Ideals in Commutative Rings Containing a Field

Asked Jul 03 '18 at 21:57

Active Jul 03 '18 at 22:00

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This is a question that always stumped me. It is from an undergrad book (Sharp, Steps in Commutative Algebra) and apparently concerns the Primary Decomposition/Prime Avoidance Theorem, so I expect the solution to be only a few lines, but I can't get it.

Suppose that $R$ is a commutative ring containing an infinite field $F$. Suppose that $I, J_1, \dots, J_n$ are ideals of $R$. Suppose that $I \subset \bigcup\limits_n J_n$. Then prove that $I$ is contained in some $J_k$.

I can't figure out how to use the presence of the field to turn it into a situation where the Prime Avoidance Theorem can be applied, but maybe this isn't the right method.

edited Jul 03 '18 at 22:00

Bernard

175,478

asked Jul 03 '18 at 21:57

John Samples

I'm sure this is what you meant, but for the sake of avoiding pedantry you should mention that $ F $ is a subring, not just contained in $ R $. – paul blart math cop Jul 03 '18 at 22:05
Are ${J_i}_{i=1}^n$ all prime ideal? – Andrew Shedlock Jul 03 '18 at 22:05
1

This is because $R, I,J_1,\dots,J_n: $ are also $F$-vector (sub)spaces, and we have the Avoidance lemma for subspaces.
For a proof, please see my answer to A vector space over an infinite field is not a finite union of proper subspaces
– Bernard Jul 03 '18 at 22:12

Ideals in Commutative Rings Containing a Field

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