### Some Interesting Rings

Monday, July 30th, 2012 | Author:

You probably know the integers $\mathbb{Z}$ which can be defined as the Grothendieck group of the monoid of natural numbers $\mathbb{N}$ which exists by the axioms of Zermelo-Fraenkel set theory and probably in almost all other axiom systems as well. So the integers are a quite fundamental object in mathematics (did I really just argue for that? Well, now I did).

It is also natural to look what one can derive from the integers. In this short article I want to describe some rings that can be obtained from the integers by quotients, subgroups, products, and in general (co)limits and combinations thereof. The purpose might not be so clear, but it was a by-product of other investigations and I hope it could be interesting to say something about some concrete rings.

• $\mathbb{Z}$ is a torsion-free ring which has residue fields in any characteristics. It is a noetherian principal ideal domain. The abelian group $\mathbb{Z}$ generates the category of abelian groups, since every abelian group can be written by generators and relations and this corresponds to a surjection of some power of $\mathbb{Z}$ onto the group (and the relations generate the kernel). Every abelian group is a $\mathbb{Z}$-module just by the operation $n.x = x + x + \cdots + x$ ($n$ times).
• $\mathbb{Z}/n\mathbb{Z}$ is a finite (hence noetherian) torsion ring, and we have maps $\mathbb{Z}/n^k\mathbb{Z} \to \mathbb{Z}/n^{k-1}\mathbb{Z}$ given by reduction modulo $n$. One can also write down (non-canonically) an isomorphism of groups $\mathbb{Z}/n\mathbb{Z} \to \zeta_n$, the group of $n$-th roots of unity, by mapping a generator to a primitive $n$-th root of unity. Under this isomorphism, the map "reduction mod $n$" corresponds to taking the $n$-th power of a $n^k$-th root of unity to obtain a $n^{k-1}$-th root of unity.
• $\mathbb{Q}$ is a field of characteristics 0 with ring of integers $\mathbb{Z}$, i.e. it is the field of fractions of $\mathbb{Z}$. As an abelian group, $\mathbb{Q}$ is divisible, so it is an injective object in the category of abelian groups. $\mathbb{Q}$-modules are the same as $\mathbb{Q}$-vector spaces.
• $\mathbb{Q}/\mathbb{Z}$ is a ring in which every element is torsion, since $a/b + \cdots + a/b$, with $b$ many summands, equals $a \in \mathbb{Z}$, so is $0$ in $\mathbb{Q}/\mathbb{Z}$. As a quotient of a divisible group, $\mathbb{Q}/\mathbb{Z}$ is divisible. It is a cogenerator of the category of abelian groups, i.e. every abelian group maps injectively in a sufficiently large power of $\mathbb{Q}/\mathbb{Z}$ (this gives us the fact, that the category of abelian groups has enough injectives).
• $\mathbb{Z}_\ell$ is the completion of $\mathbb{Z}$ at the prime $\ell$, i.e. at the $\ell$-adic valuation (hence has a nontrivial topology). One can also define equivalently $\mathbb{Z}_\ell = \lim_{\rightarrow} \mathbb{Z}/\ell^n\mathbb{Z}$ and the corresponding limit topology is the same as the completion topology. It contains $\mathbb{Z}$. It is also torsion-free of characteristics 0 and has residue field $\mathbb{Z}_\ell/\ell\mathbb{Z}_\ell \simeq \mathbb{Z}/\ell\mathbb{Z} = \mathbb{F}_\ell$ (did I mention that $\ell$ is prime? $\ell$ is always prime).
• $\mathbb{Q}_\ell$ is the field of fractions of $\mathbb{Z}_\ell$, and one can also write $\mathbb{Q}_\ell = \mathbb{Z}_\ell \otimes_\mathbb{Z} \mathbb{Q}$.
• $\hat{\mathbb{Z}}$ is the profinite completion of the integers $\mathbb{Z}$. It is torsion-free.
• $\mathbb{A}_f = \prod_{p \text{ prime}}' \mathbb{Q}_p$, where the $\prod'$ means the restricted product, which is a subset of the product given by elements which have only finitely many entries not in $\mathbb{Z}_p$. It has a natural topology making it a topological ring, which is different from the subspace topology. Oh, by the way, this ring is called the finite Adèles. The product $\mathbb{A}_f \times \mathbb{R} =: \mathbb{A}$ is called the Adèles (of $\mathbb{Q}$).
• $\mathbb{Z}[x_1,\dots]$, a polynomial ring in countably infinite many variables, has every finitely generated ring as quotient ring. This generalizes to higher cardinalities. @Jan: thanks for this addition to the list.

Do you know more interesting rings derived from these? Did I miss interesting properties?

Category: English, Mathematics

1. Any ring is a quotient of a polynomial ring over $\mathbb Z$ in infinitely many variables.