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

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I Finitely Generated Ideals and Noetherian Rings

Have something to add?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**