Corollary 4

Let $S$ be a subset of a ring $A$, $B$ a subring of $A[S^{-1}]$ containing $i_A^S(A)$ and $S'$ the set $i_A^S(A)$. Let $j$ be the canonical injection of $B$ into $A[S^{-1}]$; the unique homomorphism $g$ from $B[{S'}^{-1}]$ to $A[S^{-1}]$ such that $g \circ i_A^S = j$ is an isomorphism.

The mapping $g$ is injective by Corollary 1; the ring $g(B[{S'}^{-1}])$ contains $i_A^S(A)$ and the inverse of the elements of $S'$; hence it is equal to $A[S^{-1}]$.

If $A$ is an integral domain and $0 \notin S$, the notation $A[S^{-1}]$ agrees with that of Algebra, Chapter IV, § 2, no. 1; also, if $S$ is multiplicative, $A[S^{-1}]$ coincides in this case with the set denoted by $S^{-1}A$ in Algebra, Chapter I, § 1, no. 1.

As an extension of notation, for any multiplicative subset $S$ of a ring $A$, we henceforth denote by $S^{-1}A$ the ring of fractions $A[S^{-1}]$. If $S$ is the complement of a prime ideal $p$ of $A$, we write $A$, instead of $S^{-1}A$.

If $A$ is an integral domain and $0 \notin S$, $S^{-1}A$ is always identified with a subring of the field of fractions of $A$, containing $A$ (Remark 7).