Finitely Generated Modules and Artinian Rings

[Sudharaka]

Hi everyone, :)

Here's another question that I am struggling to complete. If you have any hints or suggestions for this one, I would be so grateful. :)

Question:

Let $S\subseteq R$ be rings and assume that $R_S$ is a finitely generated $S$-module. If $S$ is Artinian prove that $R$ is also Artinian.

 

[Sudharaka]

Hi everyone, :)

Here's another question that I am struggling to complete. If you have any hints or suggestions for this one, I would be so grateful. :)

Question:

Let $S\subseteq R$ be rings and assume that $R_S$ is a finitely generated $S$-module. If $S$ is Artinian prove that $R$ is also Artinian.

I came up with an answer and it would be nice if someone can confirm it, or show mistakes in it. :)

There is a lemma that says, "Let $V$ be a finitely generated $R$-module. If $R$ is Artinian, then so is $V$." By this lemma, we know that $R_S$ ($R$ as a $S$-module) is Artinian. So if we take any decreasing chain of $R$-submodules of $R_R$;

\[R_R \supseteq W_1\supseteq \cdots \supseteq W_n \supseteq \cdots\]

Each $R$-submodule is also a $S$-submodule. Since $R_S$ is Artinian the above chain should stabilize at some point and therefore $R_R$ is also Artinian.

 

[Deveno]

I don't see a problem with this, but...

Can you prove the lemma?

 

[Sudharaka]

I don't see a problem with this, but...

Can you prove the lemma?

Thank you so much for confirming. :) Yes I can prove it since I went through the proof which is in the textbook I am referring for Ring Theory (A Course in Ring Theory by Passman). The proof in lengthy and uses several other lemmas which are mentioned previously in the book, so I am not going to write down it here.

Thanks again for all your help. I sometimes wonder how some people (like you) grasp hard concepts in Ring Theory very easily whereas I have to go through the textbooks, internet, forums etc, for hours to figure them out. :)

 

[Deveno]

Don't get discouraged...I found rings very hard, too.

 

[Math Amateur]

Thank you so much for confirming. :) Yes I can prove it since I went through the proof which is in the textbook I am referring for Ring Theory (A Course in Ring Theory by Passman). The proof in lengthy and uses several other lemmas which are mentioned previously in the book, so I am not going to write down it here.

Thanks again for all your help. I sometimes wonder how some people (like you) grasp hard concepts in Ring Theory very easily whereas I have to go through the textbooks, internet, forums etc, for hours to figure them out. :)

Thank you for this post Sudharaka.

It is heartening to find that other members find ring theory a challenge ...

... very rewarding when you achieve understanding of such a wonderful theory though ...
Peter

 

[Sudharaka]

Thank you for this post Sudharaka.

It is heartening to find that other members find ring theory a challenge ...

... very rewarding when you achieve understanding of such a wonderful theory though ...
Peter

Hi Peter, :)

You are welcome, surely you'll see me posting a lot of ring theory questions.