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My question is: given an abelian group $G$ with addition $+$, is there some natural multiplicative structure that arises so that we can define a ring $(G, +, \cdot)$. For instance, multiplication on $\mathbb{Z}$ and $\mathbb{Z}_n$ are entirely determined by addition, since it must be that $ma = a + \ldots + a$, where the addition is $m$ times.

For finitely generated abelian groups $G$, we know that its representation according to the Fundamental Theorem of Finitely Generated Abelian Groups is

$$G = \mathbb{Z}_{p_1^{r_1}} \times \ldots \times \mathbb{Z}_{p_n^{r_n}} \times \mathbb{Z} \times \ldots \times \mathbb{Z}$$

Since each of those factors has a natural ring structure, we can define a ring structure on $G$ as the product of these ring structures. That leaves the question: can we define a "natural" ring structure on infinitely generated abelian groups $G$?

Joshua Mundinger
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    $ma$ is not actually the same as $ba$ were $b\in\mathbb Z_n$. In fact, in $\mathbb Z_n$, any generator could be used as the identity of the ring.There is no "natural" choice for the multiplicative identity. (Also, bad to talk about $na$ in the context of $\mathbb Z_n$.) – Thomas Andrews Nov 05 '14 at 00:41
  • Also, while the Fundamental Theorem is true, there isn't a "natural" representation of the group in this form. There are plenty of representations in this form... – Thomas Andrews Nov 05 '14 at 00:45
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    Is it not true, however, that no matter the choice of generator as multiplicative identity on $\mathbb{Z}_n$, that the resulting ring structures will be isomorphic? Thus can we assume $1$ is unity? – Joshua Mundinger Nov 05 '14 at 00:46
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    Let's say someone gives you a group ${0,a,b}$ with $a+a=b$, $b+b=a$, $a+b=0$. This is isomorphic to $(\mathbb Z_3,+)$ in two different ways, one with $a$ sent to $1$, one with $b$ set to $1$. So there is no "natural" way to create a ring out of this group, only if we choose arbitrarily from the isomorphisms with $\mathbb Z_3$ is there an isomorphism. – Thomas Andrews Nov 05 '14 at 00:47
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    There's no ring structure on the Prüfer groups, IIRC. – egreg Nov 05 '14 at 00:47
  • @ThomasAndrews Alright, so there may be no "natural" way to impose a ring structure, but we can certainly choose one arbitrarily in order to define ring multiplication in an arbitrary finitely generated abelian group. – Joshua Mundinger Nov 05 '14 at 00:49
  • In an arbitrary finitely generated abelian group, you mean. @Alqatrkapa If you allow rings without unit, you can take $a\cdot b=0$ for all $a,b$, of course. – Thomas Andrews Nov 05 '14 at 00:52
  • I suspect there is no ring structure on the full circle group, either, @egreg – Thomas Andrews Nov 05 '14 at 00:53
  • @ThomasAndrews Indeed. Edited to reflect that. – Joshua Mundinger Nov 05 '14 at 00:53
  • And yes, I'm just quibbling about the word "natural." Mathematicians actually have a meaning for that term... – Thomas Andrews Nov 05 '14 at 00:53

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Expanding on egreg's comment.

The Prüfer group for $p$ has the property that all elements are of finite order, and there is an element of order $p^k$ for every $k$.

Now, in a ring, every element must be of additive order equal to a divisor of the additive order of $1$, because if $n1=0$ then $nr=n(1r)=(n1)r=0$ for any $r\in R$. So the Prüfer group cannot be made a ring.

On a side note, your insistence on using the word "natural" muddles the question considerably, since the word "natural" has a very specific meaning in category theory, and none of the constructions you've defined above are "natural" in the sense of category theory.

Thomas Andrews
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