Definition 8

The ring defined in Theorem 4 is called the ring of fractions associated with $S$, or with denominators in $S$, and is denoted by $A[S^{-1}]$.

The zero of $A[S^{-1}]$ is $0/1$, the unit of $A[S^{-1}]$ is $1/1$.

We shall return to the properties of $A[S^{-1}]$ in Commutative Algebra, Chapter II, § 2.

If $S$ is the set of cancellable elements of $A$, the ring $A[S^{-1}]$ is called the total ring of fractions of $A$. $A$ is then identified with a subring of $A[S^{-1}]$ by means of the mapping $\varepsilon$, which is then injective (I, § 2, no. 4, Proposition 6).