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As I learned here, since the very beginning of humanity/Kummer's development of ideals, the idea that they are "numbers" that we can "modulo by"/do "modular arithmetic by" has been central to the concept. An example that was given in the linked post is that the ideal $(2,1+\sqrt{-5})$ in $\mathbb Z[\sqrt{-5}]$ can be thought of as (the kernel of) the ring homomorphism $$f:\mathbb Z[\sqrt{-5}] \to \mathbb Z/2\mathbb Z, f(a+b\sqrt{-5}) := a+b+2{\mathbb Z}.$$

It took until the work of Dedekind (and even then, the later revisions) that the notion of ideals as sets of elements of a ring took center stage, especially in the seemingly-naive but miraculously-suitable definition of multiplying 2 ideals. (Indeed, somehow all the stars align and we get unique factorization of ideals in terms of the aforementioned definition of product ideal in, for instance, all number rings. Although using Kummer's ring isomorphism conception of ideals we can talk about ideals dividing/divisible by other ideals, we seem to be missing the notion of multiplicity of "how many times" an ideal divides another --- perhaps this is some intuition for the alternative definition of Dedekind domain: Noetherian domain s.t. all localizations at prime ideals are DVRs, where discrete valuations are exactly the framework for talking about multiplicities [and Noetherianity is there to guarantee finiteness of the prime factorization process]).

Question: I'm wondering if there's any way to interpret the multiplication of ideals "in Kummer's original conception of the ideal", or more concretely, is there somehow a way to construct, using ring homomrphisms that define $I_1,I_2$, a ring homomorphism whose kernel is $I_1\cdot I_2$?

I'm aware that the Chinese remainder theorem does this if $I_1,I_2$ are relatively prime, but it doesn't seem to apply in general.

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    $\begingroup$ I'm being cheeky and this is certainly cheating: what about the canonical homomorphism $\pi:R\to R/(I_1\cdot I_2)$? $\endgroup$
    – wormram
    Sep 26, 2023 at 23:49
  • $\begingroup$ @morrowmh yes, this is why I specify in the statement of my question that the ring hom with kernel $I_1I_2$ should somehow be "built out of"/"constructed from" ring homs whose kernels are $I_1,I_2$. $\endgroup$
    – D.R.
    Sep 27, 2023 at 1:04

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I don't believe there is a way to do this. The natural thing to do with two quotients $R \to R/I_1$ and $R \to R/I_2$ is to consider their product $R \to R/I_1 \times R/I_2$, and of course what this produces is the intersection.

Years ago I asked a related question on MathOverflow about whether the ideal product was some categorical operation on the poset of ideals (the intersection and the sum are the meet and the join respectively), and the answer turns out to be no.

Honestly I've always thought of the ideal product as sort of an anomaly for this reason. The categorically natural operations are the intersection, the sum, and the tensor product, and I don't think the ideal product is a particularly natural thing to look at unless it agrees with one of these. (In the case of fractional ideals of a Dedekind domain $D$, ideal product agrees with the tensor product, at least up to isomorphism of modules; in fact the ideal class group is naturally isomorphic to the Picard group of line bundles on $\text{Spec } D$, with group operation the tensor product.)

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  • $\begingroup$ How interesting. Basically the first operation on ideals people worked with historically is also in some sense the weirdest. Does the fact that this "algebraic number theory" object the ideal class group can be thought of as this "algebraic geometry" object the Picard group of line bundles mean that there is some "algebraic geometry" proof of "algebraic number theory" results like number rings are Dedekind domains? $\endgroup$
    – D.R.
    Sep 27, 2023 at 3:19
  • $\begingroup$ @D.R.: I don't know. $\endgroup$ Sep 27, 2023 at 3:44
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About products of ideals, you might find interesting the characterization of Dedekind rings as domains such that whenever we have ideals $\frak{b} \supset \frak{a}$ ( also written $\frak{b}\mid \frak{a}$), there exists $\frak{c}$ such that $\frak{b}\cdot \frak{c} = \frak{a}$ ( in Milnor's K-theory book).

$\bf{Added:}$ In Noetherian rings we have the primary decomposition. In the case of Dedekind rings the $\ne 0$ primary ideals are in fact the power of the primes ( in fact maximals).

I think the product of ideals ( or more generally $I M$, where $M$ an $R$ module, and $I$ an ideal) are not that strange. Now also $I( J M) = (IJ) M$, so the products of ideals appear. Then think of $I$-adic topology and other things. Intersecting $I$ with itself does not give anything new, but we are interested in $I^n$'s. Then there is Artin-Rees lemma, completions, etc. Also, Nakayama lemma. So products of ideals appear in many forms. Just some thoughts...

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    $\begingroup$ I am aware of that definition; the issue is that that def has at its core the ideal product, but my entire issue is that I find the ideal product to be unnatural (an "anomaly", as Qiaochu puts it above), so in my mind this def of Dedekind domains (though the standard one) "rests upon shaky foundations". I am also not quite (emotionally) satisfied with the fact that this "unnatural" definition of product is somehow exactly the correct one to give us unique factorization of ideals, in all rings of integers. There are just too many miracles in algebraic number theory, even at just its base. $\endgroup$
    – D.R.
    Sep 27, 2023 at 4:06
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    $\begingroup$ @D.R. Many definitions of Dedekind (and Prufer) domains don't explicitly involve ideal products. $\endgroup$ Sep 27, 2023 at 20:22
  • $\begingroup$ @BillDubuque I am aware; in fact I discussed one such alternative definition in my original post (the one involving localizations at prime ideals being DVRs). But still, the ideal product is in some sense deeply ingrained into the 'DNA' of Dedekind domains. And also, yes the abundance of definitions ensures that Dedekind domains are fundamental to mathematics and would have been 'discovered' no matter how history turned out; yet this "remarkable confluence" is itself quite miraculous and 'mysterious'. $\endgroup$
    – D.R.
    Sep 28, 2023 at 3:20

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