Theorem 2

Let $\mathbf{K}$ be a field and $\mathbf{A}$ a non-zero ring. If $f$ is a homomorphism of $\mathbf{K}$ into $\mathbf{A}$, then the subring $f(\mathbf{K})$ of $\mathbf{A}$ is a field and $f$ defines an isomorphism of $\mathbf{K}$ onto $f(\mathbf{K})$.

Let $a$ be the kernel of $f$. Then $1 \notin a$ for $f(1) = 1 \neq 0$ in $\mathbf{A}$ and, as $a$ is a left ideal of $\mathbf{K}$, $a = \{0\}$ by Theorem 1. Therefore $f$ is injective and hence an isomorphism of $\mathbf{K}$ onto the subring $f(\mathbf{K})$ of $\mathbf{A}$; the latter ring is therefore a field.