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.

Maximal Right Ideal implies Quotient Ring is Division Ring
The proof for Maximal Right Ideal implies Quotient Ring is Division Ring is similar to the proof for Maximal Left Ideal implies Quotient Ring is Division Ring with product orders reversed. It can be found here.

Quotient Ring is Division Ring implies Maximal Right Ideal
The proof for Quotient Ring is Division Ring implies Maximal Right Ideal is similar to the proof for Quotient Ring is Division Ring implies Maximal Left Ideal with product orders reversed. It can be found here.