Ideals Containing Ideal Form Lattice

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

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

Then the poset $\left({\mathbb L_J, \subseteq}\right)$ is a lattice.

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

Then from Set of Ideals forms Complete Lattice:


 * 1) $b_1 + b_2 \in \mathbb L_J$ and is the supremum of $b_1$ and $b_2$;
 * 2) $b_1 \cap b_2 \in \mathbb L_J$ and is the infimum of $b_1$ and $b_2$.

Thus $\left({\mathbb L_J, \subseteq}\right)$ is a lattice.