MHB Noetherian Modules - Bland - Proposition 4.2.3

[Math Amateur]

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 $$ (2) \Longrightarrow (3) $$ in Proposition 4.2.3.

Proposition 4.2.3 and its proof read as follows:

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

" ... ... If $$N^* \ne N$$, let $$x \in N - N^*$$.

Then $$N^* + xR$$ is a finitely generated submodule of N that properly contains $$N^*$$ ... ..."


My question is as followsow does it follow from $$N^* \ne N$$ and $$x \in N - N^*$$ ... ...

... that ...

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

Peter***EDIT***

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

It certainly also seems that $$N^* + xR$$ properly contains $$N^*$$ ...

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

Can someone please critique my thinking?

 

[Fallen Angel]

It's a submodule of $N$ because $x$ is taken in $N$, actually $x\in N-N^{*}$