Intersection of Subrings is Largest Subring Contained in all Subrings

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Theorem

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

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


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


Proof

Let $\displaystyle L = \bigcap \mathbb L$.

From Intersection of Subrings is Subring, $L$ is indeed a subring of $R$.


By Intersection of Subgroups is Subgroup, $\struct {L, +}$ is the largest subgroup of $\struct {R, +}$ contained in each member of $\mathbb L$.

By Intersection of Subsemigroups, $\struct {L, \circ}$ is the largest subsemigroup of $\struct {R, \circ}$ contained in each member of $\mathbb L$.


Let $S$ be a subring of $R$ such that:

$\forall K \in \mathbb L: S \subseteq K$

By Intersection is Largest Subset, $S \subseteq L$.

Thus $L$ is the largest subring of $R$ contained in each member of $\mathbb L$.

$\blacksquare$


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