In many places in mathematics, we see some variant of the following simple construction. The first time we see it is in constructing the integers from the natural numbers:

Consider pairs $(a,b)$ of elements of $\mathbb{N}$ (which includes zero for our purposes, but it doesn’t really matter this time). We form equivalence classes out of these pairs by saying that $(a,b)\sim(c,d)$ if $a+d=b+c$. We can create a group structure on these pairs by setting $(a,b)+(c,d)=(a+c,b+d)$. The resulting group is isomorphic to $\mathbb{Z}$. So that’s how to construct the integers from the natural numbers.

We see a similar construction when we discuss localizations of rings. Let $R$ be a commutative ring and $S$ a multiplicative subset containing 1. (If $R$ is noncommutative, you can still localize provided that $S$ is an Ore set, but I don’t feel like going there now.) We now consider pairs $(r,s)\in R\times S$ under the equivalence relation $(r_1,s_1)\sim(r_2,s_2)$ if there is some $t\in S$ so that $tr_1s_2=tr_2s_1$. The set of equivalence classes has the structure of a ring, called the localization of $R$ at $S$, and denoted by $R_S$. This construction is generally seen with $S=R\setminus\mathfrak{p}$, where $\mathfrak{p}$ is a prime ideal of $R$. The resulting ring is then local (meaning that it has a unique maximal ideal, namely $\mathfrak{p}R_S$. (We generally write $R_{\mathfrak{p}}$ rather than $R_S$ in this situation.) Anyway, this construction is really useful because localizations at prime ideals are frequently principal ideal domains, and we know all sorts of interesting theorems about finitely generated modules over principal ideal domains. And then we can use some Hasse principle-type result to transfer our results back to the original ring.

Notice that I allowed the multiplicative set of localization to contain zero. However, in this case, the localization becomes the trivial ring (or not a ring, if you require that $0\neq 1$ in your definition of a ring, as many people do). More generally, allowing zero divisors in the multiplicative set causes various elements in $R$ to become zero in the localization.

A similar construction shows up in $K$-theory. Suppose $A$ is any commutative semigroup. We consider pairs $(a,b)\in A\times A$ under the equivalence relation $(a,b)\sim(c,d)$ if there is some $e\in A$ so that $a+d+e=b+c+e$. (This is necessary since we do not assume that $A$ satisfies the cancellation property.) The resulting equivalence classes form a group called the Grothendieck group of $A$ and denoted by $K_0(A)$.

The Grothendieck group satisfies the following universal property. Let $\phi$ be the map sending $a\in A$ to $(a+b,b)$ for any $b\in A$. (This is easily seen to be well-defined.) Now let $G$ be any abelian group and $\psi:A\to G$ any semigroup homomorphism. Then there is a (unique) map $\theta:K_0(A)\to G$ so that $\theta\phi=\psi$.

Grothendieck groups can be very helpful for studying rings. Let $R$ be a commutative ring, and let $A$ denote the semigroup of isomorphism classes of projective $R$ modules (under the operation of direct sum). Then $K_0(A)$ (or $K_0(R)$, as people often write) is an important object of study. If $R$ is a field, for instance, then $K_0(R)\cong\mathbb{Z}$. However, if $R$ is the ring of integers of a number field $K$, then $K_0(R)\cong\mathbb{Z}\oplus C(K)$, where $C(K)$ is the ideal class group.

Perhaps more interesting is Swan’s Theorem, which relates vector bundles over a compact topological space to the projective modules over its ring of continuous functions: they have isomorphic Grothendieck groups. But that’s probably the subject of another post, especially if I can manage to understand my notes from Max Karoubi’s lecture series in Edmonton.