, but I believe her interest in it was as a method of studying representations of an algebra, not as a method of embedding the algebra into a nicer space of objects.

What I can say for sure is that the idea of embedding rings into the category of modules and using the methods of homological algebra were definitely not motivations in the initial development of module theory. Category Theory was first defined by Eilenberg and Mac Lane [General Theory of Natural Equivalences, 1945]. You can trace some of the ideas back to an earlier paper from 1942, but they definitely weren't around in the 20's. Homological algebra was around as early as the 20's, and Noether played a part in it, but the only notion that existed at the time was that of the homology groups of a topological space. So homological algebra was 100% a subset of the study of the topology of spaces. It did not play a role in the study of purely algebraic objects until around the 40's. So neither of those concepts would have been a factor in the first developments of module theory.

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  • $\begingroup$ Ok, thanks so far for the historic part. Looking for answers to the other part of the question... $\endgroup$ – user302982 Nov 25 '17 at 21:26

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