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Any homomorphic image of a Noetherian ring is Noetherian. Furthermore, if $R_0$ is a Noetherian ring, and $R$ is a finitely generated algebra over $R_0$, then $R$ is Noetherian.

Noetherian is a definition ascribed for a ring. But in the second sentence, it says that $R$ is an algebra. And a ring is a special case of algebra(not the other way around). What does it mean by $R$ being Noetherian?

jk001
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An algebra over $R$ is a ring $A$ which also has a module structure over $R$, such that $r(ab)=(ra)b=a(rb)$ holds for all $a,b\in A$ and $r\in R$. So an algebra is in particular a ring, and so you can talk about it being Noetherian.

Mark
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  • Thanks so much. I know that algebra is an action of a ring over a ring. So an algebra being ring means that we are disregarding the action? Is it the same as viewing module as an underlying abelian group? – jk001 Jun 25 '21 at 15:26
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    An algebra is both a ring and a module over $R$. The definition of it being Noetherian is the same as for any ring. However, note that the theorem you mentioned states that if $R$ is a Noetherian ring and $A$ is a finitely generated algebra over it then $A$ is Noetherian. So for this theorem it is obviously important that $A$ also has a module structure, this is what connects it to $R$. – Mark Jun 25 '21 at 15:29
  • Makes sense. Thanks! – jk001 Jun 25 '21 at 15:46
  • Could you please help me with the following statement if possible? I think it's the same context but I have trouble understanding it... If $R$ is a Noetherian ring and $M$ is a finitely generated $R$-module, then $M$ is Noetherian. How can I again see module as a ring? – jk001 Jun 25 '21 at 15:55
  • Oh sorry nvm! Module being Noetherian was defined in the book. – jk001 Jun 25 '21 at 15:56
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    Yes, the general term is a Noetherian module. $R$ is called a Noetherian ring if the module $R$ (where the action is usual multiplication) is Noetherian, so it's a special case. – Mark Jun 25 '21 at 15:59