Ideals Containing Ideal Isomorphic to Quotient Ring

Theorem
Let $J$ be an ideal of a ring $R$.

Let $\mathbb L_J$ be the set of all ideals of $R$ which contain $J$.

Let the poset $\left({\mathbb L \left({R / J}\right), \subseteq}\right)$ be the set of all ideals of $R / J$.

Let the mapping $\Phi_J: \left({\mathbb L_J, \subseteq}\right) \to \left({\mathbb L \left({R / J}\right), \subseteq}\right)$ be defined as:


 * $\forall a \in \mathbb L_J: \Phi_J \left({a}\right) = q_J \left({a}\right)$

where $q_J: a \to a / J$ is the canonical epimorphism from $a$ to $a / J$ from the definition of quotient ring.

Then $\Phi_J$ is an isomorphism.

Proof

 * Let $b \in \mathbb L_J$.

From the way $\mathbb L_J$ is defined, $J \subseteq b$.

Thus $q_J^{-1} \left({q_J \left({b}\right)}\right) = b + J = b$ by Ring Epimorphism Composite with Inverse of Subring.

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

Then, by Ring Epimorphism Composite Of Subring with Inverse, $q_J \left({q_J^{-1} \left({c}\right)}\right) = c$.

Thus by Bijection iff Left and Right Inverse, $\Phi_J$ is a bijection, and $\forall c \in \mathbb L \left({R / J}\right): q_J^{-1} \left({\Phi_J}\right) c = q_J^{-1} \left({c}\right)$.


 * Now to show that $\Phi_J$ is an isomorphism.

Let $b_1, b_2 \in \mathbb L_J$.

If $b_1 \subseteq b_2$, then from Subset Maps to Subset, $q_J \left({b_1}\right) \subseteq q_J \left({b_2}\right)$.

Conversely, suppose $q_J \left({b_1}\right) \subseteq q_J \left({b_2}\right)$.

By what we have just proved, $b_1 = q_a^{-1} \left({q_a \left({b_1}\right)}\right) \subseteq q_a^{-1} \left({q_a \left({b_2}\right)}\right) = b_2$.

Thus $\Phi_J$ is an isomorphism.