- #1

Math Amateur

Gold Member

MHB

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I am currently studying Chapter 2: Prime Ideals and Primary Submodules.

I need help with an aspect of Proposition 3 in Chapter 2.

Proposition 3 and its proof read as follows:

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In the last sentence of the statement of Proposition 3, we read the following:

" ... ... Then \(\displaystyle \Omega\), together with the relation \(\displaystyle \le \) is a non-empty inductive system and all its maximal elements are prime ideals."My question/problem is as follows:

\(\displaystyle \Omega\) is a set of ideals and so its maximal elements would be maximal ideals ... ... BUT ... in the Corollary to Proposition 1 in this chapter, Northcott has

*already*shown that "Every maximal ideal is a prime ideal" ... SO ... it appears that this part of Proposition 3 is unnecessary/redundant ...

But this makes me feel I am missing something or misunderstanding something ... ...

Can someone please clarify this situation ... that is why is Northcott apparently proving that all maximal ideals are prime ideals twice over ... ?

Hope someone can help ... ...

In order for MHB members to fully understand the above, I am providing Proposition 1 and its Corollary, as follows:

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