Noetherian Modules - Bland - Proposition 4.2.3

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In summary: N$ but not in $N^{*}$. And as for it being a properly generated submodule, since $N^{*}\neq N$, there must be some element $y\in N-N^{*}$, which means that $N^{*}+xR$ is a submodule of $N$ that contains both $N^{*}$ and $x$, which means it properly contains $N^{*}$. In summary, the proof of (2) \Longrightarrow (3) in Proposition 4.2.3 states that if $N^{*}\neq N$ and $x\in N-N^{*}$, then $N^{*}+
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I am reading Paul E. Bland's book, "Rings and Their Modules".

I am trying to understand Chapter 4, Section 4.2 on Noetherian and Artinian modules and need help with the proof of \(\displaystyle (2) \Longrightarrow (3) \) in Proposition 4.2.3.

Proposition 4.2.3 and its proof read as follows:

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In the proof of \(\displaystyle (2) \Longrightarrow (3) \) we read:

" ... ... If \(\displaystyle N^* \ne N\), let \(\displaystyle x \in N - N^*\).

Then \(\displaystyle N^* + xR\) is a finitely generated submodule of N that properly contains \(\displaystyle N^*\) ... ..."


My question is as follows:How does it follow from \(\displaystyle N^* \ne N\) and \(\displaystyle x \in N - N^*\) ... ...

... that ...

... ... \(\displaystyle N^* + xR\) is a finitely generated submodule of N that properly contains \(\displaystyle N^*\)?
Hope someone can help ... ...

Peter***EDIT***

It is certainly the case that \(\displaystyle N^*\) is finitely generated since it belongs to \(\displaystyle \mathscr{S}\) and so it seems obvious that \(\displaystyle N^* + xR\) is finitely generated ... but is it a submodule? Presumably it is a module because \(\displaystyle N^*\) and \(\displaystyle xR\) are modules and the sum of two modules is a module ... but how are we sure that it is a submodule of \(\displaystyle N\) ...

It certainly also seems that \(\displaystyle N^* + xR\) properly contains \(\displaystyle N^*\) ...

... so I am really close to feeling I understand the answer to my question ...

Can someone please critique my thinking?
 
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It's a submodule of $N$ because $x$ is taken in $N$, actually $x\in N-N^{*}$
 

Related to Noetherian Modules - Bland - Proposition 4.2.3

1. What is Proposition 4.2.3 in Bland's paper on Noetherian Modules?

Proposition 4.2.3 in Bland's paper on Noetherian Modules states that every submodule of a finitely generated module over a Noetherian ring is also finitely generated.

2. What does it mean for a ring to be Noetherian?

A ring is considered Noetherian if it satisfies the ascending chain condition for ideals, which means that there is no infinite chain of ideals I1 ⊂ I2 ⊂ ... ⊂ In ⊂ ... that continues indefinitely.

3. How does Proposition 4.2.3 relate to Noetherian rings and modules?

Proposition 4.2.3 states that every submodule of a finitely generated module over a Noetherian ring is also finitely generated. This relates to Noetherian rings and modules because it shows that the property of being Noetherian is preserved under taking submodules.

4. What is the significance of Proposition 4.2.3 in the study of Noetherian modules?

Proposition 4.2.3 is significant because it allows us to prove the finiteness of certain submodules without having to explicitly construct them. This can be helpful in many applications, such as in algebraic geometry and commutative algebra.

5. Can Proposition 4.2.3 be generalized to other types of modules?

Yes, Proposition 4.2.3 can be generalized to other types of modules such as Artinian modules, which satisfy the descending chain condition for submodules. In this case, every quotient module of a finitely generated Artinian module is also finitely generated.

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