Corollary 1

Let $\mathbf{A}$ be a ring and $a$ a two-sided ideal of $\mathbf{A}$. For the ring $\mathbf{A}/a$ to be a field, it is necessary and sufficient that $a$ be a maximal left ideal of $\mathbf{A}$.

The left ideals of $\mathbf{A}/a$ are of the form $b/a$ where $b$ is a left ideal of $\mathbf{A}$ containing $a$ ($§ 8$, no. 8, Corollary to Proposition 5). To say that $\mathbf{A}/a \neq 0$ means that $a \neq \mathbf{A}$. Under this hypothesis, $\mathbf{A}/a$ is a field if and only if the only left ideals of $\mathbf{A}$ containing $a$ are $a$ and $\mathbf{A}$ (Theorem 1), whence the corollary.