On non-maximal prime ideals of $C[0,1]$

TL;DR

This paper demonstrates that non-maximal prime ideals exist in the ring of continuous real-valued functions on [0,1], showing they are contained within unique maximal ideals through geometric analysis of zero sets.

Contribution

It provides a simple, geometric proof of the existence and structure of non-maximal prime ideals in C[0,1], a well-known but counterintuitive result.

Findings

Non-maximal prime ideals exist in C[0,1]

Each non-maximal prime ideal is contained in a unique maximal ideal

Zero set analysis reveals ideal structure

Abstract

We first show a counter intuitive result that in the ring of real valued continuous functions on $[0,1]$ non maximal prime ideals exist. This is a standard proof and a well known result. Interestingly, a non maximal prime ideal in this ring is actually contained inside a unique maximal ideal. We arrive at this result merely by looking at the zero set of ideals in this ring and by making simple geometrical observations. We end by leaving the reader with an interesting open problem that logically follows from this article.