Noetherian Modules - Maximal Condition - Berrick and Keating Ch. 3, page 111

In summary: The attachment with figure 1 has disappeared, I try to give an answer anyway.If you want to apply Zorn’s Lemma on a collection $\hat S$ of sets, which is ordered by inclusion, then you have to prove that every nonempty chain in $\hat S$ has an upper bound. So, if $\hat C$ is a chain in the set $\hat S$, then there must be an $S \in \hat S$ such that for every $C \in \hat C$ we have $C \subset S$. Then $S$ is an upper bound of $\hat C$ and $S$ does NOT need to be an element of $\hat C$.A maximal element is
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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
 
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The attachment with figure 1 has disappeared, I try to give an answer anyway.
If you want to apply Zorn’s Lemma on a collection $\hat S$ of sets, which is ordered by inclusion, then you have to prove that every nonempty chain in $\hat S$ has an upper bound. So, if $\hat C$ is a chain in the set $\hat S$, then there must be an $S \in \hat S$ such that for every $C \in \hat C$ we have $C \subset S$. Then $S$ is an upper bound of $\hat C$ and $S$ does NOT need to be an element of $\hat C$.
A maximal element is something else.
If $\hat A$ is a collection of sets (for instance, submodules of a module $P$), then $M \in \hat A$ is maximal in $\hat A$ if the condition [$X \in \hat A$ AND $M \subset X$] implies $X=M$. So notice that the maximal element $M$ of $\hat A$ is an element of $\hat A$ and that $\hat A$ need not be a chain.

The module $M$ is said to satisfy the maximum condition if any nonempty set of submodules of M has a maximal member (with respect to inclusion). So if $\hat S$ is the collection of all submodules of $M$, ordered by inclusion, and every nonempty subset $\hat A \subset \hat S$ has a maximal element, then the module $M$ is said to satisfy the maximal condition.

I am sorry for my clumsy notation, I was looking for $\mathscr{C}$ and $\displaystyle \mathscr{S}$, but could not find them.
 
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  • #3
steenis said:
The attachment with figure 1 has disappeared, I try to give an answer anyway.
If you want to apply Zorn’s Lemma on a collection $\hat S$ of sets, which is ordered by inclusion, then you have to prove that every nonempty chain in $\hat S$ has an upper bound. So, if $\hat C$ is a chain in the set $\hat S$, then there must be an $S \in \hat S$ such that for every $C \in \hat C$ we have $C \subset S$. Then $S$ is an upper bound of $\hat C$ and $S$ does NOT need to be an element of $\hat C$.
A maximal element is something else.
If $\hat A$ is a collection of sets (for instance, submodules of a module $P$), then $M \in \hat A$ is maximal in $\hat A$ if the condition [$X \in \hat A$ AND $M \subset X$] implies $X=M$. So notice that the maximal element $M$ of $\hat A$ is an element of $\hat A$ and that $\hat A$ need not be a chain.

The module $M$ is said to satisfy the maximum condition if any nonempty set of submodules of M has a maximal member (with respect to inclusion). So if $\hat S$ is the collection of all submodules of $M$, ordered by inclusion, and every nonempty subset $\hat A \subset \hat S$ has a maximal element, then the module $M$ is said to satisfy the maximal condition.

I am sorry for my clumsy notation, I was looking for $\mathscr{C}$ and $\displaystyle \mathscr{S}$, but could not find them.
Thanks for the help, Steenis ...

Just reflecting on what you have written and revising the issue now ...

Peter
 

FAQ: Noetherian Modules - Maximal Condition - Berrick and Keating Ch. 3, page 111

What is the definition of a Noetherian module?

A Noetherian module is a module that satisfies the ascending chain condition, which means that every increasing sequence of submodules eventually stabilizes.

What is the maximal condition for a module?

The maximal condition for a module is a property that states that if a submodule M of a module N is not equal to N, then there exists a submodule L of N such that M is a proper submodule of L.

What is the significance of Noetherian modules?

Noetherian modules are important in many areas of mathematics, including commutative algebra and algebraic geometry. They have many useful properties and can be used to prove important theorems in these fields.

How does the maximal condition relate to Noetherian modules?

The maximal condition is a stronger version of the ascending chain condition, which is a defining property of Noetherian modules. In fact, a module is Noetherian if and only if it satisfies the maximal condition.

What is the main result of Berrick and Keating's Chapter 3 on Noetherian modules?

The main result of Berrick and Keating's Chapter 3 is the Krull-Schmidt theorem, which states that every finitely generated module over a Noetherian ring can be uniquely decomposed into a direct sum of indecomposable modules.

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