# Tagged Questions

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### $f:M\rightarrow N$ module homomorphism, $(N/\mathrm{Im}f)_m=N_m/\mathrm{Im}f_m$

$f:M\rightarrow N$ is an $R$-module homomorphism and $f_\mathfrak{m}:M_\mathfrak{m}\rightarrow N_\mathfrak{m}$ is the induced $R$-module homomorphism $$f_\mathfrak{m}(m/s)=f(m)/s$$ where ...
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### Nilpotency of finite ideal

Suppose we have a commutative local ring $R$ with unit. I'm curious about whether the following statements are correct: 1- every proper finite ideal is nilpotent. 2-every proper finitely generated ...
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### Monic irreducible polynomial over an integral domain

These days, I have some basic problem in abstract algebra. I know that in any integral domain, any prime element must be an irreducible element. Moreover, if $A$ is a UFD, then an element $a \in A$ is ...
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### Exact sequence out of commutative exact diagram

I'm trying to get grip on the following commutative exact diagram: I know where the maps come from and could verify the exactness and the other maps. (It is induced by the long exact sequence of ...
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### A question in Chapter III.4 of Dino Lorenzini's “An Invitation to Arithmetic Geometry”

Question 1 I am studying in the book "An Invitation to Arithmetic Geometry" by Prof. Dino Lorenzini. In Chapter III Section 4, we consider the following condition: Let $A$ be a Dedekind domain ...
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### Hom functor and localization

I saw in some book the following question Let $R$ be a commutative ring and $P$ a prime ideal of $R$. Prove that $$\mathrm{Hom}_R(M,N)_P \simeq \mathrm{Hom}_{R_P}(M_P,N_P)$$ if $M$ is finitely ...
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### When is a local, reduced, (commutative) ring an integral domain?

Question I am wondering whether or not it is true that if $A$ is a reduced ring, then is it the case that the localization of $A$ at any of its prime ideals is an integral domain? Discussion ...