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:

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

 

[andrewkirk]

He just hasn't described $J$ very well. Since $M$ is finitely-generated, let $G$ be a finite generating set for it. Then define $J$ to be the ideal generated by $G\cup\{a\}$. Then $J$ is finitely generated, is an ideal of $I$ (because $G\subseteq M\subseteq I$ and $a\in I$), and properly contains $M$, thereby contradicting our assumption of the maximality of $M$.

It can be shown that my definition of $J$ matches his but we don't need to do that, as the above two lines proves what's needed.

 

[mathwonk]

the proof is easy. choose any element of a given ideal. if it generates the ideal you are done and the ideal has one generator. If not, choose any new element not in the ideal gehnerated by that one and add it in as a generator. This generates a larger ideal. If that is the whole ideal you are done and it is generated by 2 elements. If not, choose another element not in the ideal generated by those 2, and add it in. That generates another larger ideal. continue... since you are creating a sequence of larger and larger ideals, and every such sequence has a maximal element, at some point this process must stop, and at that point you must have generated the whole ideal by a finite set of generators.

If I may make a suggestion, it is better for you to try to prove such a statement for yourself than to try to puzzle out the proof in the book. this one for example is so easy you would probably have gotten it much easier that way than by reading it.

 

[Math Amateur]

Thanks to Andrew and mathwonk for the help and advice ...

I appreciate both the help and advice ...

Thanks again,

Peter

 

[andrewkirk]

I second mathwonk's advice. Many theorems in texts are not that difficult to do on one's own and, if they are not well written, it is sometimes much more difficult to follow the proof than it is to figure it out oneself, as well as being less satisfying.

On the other hand, a text on a nontrivial subject would nearly always have some theorems that are hard, in the sense of being long and complex, or requiring a completely novel trick that very few people would be capable of thinking up on their own. Dirac's ladder operators that he uses to solve the Schrodinger equation for a harmonic oscillator come to mind. A really good text will warn the reader when a really difficult theorem is coming up, but many don't do that. So it's not always easy to tell whether it would be easier to construct the proof oneself or to follow the one in the text.

One pragmatic approach is to alternate. Perhaps try proving every second theorem yourself, or perhaps one in three. Often you can get a clue by a quick glance over the proof as to whether it will be hard to do on your own. If it is short and does not appear to introduce any new concepts (like ladder operators!) then the chances are good that one could do it without the text.

Also, when trying to follow a proof in a text, if you reach an impasse, you could try taking a step back and thinking 'If I were to try to prove this, how might I go about it?'. There might even be some hints in the non-stuck parts of the text proof that one could use. For instance in this one, without getting into the detail, one can see that the author is trying to derive a contradiction by choosing a maximal ideal M of the family of finitely-generated ideals and then, by choosing an element that is not in that ideal, constructing an ideal that contradicts the maximality of M. I think it's easier to make one's own proof with that broad structure than to follow step by step the one written in the text.

This is probably all superfluous to what MW has already written. But clarity in teaching is a subject that's dear to my heart, so I couldn't help having a bit of a babble.

 

[Math Amateur]

Thanks for those thoughts, Andrew ...

Helpful and informative ...

Peter

 

[mathwonk]

one more remark. if you start trying to prove the theorems yourself, you not only bypass bad proofs and understand more, but you break free from the dependence on books, and begin to make your own way in the forest of learning, this is the beginning of independent learning, wherein you realize that math is a flexible subject that you can be part of and not a fixed package that is codified in books.

 

[Math Amateur]

one more remark. if you start trying to prove the theorems yourself, you not only bypass bad proofs and understand more, but you break free from the dependence on books, and begin to make your own way in the forest of learning, this is the beginning of independent learning, wherein you realize that math is a flexible subject that you can be part of and not a fixed package that is codified in books.

Thanks for the advice and analysis regarding approaches to learning mathematics and the higher levels one can reach, mathwonk ...

Mind you ... whether I have the confidence and ability to reach these levels is another question ... but I have learned quite a bit from you, fresh_42 and Andrew ... Indeed, learning mathematics is a wonderful activity ...

Peter