Increasing Union of Ideals is Ideal

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 $R$.