I am reading the book "An Introduction to Rings and Modules with K-theory in View" by A.J. Berrick and M.E. Keating ... ...(adsbygoogle = window.adsbygoogle || []).push({});

I am currently focused on Chapter 3; Noetherian Rings and Polynomial Rings.

I need someone to help me to fully understand the maximal condition for modules and its implications ...

On page 111, Berrick and Keating state the following:

"The module M is said to satisfythe maximum conditionif any nonempty set of submodules of M has a maximal member (with respect to inclusion) "

It seems to me that this definition, when it is satisfied, means that all the submodules of M must be in a chain of inclusions ... so we cannot have a situation like that depicted in Figure 1 below:

Can someone confirm that my basic understanding of the implication of the definition mentioned above is correct ... ... and/or ... ... give a simple explanation of the maximal condition ...

Help will be appreciated ...

Peter

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*** EDIT ***

Just reflecting a bit on this matter ... ...

Maybe the maximal condition means that for any collection of submodules IF one can arrange a subset into an ascending chain of inclusions then that chain will have a maximal member ... ... ?

Mind you if that is true ... then you can have several maximal members ...

Hope someone can clarify this issue for me ....

Peter

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# Noetherian Modules - Maximal Condition - Berrick and Keating

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