Intersection of Ring Ideals is Largest Ideal Contained in all Ideals

Theorem

Let $\struct {R, +, \circ}$ be a ring

Let $\mathbb L$ be a non-empty set of ideals of $R$.

Then the intersection $\bigcap \mathbb L$ of the members of $\mathbb L$ is the largest ideal of $R$ contained in each member of $\mathbb L$.

Proof

Let $L = \bigcap \mathbb L$.

From Intersection of Ring Ideals is Ideal $L$ is indeed an ideal of $R$.

Let $L = \bigcap \mathbb L$.

From Intersection of Subrings is Largest Subring Contained in all Subrings, we have that $L$ is the largest subring of $R$ contained in each member of $\mathbb L$.

As $L$ is the largest subring of $R$ contained in each member of $\mathbb L$, and it is an ideal of $R$, there can be no larger ideal as it would then not be a subring.

So $L$ is the largest ideal of $R$ contained in each member of $\mathbb L$.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old: Theorem $22.4$