# Tagged Questions

Questions about commutative rings, their ideals, and their modules.

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### Ramification in the ring of all algebraic integers

If $F$ is a finite extension of $\mathbb{Q}$ then its of integers $R$ is a Dedekind domain, and has unique factorization of ideals into powers of prime ideals. For each prime number $\ell$, you can ...
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### Homomorphisms of a field into its valuation ring

Let $R$ be a discrete valuation ring with quotient field $K$. Let $k$ be a field contained in $R$. What are the $k$-algebra homomorphisms $\operatorname{Hom}_k(K, R)$? Are they all trivial?
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### Divisor on curves, Proposition (II.6.9) from Hartshorne

I have some question related to the proof of Proposition (II.6.9) from Hartshorne's book: Let $f:X \rightarrow Y$ be a finite morphism of nonsingular curves over a field $k$. Then for any divisor ...
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### If $R$ is a regular local ring module finite over $k[[x,y]]$ is $x-y^2$ irreducible in $R$?

This user deleted the following question which I think deserves to be here: Let $R$ be a regular local ring module finite over $k[[x,y]]$. Does $x-y^2$ remain irreducible over $R$? (We may assume ...
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### Universal property of de Rham differential.

Suppose $A$ is a commutative algebra over a field $k$. It is well known that there is a module that generalizes the notion of differential $1$-forms. It is denoted $\Omega^1_{k}(A)$ and is called the ...
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### Understanding the right-exactness of the tensor product using *only* its universal property and the Yoneda lemma

I would like to get an intuition for why $(-)\otimes N$ is right-exact using its universal property involving bilinear maps, not by appealing to higher-level observations such as "left-adjoints ...
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### If $R$ is an integral domain, then $R[[x]]$ is an integral domain

While solving another problem (specifically Exercise 7.2 in Atiyah & Macdonald's Introduction to Commutative Algebra), I got stuck in the following step: If $R$ is an integral domain, how I ...
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### A commutative ring in which every prime ideal is 2-generated

Suppose $R$ is a commutative ring with 1. There are some statements that tells us if prime ideals behave in certain way, then all the ideals will behave in that way. For example, If every prime ...
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### Isomorphism of formal power series factorrings over polynomials

This problem is taken from the Hartshorne's book Algebraic Geometry, Chapter 1, Section 5, Problem 14(a). Two polynomials $f(x,y)$ and $g(x,y)$ are written in the form f(x,y) = f_{r}(x,y) + f_{r+...
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### Isomorphic factor rings of polynomial rings does imply isomorphic ideals?

Let $k$ be a field, $I$ and $J$ are ideals of $R=k[x_1,\dots,x_n]$. If $R/I\simeq R/J$ as rings, then $I \simeq J$ as $R$-modules holds? Thanks in advance!