Exercise 15 of section 2-1 of Kaplansky's Commutative Rings is to show that if $T$ is a Noetherian ring and is finitely generated module over a subring $R$ of $T$, then $R$ is Noetherian. Kaplansky says that the problem can be reduced to the case where $T$ is a domain and $T/J$ is a Noetherian $R$-module for every nonzero ideal $J$, using the following:

If $T$ is a ring with a subring $R$, and $I$ is an ideal in $T$ maximal with respect to the property that $T/I$ is not a Noetherian $R$-module, then $I$ is a prime ideal of $T$.

Now, since $T_{0}=T/I$ is a domain and $T_{0}/J_{0}$ is Noetherian for every nonzero ideal $J_{0}$ of $T_{0}$, we have that $R/R\cap I$ is Noetherian, assuming what Kaplansky said above. But I still don't know how to show that $R$ is Noetherian from that. Am I missing something?


Although I don't understand very well your question, let me try to sketch the proof of the theorem (as it is given in Kaplansky's book):

Reduce the proof to the case $T$ domain and $T/J$ noetherian $R$-module for any $J\subset T$ non-zero ideal: if $T$ is a noetherian $R$-module, then $R$ is a noetherian ring since a ring which has a noetherian faithful module is noetherian (why?). Then suppose that $T$ is not a noetherian $R$-module and take an ideal $K$ in $T$ maximal with the property that $T/K$ is not a noetherian $R$-module. Prove that $K$ is prime.
Consider a non-zero ideal $I$ of $R$. From a previous exercise one knows that there is $J\subset T$ non-zero ideal such that $J\cap R\subseteq I$.
Since $T/J$ is a noetherian $R$-module, then it is also a noetherian $(R/J\cap R)$-module. But a ring which has a noetherian faithful module is noetherian (why?). In our case the module is $T/J$ and the ring is $R/J\cap R$. We deduce that $R/J\cap R$ is noetherian, and therefore $R/I$ is noetherian.

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  • $\begingroup$ I think I haven't really clarified what I'm asking for, and I apologize that. What I meant was, to justify that reduction at the beginning of the proof, one must show that when the theorem holds for T is a domain and T/J is Noetherian for every ideal J of T, then it holds for the general case. But even after assuming it holds for the domain case, I couldn't show the theorem for the general case, and that's where I'm lost. $\endgroup$ – user124816 Jul 2 '14 at 22:12
  • $\begingroup$ I'm really sorry, now I got it. THank you so much! $\endgroup$ – user124816 Jul 2 '14 at 22:27
  • $\begingroup$ I got the same problem as user124816. By previous excercise we reduced to the case $T=T/K$ and $R=R/(R\bigcap K) $ be domains ( the result about contracted ideal holds provided both ring are domains) and $T$ is $R$-finite module, follow the fashion of this answer we conclude $R/(R\bigcap K) $ is noetherian. The only problem is: How can we know if $R/(R\bigcap K) $ is noetherian then so is $R$. ? $\endgroup$ – Anh_Rose 1210 Feb 20 '17 at 16:06

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