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Given an abelian group $(A,+)$, what are conditions on $A$ that ensure there is or isn't a unitary ring structure $(A,+,*)$? That is, an associative bilinear operation $* : A^2 \to A$ with an identity $1_A \in A$.

A possible follow-up question would be conditions for a nontrivial rng structure on $A$ (indeed, the multiplication $ab = 0$ always gives a rng structure).

For example, if $A$ is finitely generated, then by the structure theorem on finite type abelian groups, $A \simeq \mathbb Z^r \times \prod \mathbb Z/p_i^{a_i} \mathbb Z$ and it has a product ring structure.

On the other hand, $\mathbb Q / \mathbb Z$ doesn't: if it did, the unit $u=[p/q]$ would have finite order $q$, and $\forall x \in \mathbb Q / \mathbb Z, qx = q(u*x) = (qu)*x = 0$, but $q(1/2q) = 1/2 \neq 0$. More generally, by this question, if every element has finite order but the orders are not bounded, then there is no ring structure on $A$.

Is this condition also necessary? I doubt it: I have trouble envisioning a ring structure on $(\mathbb R / \mathbb Z, +) = (S^1, \cdot)$, but not every element has finite order in $S^1$. The argument also completely fails for rngs.

(Note that my question is different from the one I linked: this one simply asked for a counterexample, I'm asking for general conditions)

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Weak evidence that this is an open problem: mathoverflow.net/questions/92557/… –  Qiaochu Yuan Jun 30 '13 at 9:00
    
A partial answer is given in Infinite abelian groups by Kaplansky: An abelian group of bounded order is a direct sum of cyclic groups (theorem 6 page 17). –  Seirios Jun 30 '13 at 9:03
    
@QiaochuYuan: I don't know if this says anything about openness, they are talking about first-order axiomatization, not recognition in general. But it's possible that the problem is much more difficult than expected. –  Najib Idrissi Jun 30 '13 at 10:17
    
@Seirios: Thanks for the reference. –  Najib Idrissi Jun 30 '13 at 10:17
    
@nik: I'd be more likely not to see an answer to that MO question in worlds where this is an open problem than in worlds where it's not (e.g. in some worlds it's a very easy problem, the answer can be first-order axiomatized, and someone knew that and posted an answer to the MO problem). Are you familiar with the Bayesian definition of evidence? –  Qiaochu Yuan Jun 30 '13 at 19:31
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1 Answer

up vote 3 down vote accepted

In a unitary ring you always have the fundamental concept of "characteristic". If you have a torsion Abelian group $G$, and you want a unitary ring structure on $G$, then first of all you need to choose a unit. If $g\in G$ is such unit, then you are force to have a ring of characteristic $n=|g\mathbb Z|$. This says you that $nh=0$ for all $h\in G$... so if $G$ has not finite exponent you cannot have a ring structure on $G$.

If you want a source for many torsion-free counterexamples you can proceed as follows. You can use a famous theorem of Saharon Shelah to find, for any infinite cardinal $\alpha$, $2^\alpha$ many non-isomorphic torsion free abelian groups of cardinality $\alpha$ and whose endomorphism ring is a subring of the rationals. In particular, given such a $G$, $End(G)$ is countable. Notice that, if you have a ring structure on $G$, then there is an immersion $G\to End(G)$ sending $g$ to left-multiplication by $g$.

Let me add that these matters are nicely discussed in Chapter XVII of the second volume of "Infinite Abelian Groups" by Laszlo Fuchs (the title of the chapter is "Additivie groups of rings").

EDIT: I briefly sum up some of the matters treated by Fuchs. Let me start quoting his final "note": "The problem of defining ring structures on an additive group was raised by Beaumont who considered rings on direct sums of cyclic groups. Nearly at the same time. Szele investigated zero-rings, Ridei and Szele, and Beaumont and Zuckerman described the rings on subgroups of the rationals. A rather systematic study of constructing rings on a group appeared in Fuchs where the fundamental role of the basic subgroup was pointed out. More satisfactory results have been obtained for torsion-free groups of finite rank by Bedumont and Pierce [...].

It would be a serious mistake to expect too much from a study of the additive structures of rings, as far as ring theory is concerned. In many important cases the additive structures are too trivial [e.g,torsion-freedivisibleor an elementalyp-group]to give any real information about the ring structure. This especially applies to the tonion-free case, whore a close interrelation between the additive and the multiplicative structurescan be expected only if the additive group is more complicated. One should, however, remember that there are intriguing questions even if the additive group is too easy to describe; for instance, we do not know of any uncountable Naethcrian ring whose additive group is free"

Let me say that your question is open in general but there are unswers for particular classes of groups, such as torsion free groups of rank one, where one can not just say when a ring structure exists but also encode somehow all the possible ring structure, even non associative or non-unitary. The fact is that one of the tools for studying ring structures on a group $G$ is the group of multiplications on $G$, $Mult(G)$. This is a group in general only if you include non-associative multiplications.

The chapter starts studying general properties of $Mult(G)$ (for example one can show that $Mult(G)\cong Hom(G\otimes G,G)$). After that it passes to general conditions on torsion and torsion-free groups (expetially divisible, or finite rank, or even rank 1 for the best results). Notice that the main focus is not exactly on your question but it is on an even more delicate matter, that is, given a group $G$ try to classify all the rings that have $G$ as underlying group. Proceeding with the chapter more particular question are investigated. For example, one can give a very precise form for the additive groups of Artinian rings (see Teorem 122.4). There are also precise results for regular rings and the chapter ends (as all the chapters of this book) asking some intriguing questions on the matter.

It may happen that this classical book is not up to date as it goes back to '73 but, for my experience, you can be quite sure that, if something related to Abelian groups was known before '73, then it is in this book. I was looking for more recent information about your question something like 3 years ago but, as far as I remember, I could not find anything relevant after Fuchs' book.

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@nik if you read all the paragraph you can see that I'm speaking about torsion Abelian groups... $\mathbb Z$ is obviously not in there. I can edit to make it clearer if you think it is the case –  Simone Jun 30 '13 at 10:11
    
Ah yes, I missed the adjective torsion, sorry. I was confused since I already cover this in my question. –  Najib Idrissi Jun 30 '13 at 10:14
    
yes, you are right, I just wanted to be complete... to show some counter-example in the torsion case but to show that there exist also many examples in the torsion-free case. The real answer is the reference to the book of Fuchs. –  Simone Jun 30 '13 at 10:21
    
I don't have access to the book right now, unfortunately. Could you possibly sum up the gist of it, so that I can accept your answer? –  Najib Idrissi Jun 30 '13 at 10:24
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