*Note: The definition of “ring” used here does not require an identity for multiplication.*

Let be a commutative ring, and suppose is some subset with the property that , contains no zero divisors, and is closed under multiplication. Then by considering and defining and quotienting out an equivalence relation (to “ignore” different representations of fractions), we can form something known as the *ring of quotients*, or *localization*, of elements of by elements of , denoted .

Let us assume we have this situation, and write . Then the ring has a couple of properties:

- can be embedded into , that is, it can be viewed as a subring of .
- Every , under the above identification, becomes a unit in .
- Up to isomorphism, is the unique smallest ring containing which also satisfies the previous property.

This is not a precise statement, so let us try to flesh out what we really want to say. In the first property, what is it exactly that we mean by “embedding” into ? Well, in algebra, the word *embedding* is usually used as a synonym for *injective homomorphism* (to be concise: *monomorphism*). So indeed, with the first property, we really mean to say that there exists a ring monomorphism . You can think of this map as taking the structure of and carrying it into in a rigid manner, so it is indeed justified to term it an embedding (I say “rigid” due to the map’s injectivity). This map , when instead viewed as a map from to , then becomes a ring isomorphism between and some subring of .

Now that we have properly articulated the first property, let us move on to the second. It is quite simple to restate this one: if we take any and consider it as an element of by way of our embedding , the claim is that it will be invertible. In other words, is a unit of for all .

The third property is where things begin to get interesting, because it is what is known as a universal mapping property (in disguise). If you have seen an abstract treatment of tensor products, for example, you will recognize its form. It turns out that it takes quite a bit of effort to really say what we mean here. So let’s reason this out. What do we mean by “ is the *unique smallest *ring containing satisfying the second property”? Well, of course does not, strictly speaking, “contain” , but rather, is embeddable into . Also, we want to get across the idea that can always be embedded in any ring to which admits a ring monomorphism with the second property, and this embedding is moreover uniquely determined by such a monomorphism. So our statement becomes “ is the *unique smallest* ring to which admits a ring monomorphism such that is a unit in for every “. We are getting closer, but still not quite there yet. To make the “smallest ring” bit formal, we have to observe that this means the following:

For any commutative, unital ring and ring monomorphism such that is a unit in for every , there is a unique ring monomorphism with the property that .

By now, you are almost certainly rolling your eyes at how long this is getting, but hilariously enough, we are still not quite finished. Here’s the subtlety: we really want to provide the assurance that indeed, is characterized by this property. That is, we want to say that there *do not exist two structurally different (non-isomorphic) “smallest” rings* and “containing” with the property that every becomes a unit in that ring. After all, if we’re going to use words like “smallest”, then one usually speaks of *the* smallest object in a collection, not *a *smallest object in a collection. We need to communicate that this is, indeed, the case. So, the penultimate revision of our condition is:

For any commutative, unital ring and ring monomorphism such that is a unit in for every , there is a unique ring monomorphism with the property that . Moreover, this property characterizes up to isomorphism.

What do we mean by this “characterization”? Well, we mean this:

For any commutative, unital ring and ring monomorphism such that is a unit in for every , there is a unique ring monomorphism with the property that .

Moreover, suppose and are two rings, both having the above property. (That is, suppose that the above property holds, when every occurrence of is replaced by , and also when every occurrence is replaced by ). Then there exists a ring isomorphism .

The first part is illustrated by this diagram:

And the second part is (perhaps poorly) illustrated by this one: