I am reading Joseph J. Rotman's book: Advanced Modern Algebra (AMA) and I am currently focused on Section 5.3 Noetherian Rings ...(adsbygoogle = window.adsbygoogle || []).push({});

I need some help with understanding the proof of Proposition 5.33 ... ...

Proposition 5.33 reads as follows:

In the above text from Rotman, in the proof of (ii) ##\Longrightarrow## (iii) we read the following ...

"... ... The ideal

##J = \{ m + ra \ : \ m \in M \text{ and } r \in R \} \subseteq I##

is finitely generated. ... ...

Can someone please explain to me why it follows that ##J## is finitely generated ... ...

... ... Rotman's assertion that ##J## is finitely generated puzzles me since, although ##M## is finitely generated it may have an infinite number of elements each of which is necessary to generate ##J## as they appear in the formula above ... so how can we argue that ##J## is finitely generated ... it seems it may not be if ##M## is an infinite set ...

Hope someone can help ...

Peter

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# I Finitely Generated Ideals and Noetherian Rings

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