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

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Theorem

Let $R$ be a ring with unity.

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

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


Proof

Since $J \subset R$, it follows from Quotient Ring of Ring with Unity is Ring with Unity that $R / J$ is a ring with unity.


We now need to prove that every non-zero element of $\struct {R / J, +, \circ}$ has an inverse for $\circ$ in $R / J$.

By Left Inverse for All is Right Inverse it is sufficient to show that $\struct {R / J, +, \circ}$ has a left inverse for every non-zero element.


Let $x + J \in R / J$ not be the zero element of $R / J$.

That is, $x + J \ne J$ and $x \notin J$.

Take $K \subseteq R$ such that $K = \set {j + r \circ x: j \in J, r \in R}$.

That is, $K$ is the subset of $R$ which can be expressed as a sum of an element of $J$ and a left product in $R$ of $x$.


Now $0_R \in K$ as $0_R \in J$ and $0_R \in R$, giving $0_R + 0_R \circ x = 0_R$.

So:

$(1): \quad K \ne \O$


Now let $g, h \in K$.

That is:

$g = j_1 + r_1 \circ x, h = j_2 + r_2 \circ x$

Then:

\(\displaystyle g + \paren{-h }\) \(=\) \(\displaystyle \paren{j_1 + r_1 \circ x } - \paren{j_2 + r_2 \circ x }\)
\(\displaystyle \) \(=\) \(\displaystyle \paren{j_1 - j_2 } + \paren{r_1 \circ x - r_2 \circ x }\)
\(\displaystyle \) \(=\) \(\displaystyle \paren{j_1 - j_2 } + \paren{r_1 - r_2 } \circ x\)
\(\displaystyle \) \(\in\) \(\displaystyle K\) $j_1 - j_2 \in J$ from Test for Ideal and $r_1 - r_2 \in R$

So we have:

$(2) \quad \forall g, h \in K, g-h \in K$


Now consider $g = \paren {j_1 + r_1 \circ x} \in K, y \in R$.

Then:

\(\displaystyle y \circ g\) \(=\) \(\displaystyle y \circ \paren {j_1 + r_1 \circ x}\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {y \circ j_1 } + y \circ \paren { r_1 \circ x }\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {y \circ j_1 } + \paren {y \circ r_1} \circ x\)
\(\displaystyle \) \(\in\) \(\displaystyle K\) $\paren{y \circ j_1} \in J$, as $J$ is an ideal, while $y \circ r_1 \in R$

Thus:

$(3) \quad y \circ g \in K$

So Test for Left Ideal can be applied to statements $(1)$ to $(3)$, and it is seen that $K$ is a left ideal of $R$.


Now:

\(\displaystyle j\) \(\in\) \(\displaystyle J\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle j + 0_R \circ x\) \(\in\) \(\displaystyle K\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle j\) \(\in\) \(\displaystyle K\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle J\) \(\subseteq\) \(\displaystyle K\)

As $x = 0_R + 1_R \circ x$, it follows that $x \in K$ also.


As $x \notin J$, it follows that $K$ is a left ideal such that $J \subset K \subseteq R$.

As $J$ is a maximal left ideal, it follows that $K = R$.

Thus $1_R \in K$ and thus:

$\exists j_0 \in J, s \in R: 1_R = j_0 + s \circ x$

So:

$1_R + \paren {- s \circ x} = j_0 \in J$

Hence:

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


So in the ring with unity $\struct {R / J, +, \circ}$, the left inverse of $x + J$ is $s + J$.

The result follows.

$\blacksquare$


Sources