Limit of ideals? I've been playing around with an idea in my head for a while. Consider the ideals $n \mathbb{Z}$ of $\mathbb{Z}$. As $n \to \infty$, I want to say that $n \mathbb{Z} \to 0 \mathbb{Z}$, for a couple of reasons. Firstly, the density of $n \mathbb{Z}$ in $\mathbb{Z}$ goes to 0 as $n$ grows. Secondly, it's intuitive that working (mod 0) should be equivalent to working (mod $\infty$) in a number-theoretical sense. 
I want to know whether this makes sense. If so, how would you best formalise this notion of a sequence of ideals converging to a "limiting" ideal?
 A: Note that this sequence of principal ideals, when ordered by inclusion $n{\bf Z}\subseteq d{\bf Z}$ for $d\mid n$, is not totally ordered (in the sense that $n{\bf Z}$ and $m{\bf Z}$ are incomprable when neither of $n,m$ are divisors of the other). However, we can speak of all maximal chains in this poset (lattice, actually) of ideals, and defining the 'limit' of a chain $A_1\supset A_2\supset A_2\supset\cdots$ to be the intersection $\bigcap_{n\ge0}A_n$; it is a fairly straightforward logic exercise to verify arbitrary intersections of ideals are ideals. In this case, the limit of every single maximal chain is $0{\bf Z}$ (another relatively straightforward exercise).
There are different notions of limit, however, that complicate this picture with some exotic new algebraic structures. If $n{\bf Z}\to0{\bf Z}$ as ideals, we might naively expect ${\bf Z}/n{\bf Z}\to{\bf Z}/0{\bf Z}\cong{\bf Z}$ in some or other sense as rings (informally put, 'taking quotients is continuous'). In an arbitrary category (of groups, rings, fields, modules, lattices, ...), an inverse system of objects and morphisms can define an inverse limit. For our purposes, we can consider a sequence of rings with onto homomorphisms
$$R_0\xleftarrow{\varphi_1} R_1\xleftarrow{\varphi_2} R_2\xleftarrow{\varphi_3}R_3\xleftarrow{\varphi_4}\cdots.$$
We want to view the $R_i$ terms, as $i\to1$ from $\infty$, as successively further collapsed versions (which is to say, quotients, or homomorphic images of) some limiting object. The categorical definition of the limit involves universal properties and commutative diagrams, but for our purposes it suffices to look at the explicit construction: Define $\Pi:=\prod_{n\ge0}R_n$ to be the infinite product of the $R_i$s, and define $\varprojlim R_i$ to be the subring of $\Pi$ comprised of those tuples that are 'coherent,' in the sense that for these tuples $(r_0,r_1,r_2,\cdots)$, for each coordinate $r_n$ ($n>0$), $\varphi_n(r_n)=r_{n-1}$. That is, each coordinate is the homomorphic image of the next coordinate in line.
It makes sense that this is called an inverse limit, since the morphisms are flowing from right to left, and so to take the limit we are swimming upstream so to speak. It is also called the projective limit, which also makes sense because these transition morphisms may be called projections.
Some fun inverse systems to consider:
$${\bf Z}/p^0{\bf Z}\leftarrow{\bf Z}/p^1{\bf Z}\leftarrow {\bf Z}/p^2{\bf Z}\leftarrow\cdots $$
$$F\leftarrow F[T]/(T)\leftarrow F[T]/(T)^2\cdots$$
The inverse limit of the first is ${\bf Z}_p$, the $p$-adic integers, which has no zero divisors at all (so it is not to be confused with the integers modulo $p$). The inverse limit of the second, with $F$ an arbitrary field (actually it could be any ring), is $F[[T]]$, the ring of formal power series with coefficients taken from $F$. The latter is actually the "function field" analogue of the former (which is a "number field" fact); we use the term "global" fields for those that are either function fields or number fields, and these inverse limits are completions of them (or technically, of their integers). The fraction fields of the rings ${\bf Z}_p$ and $F[[T]]$, which are ${\bf Q}_p$ and $F((T))$ respectively, are called local fields.
Note that $\varprojlim{\bf Z}/n{\bf Z}$, taken over all $n\ne0$ (so our inverse system is no longer totally ordered, and our original sequence definition would need upgrading), is the direct product $\prod_p{\bf Z}_p$ over all prime numbers $p$. This is, arguably, high-tiered language for Sun-Ze's theorem (aka CRT).
There is a notion dual to inverse limits, called direct limits (or colimits). If we have a sequence
$$A_0\to A_1\to A_2\to A_3\to\cdots, $$
we can, roughly speaking, consider each morphism a sort of embedding of each term into the next, in which case the direct limit is the 'final object' that they are all embedded in, intuitively. For our algebraic-number-theoretic example, we have (as additive groups, not rings)
$$\varinjlim\, {\bf Z}/p^k{\bf Z}={\bf Z}(p^\infty)\cong{\bf Z}[1/p]/{\bf Z}\cong{\bf Q}_p/{\bf Z}_p$$
These are called the Prufer $p$-groups, and they occur in the $p$-primary decomposition of ${\bf Q}/{\bf Z}$; an analogous picture manifests when we look towards function fields, which tells us these Prufer groups encode the number field analogue of partial fraction decomposition in function fields.
Funnily enough, the Prufer $p$-groups ${\bf Z}(p^\infty)$ and $p$-adic integers ${\bf Z}_p$ are each other's dual groups.
This was rather tangential, so perhaps only the first paragraph was relevant to you, OP. It was also written summarily, so it'd be perfectly expected that others might be interested in expanding on any of the above points. (Mostly, it's an advertisement for studying number theory, I guess.)
A: Your intuition is correct.  If turn the (non-negative) integers into a partially ordered set via divisibility, $1$ is initial and $0$ is final.  In other words,
$$ 1 \,|\, n \qquad \text{and} \qquad n \,|\, 0 \qquad \text{for all } n \in \mathbb{Z}_{\ge0}$$
As ideals, under set containment, we get the opposite poset.
$$ 0 = 0\mathbb{Z} \subset n\mathbb{Z} \subset 1\mathbb{Z} = \mathbb{Z} \qquad \text{for all }n. $$
Finally, if we consider the quotients, the order reverses again, so
$$ 0 = \mathbb{Z}/1\mathbb{Z} \subset \mathbb{Z}/n\mathbb{Z} \subset \mathbb{Z}/0\mathbb{Z} = \mathbb{Z} \qquad \text{for all }n. $$
