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I'm a beginner in commutative algebra, and I'm self-studying this subject via R.Y. Sharp's textbook.

Let $R$ be a commutative ring with $1$ which contains an infinite field as a subring. Let $I$ and $J_1,\ldots,J_n$ where $n\ge2 $, be ideals of $R$ such that:

$$I\subseteq J_1\cup J_2\cup\cdots\cup J_n.$$

Prove that $I \subseteq J_j$ for some $j$ with $1\le j \le n$.

It looks similar to the Prime Avoidance Theorem, but in this case each $J_i$ just to be ideals and I can't figure out what the role of "contains an infinite field as a subring" in this situation.

I expect for some idea / suggestion. Thank in advance.

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HINT: The ring $R$ can be thought of as vector space over the infinite field. Then prove that a vector space is not a finite union of subspaces.

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  • $\begingroup$ Well, a vector space can be a finite union of subspaces, but not when the field is infinite. $\endgroup$ – user26857 Aug 26 '16 at 19:39
  • $\begingroup$ yeah,in the case infinite field I use pigeonhole principle to enter the contradiction. $\endgroup$ – Anh_Rose 1210 Aug 26 '16 at 20:56

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