Uncountably Generated Ideals of Functions

by B. Sury

This article originally appeared in:
College Mathematics Journal
November, 2011

Subject classification(s): Algebra and Number Theory | Abstract Algebra | Rings and Ideals
Applicable Course(s): 4.2 Mod Algebra I & II

It is well known that maximal ideals in the ring of continuous functions on the closed interval \([0, 1]\) are not finitely generated. Less well known is the fact that these ideals are not countably generated, although the proof is not more difficult. The author proves this, and uses the result to produce some non-prime ideals in the ring of continuous functions on the open interval \((0, 1)\) which also cannot be countably generated.

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