Increasing Union of Ideals is Ideal

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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 $S$:

$\displaystyle S = \bigcup_{i \mathop \in \N} S_i$

is an ideal of $R$.


Chain of Ideals

Let $R$ be a ring.

Let $\left({P, \subseteq}\right)$ be the ordered set consisting of all ideals of $R$, ordered by inclusion.

Let $\left\{{I_\alpha}\right\}_{\alpha \in A}$ be a non-empty chain of ideals in $P$.

Let $\displaystyle I = \bigcup_{\alpha \in A} I_\alpha$ be their union.


Then $I$ is an ideal of $R$.