Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Quotient Ring is Division Ring implies Maximal Right Ideal

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Let $R$ be a ring with unity.

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

If the quotient ring $R / J$ is a division ring then $J$ is a maximal right ideal.


Let $K$ be a right ideal of $R$ such that $J \subsetneq K \subset R$.

Let $x \in K \setminus J$.

As $x \notin J$ then $x + J \ne J$, the zero in $R / J$.

As $R / J$ is a division ring then $x + J \in R / J$ has an inverse, say $s + J$.

That is:

$1_R + J = \paren {x + J} \circ \paren {s + J} = \paren {x \circ s } + J$

By Left Cosets are Equal iff Product with Inverse in Subgroup then:

$1_R - x \circ s \in J \subsetneq K$

By the definition of a right ideal:

$x \in K$ and $s \in R \implies x \circ s \in K$
$1_R - x \circ s \in K$ and $x \circ s \in K \implies \paren {1_R - x \circ s } + \paren {x \circ s} = 1_R \in K$
$1_R \in K \implies \forall y \in R, 1_R \circ y = y \in K$

Hence $K = R$

The result follows.