# Maximal Ideal iff Quotient Ring is Division Ring

## Theorem

Let $R$ be a ring with unity.

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

The following are equivalent:

$(1): \quad$ $J$ is a maximal ideal.
$(2): \quad$ The quotient ring $R / J$ is a division ring.

## Proof

By Quotient Ring of Ring with Unity is Ring with Unity, $R / J$ is a ring with unity.