Corollary 2

Let $A$ be a ring and $S$ and $T$ two subsets of $A$ such that $S \subset T$. There exists a unique homomorphism $i_A^{T,S}$ from $A[S^{-1}]$ to $A[T^{-1}]$ such that $i_A^T = i_A^{T,S} \circ i_A^S$.

For all $a \in A$, $i_A^{T,S}$ then maps the element $a/s$ in $A[S^{-1}]$ to the element $a/s$ in $A[T^{-1}]$.

Remark (7). Note that, if $i_A^T$ is injective, so is $i_A^{T,S}$ (Corollary 1). This is what happens if $T$ is the set $R$ of elements of $A$ which are not divisors of 0; then it is possible to identify $A[S^{-1}]$ with the subring of the total ring of fractions $A[R^{-1}]$ generated by $A$ and the inverses in $A[R^{-1}]$ of the elements of $S$.