Maximal Ideal iff Quotient Ring is Division Ring

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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.



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