Increasing Union of Ideals is Ideal
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Theorem
Sequence of Ideals
Let $R$ be a ring.
Let $S_0 \subseteq S_1 \subseteq S_2 \subseteq \dotsb \subseteq S_i \subseteq \dotsb$ be ideals of $R$.
Then the increasing union $S$:
- $\ds S = \bigcup_{i \mathop \in \N} S_i$
is an ideal of $R$.
Chain of Ideals
Let $R$ be a ring.
Let $\struct {P, \subseteq}$ be the ordered set consisting of all ideals of $R$, ordered by inclusion.
Let $\set {I_\alpha}_{\alpha \mathop \in A}$ be a non-empty chain of ideals in $P$.
Let $\ds I = \bigcup_{\alpha \mathop \in A} I_\alpha$ be their union.
Then $I$ is an ideal of $R$.