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 Sum and Union of Ideals: Corollary 2:


 * 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.