Ideals Containing Ideal Form Lattice

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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 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.

$\blacksquare$


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