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 ordered set $\struct {\mathbb L_J, \subseteq}$ 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 $\set {b_1, b_2}$

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

Thus $\struct {\mathbb L_J, \subseteq}$ is a lattice.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old: Theorem $22.7$