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I am reading the book "An Introduction to Rings and Modules with K-theory in View" by A.J. Berrick and M.E. Keating ... ...
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 \(\displaystyle M\) is said to satisfy the maximum condition if any nonempty set of submodules of \(\displaystyle M\) has a maximal member (with respect to inclusion)"It seems to me that this definition, it 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:https://www.physicsforums.com/attachments/4883Can someone confirm that my basic understanding of the implication of the definition mentioned above is correct?
Peter
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 \(\displaystyle M\) is said to satisfy the maximum condition if any nonempty set of submodules of \(\displaystyle M\) has a maximal member (with respect to inclusion)"It seems to me that this definition, it 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:https://www.physicsforums.com/attachments/4883Can someone confirm that my basic understanding of the implication of the definition mentioned above is correct?
Peter