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Let $R$ be a commutative ring with unit. Is it true that given an ideal $I$ and $r \geq 2$ principal ideals $P_1, ..., P_r$, then if $I \subseteq \bigcup_{i = 1}^r P_i$, $I \subseteq P_i$ for some $i$? This would be the analogous of the Prime Avoidance Lemma for principal ideals.

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marked as duplicate by rschwieb abstract-algebra Sep 24 '18 at 10:35

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    $\begingroup$ As Eric Wofsey has cleverly remarked in his answer here, this would imply that all finite rings are principal rings, which is not the case. $\endgroup$ – Saucy O'Path Sep 24 '18 at 10:14