Proposition 3

*Let $\mathbf{B}$ be a non-zero ring and $\mathbf{A}$ a commutative subring of $\mathbf{B}$ such that every non-zero element of $\mathbf{A}$ is invertible in $\mathbf{B}$. (a) $\mathbf{A}$ is an integral domain. (b) *Let $\mathbf{A}'$ be the field of fractions $\mathbf{A}$. The canonical injection of $\mathbf{A}$ into $\mathbf{B}$ can be extended uniquely to an isomorphism $f$ of $\mathbf{A}'$ onto a subfield of $\mathbf{B}$. (c) The elements of $f(\mathbf{A}')$ are the $xy^{-1}$ where $x \in \mathbf{A},\ y \in \mathbf{A},\ y \neq 0$.

Assertion (a) is obvious. The canonical injection of $\mathbf{A}$ into $\mathbf{B}$ extends uniquely to a homomorphism $f$ of $\mathbf{A}'$ into $\mathbf{B}$ (§ 8, no. 12, Theorem 5). Assertion (b) then follows from no. 1, Theorem 2. The elements of $\mathbf{A}'$ are the fractions $x/y$ with $x \in \mathbf{A},\ y \in \mathbf{A},\ y \neq 0$ and $f(x/y) = xy^{-1}$, whence (c).