I is a maximal ideal iff R/I is a field

Dependencies:

1. Ideal
2. Field
3. Quotient Ring
4. gH = H iff g in H

Let $R$ be a commutative ring with unity. Let $I$ be a proper ideal of $R$.

$I$ is defined to be a maximal ideal iff no proper ideal of $R$ is a proper superset of $I$.

$I$ is a maximal ideal iff $R/I$ is a field.

Proof

Since $R$ is a commutative ring with unity, $R/I$ is a commutative ring with unity $1+I$.

Proof of 'only-if' part

Let $a+I \in R/I$ be a non-zero element. Thus, $a \not\in I$. Since $0 \in I$, $a \neq 0$.

Let $J = \{ra + i: r \in R \wedge i \in I\}$.

• $0 = 0a + 0 \in J$, so $J$ is non-empty.
• Let $r_1a + i_1, r_2a + i_2 \in J$. Then $(r_1a + i_1) - (r_2a + i_2) = (r_1 - r_2)a + (i_1 - i_2) \in J$.
• Let $r_1a + i_1, r_2a + i_2 \in J$. Then $(r_1a + i_1)(r_2a + i_2) = (r_1ar_2)a + (r_1ai_2 + i_1r_2a + i_1i_2) \in J$.
• Let $r' \in R \wedge ra + i \in J$. Then $r'(ra + i) = (r'r)a + r'i \in J$ and $(ra + i)r' = (rr')a + ir' \in J$.

Therefore, $J$ is an ideal of $R$.

Let $i \in I$. Then $i = 0a + i \in J$. Therefore, $I \subseteq J$. $a = 1a + 0 \in J$, but $a \not\in I$. Therefore, $I \subset J$ and $I \neq J$.

Let $I$ be a maximal ideal. This means that $J = R$.

Since $1 \in R, 1 \in J$, which means $\exists b \exists i, 1 = ba + i$. $\Rightarrow 1 + I = ba + I = (b + I)(a + I)$. Therefore, every non-0 element $a+I$ in $R/I$ has an inverse. Therefore, $R/I$ is a field.

Proof of 'if' part

Let $R/I$ be a field. So $0 + I, 1 + I \in R/I$. Therefore, $I \neq R$.

Since $I$ is not a maximal ideal, let $J$ be an ideal of $R$ which is a superset of $I$. Let $a \in J-I$. Since $a \not\in I, a + I \neq 0 + I$.

Since $R/I$ is a field and $a+I$ is non-0, $a+I$ has an inverse. Therefore, $\exists b, (a+I)(b+I) = ab + I = 1 + I$. Therefore, $\exists i, ab + i = 1$.

$ba \in J$ because $a \in J$ and absorption. $ba + i \in J$ because $I \subset J$ and closure of $J$. Therefore, $1 \in J$. By absorption, $\forall r \in R, r \in J$. Therefore, $R = J$. Therefore, $I$ is maximal.

Dependency for: None

Info:

• Depth: 6
• Number of transitive dependencies: 14

Transitive dependencies:

1. Group
2. Coset
3. For subset H of a group, a(bH) = (ab)H and (aH)b = a(Hb)
4. Ring
5. Ideal
6. Field
7. Subgroup
8. Normal Subgroup
9. Product of normal cosets is well-defined
10. Factor group
11. Inverse of a group element is unique
12. gH = H iff g in H
13. Product of ideal cosets is well-defined
14. Quotient Ring