Maximal Left and Right Ideal iff Quotient Ring is Division Ring

Theorem
Let $R$ be a ring with unity.

Let $J$ be an ideal of $R$.

Then the following are equivalent:
 * $(1):\quad J$ is a maximal left ideal
 * $(2):\quad J$ is a maximal right ideal
 * $(3):\quad$ the quotient ring $R / J$ is a division ring.

$(2) \implies (3)$
The proof for $(2) \implies (3)$ is similar to the proof for $(1) \implies (3)$ with product orders reversed. It can be found here.

$(3) \implies (2)$
The proof for $(3) \implies (2)$ is similar to the proof for $(3) \implies (1)$ with product orders reversed. It can be found here.