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I am now reading the book "Number Theory" by Borevich and Shafarevich. It seems to treat some topics in a way different than most modern-ish texts (I might be wrong, I have not read all the books on the topic), for example the theory of divisors in arbitrary ring, which is built before the more conventional theory of ideals in algebraic number fields.

I wanted to ask whether there are any other texts which consider this theory of divisors? I have tried to look something up on the web, but of course the term "divisor" has another, more popular, meaning, so I couldn't find anything.

Any reference, or even just giving some keywords which would help searching, will be appreciated.

Thanks in advance.

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    $\begingroup$ Unless they're talking about Arakelov divisors (which I don't recall they are) it is probably just the usual notion of divisor in algebraic geometry. Namely, a divisor on a ring $R$ is most likely, for them, just a divisor on $\text{Spec}(R)$ in the algebro-geometric sense. $\endgroup$
    – Alex Youcis
    Sep 30, 2016 at 19:26
  • $\begingroup$ @AlexYoucis It seems like something is on point there. Somehow I didn't see the algebraic-geometric notion of divisor while looking up the topic (perhaps I subconciously ignored it because I know no algebraic geometry?), but somehow I feel you might be right. $\endgroup$
    – Wojowu
    Sep 30, 2016 at 19:39

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After I studied Borevich-Shafarevich, I had the same question. After a lot of reading I found that the best (elementary) reference is chapter VII (Divisors) of Boubaki's "Commutative Algebra". It is an excelent exercise to read this and understand that both notions of divisors are basically the same. While reading Bourbaki I found very useful to complement the reading with the chapter on "Integral Algebraic Elements" of Van der Waerden's "Algebra".

Also, I found extremely useful to understand the following paper:

L. Skula, Divisorentheorie einer Halbgruppe.

He proves that one can omit the axiom that "a divisor is basically an ideal" from the axioms given in Borevich-Shafarevich. This means that the multiplicative structure "implies" the additive structure of divisors.

Nowdays, divisor theories are usually studied over monoids, and you can look at some of Franz Halter-Koch's (et al) survey articles. An old paper that I found very nice to read on this subject is:

A. H. Clifford, Arithmetic and ideal theory of commutative semigroups.

Sorry for my english. Hope this helps.

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