Ideals Containing Ideal Form Lattice

Theorem

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

Let $\mathbb L_J$ be the set of all ideal 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:

$(1): \quad b_1 + b_2 \in \mathbb L_J$ and is the supremum of $\left\{{b_1, b_2}\right\}$

$(2): \quad b_1 \cap b_2 \in \mathbb L_J$ and is the infimum of $\left\{{b_1, b_2}\right\}$.

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

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 22$: Theorem $22.7$