# Tagged Questions

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

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### There always exists a finite, increasing chain of R-submodules of M isomorphic to R/P. Can we describe P?

So I've been studying some commutative algebra and I came across the following theorem Theorem : Let R be a Noetherian ring. Let $M$ be a non trivial $R$-module, finite over $R$. There exists a ...
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### Zeros of specialization of a family of polynomials [closed]

Let $k$ be an algebraically closed field, and $K\supset k$ be an algebraically closed extension. Let $a\in K^n$ be a tuple, we call $a^\prime\in k^n$ a specialization of $a$ if for any $f(X)\in k[X]$ ...
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### Is every “prefield” a field?

Definition 0. Call a poset $P$ well-ranked iff it is well-founded, and for all $x \in P$, we have that any two maximal subchains in the lowerset generated by $x$ have the same length. Definition ...
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### Analytical isomorphism implies same multiplicities [duplicate]

I want to prove the following problem in Robin Hartshorne's Algebraic Geometry Chapter 1 exercise 5.14 If $P\in Y$ and $Q\in Z$ are analytically isomorphic plane curve singularities, show that the ...
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### Regular element of a Noetherian ring [duplicate]

Let $R$ be a Noetherian ring and $x\in R$ an $R-\mathrm{regular}$ element. Show that $\mathrm{Ass}_R(R/(x^n))=\mathrm{Ass}_R(R/(x))$ for every $n\geqslant 1$. Let $M$ be an $R-\mathrm{module}$. An ...
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### Example of a projective variety that is not projectively normal but normal

I want to prove the following statement: Let $Y$ be the quartic curve in $\mathbb{P}^3$ given parametrically by $(x,y,z,w)=(t^4,t^3u,tu^3,u^4)$. Then $Y$ is normal but not projectively normal. ...
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### Show that $S_f^{\ge0}=\bigoplus_{d\ge0}(S_f)_d$ is a normal domain, where $S$ is an $\mathbf N$-graded domain, $S_{(f)}$ a normal domain $f\in S_1$ [closed]

Let $S$ be an $\mathbf N$-graded domain with $S_{(f)}$ a normal domain for some $f\in S_1$. Then $S_f^{\geq0}=\bigoplus_{d\geq0}(S_f)_d$ is a normal domain.
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### Transitivity-like Results in Group, Ring, Module, Field and Galois Theory [closed]

I am reading Michael Atiyah and Ian Macdonald's Introduction to Commutative Algebra. On page 28, Proposition 2.16 says: Suppose $A,B$ are rings, $N$ is a finitely generated $B$-module, $B$ is ...
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### Correspondence between prime ideals and irreducible algebraic sets

Let $k$ be an algebraic closed field. The Nullstellensatz theorem prove that $$I(V(J))=\sqrt{J}$$ and we have $$V(J)\text{ irreducible }\iff I(V(J)) \text{ prime }$$ So if $J$ is prime, $I(V(J))=J$ is ...
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### Extension of DVRs and uniformizers

Let $(A,\mathfrak m)$ be a regular, Noetherian, local, domain of dimension $2$ and consider a prime ideal $\mathfrak p\subset A$ of height $1$. Moreover let $\hat{A}$ be the completion of $A$ with ...
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### $n$th root of power series when its coefficients are from a field with positive characteristic

Let $k$ be algebraically closed field of characteristic $p>0$. Let's consider a power series $f(x,y)\in k[[x,y]]$. Under what conditions (on $n$, $f$, ...) there exists $g(x,y)\in k[[x,y]]$ such ...
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### Some questions about local rings and chain rings.

(All my rings are commutative with $1$.) The notion of a field spews forth many derived concepts. For example: An integral domain is a ring that can be homomorphically injected into a field. A ...
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### Minimal presentation of a tensor product of modules

Let $(R,m)$ be a local commutative noetherian ring. Suppose I have two modules $M$ and $N$ over $R$ given in terms of minimal presentations $$M = \operatorname{coker}A,$$ and  N = \operatorname{...
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### Why is $\mathbb{F}_5[x]$ a Jacobson ring? [closed]

As the question title suggests, why is $\mathbb{F}_5[x]$ a Jacobson ring?
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### Can I use Krull dimension to test if a sequence of polynomials is regular?

A sequence $(f_1, \ldots, f_n)$ of elements of a commutative ring $R$ is said to be regular if for each $i$, $f_i$ is not a zero divisor in $R/(f_1, \ldots, f_{i-1})$. Call a sequence dimension ...
Let $R\subset T$ be two commutative rings, and $T$ is integral over $R$. Let $\mathfrak m\in \operatorname{Max} R,\mathfrak n\in\operatorname{Max}T$ such that $\mathfrak m=\mathfrak n\cap R$. Show ...
Let $A$ be an integral domain, $M$ an $A$-module, and $m\in M$. Now for all maximal ideals $\mathfrak{m}$ there exists an $n\notin \mathfrak{m}$ such that $nm=0$. Why does this mean that $m=0$?