Proposition 4

*Let $\mathbf{A}$ be a commutative ring and $\mathfrak{p}$ an ideal of $\mathbf{A}$. The following conditions are equivalent: (a) the ring $\mathbf{A}/\mathfrak{p}$ is an integral domain; (b) $\mathbf{A} \neq \mathfrak{p}$ and, if $x \in \mathbf{A} - \mathfrak{p}$ and $y \in \mathbf{A} - \mathfrak{p}$, then $xy \in \mathbf{A} - \mathfrak{p}$. (c) $\mathfrak{p}$ is the kernel of a homomorphism of $\mathbf{A}$ into a field.

The implications (c) ⇒ (b) ⇒ (a) are obvious. If $A/p$ is an integral domain, let $f$ be the canonical injection of $A/p$ into its field of fractions and $g$ the canonical homomorphism of $A$ onto $A/p$; then $p$ is the kernel of $f \circ g$, whence the implication (a) ⇒ (c).