Union of Chain of Proper Ideals is Proper Ideal
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Theorem
Let $R$ be a ring with unity.
Let $\struct {P, \subseteq}$ be the ordered set consisting of all ideals of $R$, ordered by inclusion.
Let $\sequence {I_\alpha}_{\alpha \mathop \in A}$ be a non-empty chain of proper ideals in $P$.
Let $\ds I = \bigcup_{\alpha \mathop \in A} I_\alpha$ be their union.
Then $I$ is a proper ideal of $R$.
Proof
By Union of Chain of Ideals is Ideal, $I$ is an ideal.
It remains to show that $I \subsetneq R$.
The ideals $I_\alpha$ are all proper, so none of them contain the unity.
Thus $I$ does not contain $1$, which means $I \subsetneq R$.
$\blacksquare$