Intersection of Ring Ideals Containing Subset is Smallest

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Theorem

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

Let $S \subseteq R$ be a subset of $R$.

Let $L$ be the intersection of the set of all ideals of $R$ containing $S$.


Then $L$ is the smallest ideal of $R$ containing $S$.


Proof

From Intersection of Subrings Containing Subset is Smallest, $L$ is the smallest subring of $R$ containing $S$.

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


As $L$ is the smallest subring of $R$ containing $S$, and it is an ideal of $R$, there can be no smaller ideal of $R$ containing $S$ as it would then not be a subring.


So the intersection of the set of all ideals of $R$ containing $S$ is the smallest ideals of $R$ containing $S$.

$\blacksquare$


Sources