MHB Finitely Generated Ideals and Noetherian Rings

[Math Amateur]

I am reading Joseph J. Rotman's book: Advanced Modern Algebra (AMA) and I am currently focused on Section 5.3 Noetherian Rings ...I need some help with understanding the proof of Proposition 5.33 ... ...Proposition 5.33 reads as follows:View attachment 6008
https://www.physicsforums.com/attachments/6009
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

 

[GJA]

Hi Peter,

... ... 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 ...
Peter

To help with the intuition as to how an infinite set is finitely generated, think about the Cartesian plane. There are infinitely many points (x,y) in this set, but it is finitely generated by all possible linear combinations of (1,0) & (0,1).

To answer your question specific to the text, since $M$ is finitely generated there are finitely many elements $m_{1},\ldots, m_{n}$ in $M$ such that every element of $M$ can be written as a linear combination of these elements whose coefficients come from the ring. Now add to this finite list the single element $a$. Then every element in $J$ can be expressed as a linear combination of $a,m_{1},\ldots,m_{n}.$

 

[Math Amateur]

Hi Peter,
To help with the intuition as to how an infinite set is finitely generated, think about the Cartesian plane. There are infinitely many points (x,y) in this set, but it is finitely generated by all possible linear combinations of (1,0) & (0,1).

To answer your question specific to the text, since $M$ is finitely generated there are finitely many elements $m_{1},\ldots, m_{n}$ in $M$ such that every element of $M$ can be written as a linear combination of these elements whose coefficients come from the ring. Now add to this finite list the single element $a$. Then every element in $J$ can be expressed as a linear combination of $a,m_{1},\ldots,m_{n}.$

Hi GJA ... thanks for the help ...

I must say that your explanation is many times clearer than Rotman's explanation which is somewhat misleading ...

Thanks again ... appreciate your assistance ...

Peter