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Let $\phi:R \longrightarrow S$ be a local homomorphism of local rings. Let $m_R$ and $m_S$ denote the unique maximal ideals of the local rings $R$ and $S$ respectively. Under what conditions can we say that $\phi(m_R)S$ is $m_S$-primary? Is there a way to characterize such local homomorphisms?

Shaun
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Sam
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    A sufficient condition is if the homomorphism is finite , for instance. – Bernard Sep 05 '17 at 15:25
  • @Bernard: Sorry, but I don't even see why is $\phi(m_R)S$ a primary ideal of $S$ when $\phi$ is finite. Could you explain? – Sam Sep 06 '17 at 11:52
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    That is because $S/\mathfrak m_R S$ is a finite vector space over $R/\mathfrak m_R$, hence is an artinian (local) ring. It has only one prime ideal $\mathfrak m_S/\mathfrak m_R S$. – Bernard Sep 06 '17 at 12:18
  • @Bernard Integral instead of finite works as well. – user26857 Sep 06 '17 at 12:37
  • Yes, probably. I didn't write it because there might have been problems of noetherianness, and I didn't have time to check the details. The single problem is that primary ideals (or more generally, primary submodules) are defined only for noetherian rings. – Bernard Sep 06 '17 at 12:45
  • @Bernard "The single problem is that primary ideals (or more generally, primary submodules) are defined only for noetherian rings." Pardon? – user26857 Sep 06 '17 at 20:03
  • @user26857: If $R\to S$ is only an integral extension, you're not sure that $S$ will noetherian, even if $R$ is, and the notion of primary ideal will not be defined. – Bernard Sep 06 '17 at 21:02
  • @Bernard This is the first time when I hear that one needs noetherianity for defining the notion of primary ideal. – user26857 Sep 06 '17 at 21:30
  • I'm referring to Bourbaki, Commutative Algebra, Ch. 4, Associated Prime Ideals and Primary Decomposition, §2 no 1. – Bernard Sep 06 '17 at 23:07
  • @Bernard While I'm referring to https://en.wikipedia.org/wiki/Primary_decomposition#Definitions, and to all books in Commutative Algebra I know. (As a matter of fact, I wonder how Bourbaki defines a laskerian module in Exercise 23 at the end of the same chapter?) – user26857 Sep 07 '17 at 17:44
  • @user26857: The real problem is not to know whether we can have definitions that make sense in more contexts, but whether these extensions are useful. As far as I know we have useful results with primary decompositions only in the case of noetherian rings. A typical example is the concept of associated prime ideals: it works well only in the noetherian case, albeit it makes sense for any commutative ring. It happens that in the general case, there is a concept that works: the so-called weakly associated prime ideals (taken care of in the exercises of the same chapter). – Bernard Sep 07 '17 at 17:55

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