Maximal Ideal iff Quotient Ring is Field/Proof 1/Maximal Ideal implies Quotient Ring is Field
Theorem
Let $\struct {R, +, \circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$.
Let $J$ be an ideal of $R$.
Let $J$ be a maximal ideal.
Then the quotient ring $R / J$ is a field.
Proof
Let $J$ be a maximal ideal.
Because $J \subset R$, it follows from Quotient Ring of Commutative Ring is Commutative and Quotient Ring of Ring with Unity is Ring with Unity that $R / J$ is a commutative ring with unity.
We now need to prove that every non-zero element of $\struct {R / J, +, \circ}$ has a product inverse in $R / J$.
Let $x \in R$ such that $x + J \ne J$, that is: $x \notin J$.
Thus $x + J \in R / J$ is not the zero element of $R / J$.
Take $K \subseteq R$ such that:
- $K = \set {j + r \circ x: j \in J, r \in R}$
that is, the subset of $R$ which can be expressed as a sum of an element of $J$ and a 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:
- $-h = -j_2 + \paren {-r_2} \circ x$
But $j_1 - j_2 \in J$ from Test for Ideal.
Similarly $-r_2 \in R$.
So $-h \in K$ and we have:
- $(2) \quad g + \paren {-h} = \paren {j_1 - j_2} + \paren {r_1 - r_2} \circ x$
Now consider $g \in J, y \in R$.
Then:
- $g \circ y = \paren {j_1 + r_1 \circ x} \circ y = \paren {j_1 \circ y} + \paren {r_1 \circ y} \circ x$
which is valid by the fact that $R$ is a commutative ring.
But as $J$ is an ideal:
- $\paren {j_1 \circ y} \in J$
while $r_1 \circ y \in R$.
Thus:
- $(3) \quad g \circ y \in K$
and similarly:
- $(3) \quad y \circ g \in K$
So Test for Ideal can be applied to statements $(1)$ to $(3)$, and it is seen that $K$ is an ideal of $R$.
Now:
\(\ds j\) | \(\in\) | \(\ds J\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds j + 0_R \circ x\) | \(\in\) | \(\ds K\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds j\) | \(\in\) | \(\ds K\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds J\) | \(\subseteq\) | \(\ds K\) |
and because $x = 0_R + 1_R \circ x$ (remember $0_R \in J$), then $x \in K$ too.
So, because $x \notin J$, $K$ is an ideal such that $J \subset K \subseteq R$.
Because $J$ is a maximal ideal, then $K = R$.
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 commutative ring $\struct {R / J, +, \circ}$, the product inverse of $x + J$ is $s + J$.
The result follows.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 63.1$ Construction of fields as factor rings