Exercise similar to prime avoidance theorem in R.Y. Sharp's textbook

Question

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.

Related: http://math.stackexchange.com/questions/1165705/a-vector-space-over-an-infinite-field-is-not-a-finite-union-of-proper-subspaces — user26857, Aug 26 '16 at 20:05

score 2 · Accepted Answer · edited Aug 26 '16 at 19:38

2

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.

edited Aug 26 '16 at 19:38

user26857

52,094

answered Aug 26 '16 at 18:01

Ashar Tafhim

451

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

Exercise similar to prime avoidance theorem in R.Y. Sharp's textbook

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