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Let $p_1=2,p_2=3,p_3=5, p_4=7, p_5=11,...$ the primes.

I recently had Q/A here and here about the ring$$\boxed{(\prod_{n=1}^{+\infty} \mathbb{Z} / p_n\mathbb Z,+,\times)} $$ where $\prod_p \mathbb{Z} / p $ seemed to be relatively well-known. For my part, I had never read of $$\prod_p \mathbb{Z} / p=\mathbb Z/2\times \mathbb Z/3\times \mathbb Z/5\times \mathbb Z/7\times \mathbb Z/11\times...$$

For example, $$\varphi:\mathbb Z\to \prod_p \mathbb{Z} / p$$ $$n\mapsto (n,n,n,n,n,...)$$is an embedding of $\mathbb Z$, which is injective but not surjective.

  • Let $n\neq n'$.

Then $\exists m, s.t. p_m\ge n \text { and }p_m\ge n'$

Then $\varphi(n)_m=n\neq n'=\varphi(n')_m$

Example (with $n=11,n'=2111, m=5, p_5=11$) : $$\varphi(11)=(1,2,1,4,0,...)$$ $$\varphi(2111)=(1,2,1,4,10,...)$$

  • Let $n\in \mathbb Z$. Then, $n$ has only a finite number of divisors. For example, $15=3\times 5\mapsto (1,0,0,1,4,2,15,15,15,15,15,15,...)$

So, $$(1,0,0,0,0,0,0,0,0,0...)\notin \varphi(\mathbb Z)$$

I find this interesting. And, besides the question of the post, I would like to be able to find any references about this ring. Any help will be greatly appreciated.

Stéphane Jaouen
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  • a good exercise is to show that it has infinite characteristic, maybe study the nilpotent elements. – Colver Apr 16 '24 at 07:50
  • I suppose it's related to the ring of adeles where you reduce to the residue field at each place, ignoring the infinite place. I don't know if this has any name or is particularly relevant to study in that context, but I'd guess it's overshadowed by that. – Merosity Apr 16 '24 at 07:51
  • Why the downvote ? What is unclear? What is not useful ? – Stéphane Jaouen Apr 16 '24 at 09:56
  • Are you looking for Adele ring? – Liding Yao May 06 '24 at 17:08
  • I don’t know what an “adele ring” is; I’m just looking for a name that might be given to $(\prod_{n=1}^{+\infty} \mathbb{Z} / p_n\mathbb Z,+,\times)$, else that "product of all $\mathbb Z/p$ for every prime $p$. – Stéphane Jaouen May 06 '24 at 17:30
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    @LidingYao No, the adele ring is the subring of $\prod_p \mathbb{Q}_p$ consisting of elements which are in $\mathbb{Z}_p$ (the $p$-adic integers) for all but finitely many primes $p$. The ring $\prod_p \mathbb{Z}_p$ is sometimes called the Prufer ring, and denoted $\widehat{\mathbb{Z}}$. But I agree with diracdeltafunk; I've never seen a standard name or symbol for $\prod_p \mathbb{Z}/p \mathbb{Z}$. – David E Speyer May 08 '24 at 13:52

1 Answers1

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The simple answer is that this object does not have a standard name. You could call it "the product of all finite prime fields", if you'd like. It is certainly a well-known object, but not of enough theoretical importance to have a special name.

diracdeltafunk
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