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

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1answer
38 views

Question about Matsumura Theorem 23.4

I have one question about the proof. I'm not sure why $I$ is of the form $(A/m)^t$. My thought: Here $A$ is a Noetherian local ring of dimension zero, so it's Artinian. So $m$ is nilpotent, ...
0
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1answer
23 views

For an element, Prove there exist finitely many polynomials in a set Y such that u is a 0 of them iff u is a 0 of all polynomials in Y

Let F be a field and let Y be a set of polynomials in k variables over F. Prove that there exist finitely many polynomials f_{1}, . . . , f_{m} ∈ Y such that for u_{1}, . . . , u_{k} ∈ F^k , f( u_{1} ...
1
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2answers
94 views

Showing that $R[X]/(Xf-1) \cong R[1/f]$ [duplicate]

Let $R$ be an integral domain with quotient field $K$. Let $0 \neq f \in R$. I want to prove Statement: $R[X]/(Xf-1) \cong R[1/f]$. Argument: Consider the epimorphism $\phi: R[X] \rightarrow ...
0
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1answer
34 views

isomorphism between $k[[x]]$ into $\varprojlim_n k[x]/(x^n)$ [duplicate]

i want to find isomorphism between $k[[x]]$ and $\varprojlim_n k[x]/(x^n)$ but I cant.please help me to find this.
1
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1answer
60 views

inverse limit of $k[x]/(x^n)$

I know the inverse limit of $k[x]/(x^n)$ is $k[[x]]$ but i can't show the homomorphism between these is onto. For example $\alpha:k[[X]]\to\varprojlim k[x]/(x^n)$ by F(x)=family of$ F(x)+(x^n)$. for ...
1
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0answers
18 views

Dimension of the vector space given by the quotient of an Artin ring by the product of all its maximal ideals [duplicate]

Let $\mathcal{M}_1,\dots,\mathcal{M}_r$ be all the maximal ideals of an Artin ring $A$ which is a finite $\mathbb{K}$-algebra; so let $A/\mathcal{M}_1\cdots\mathcal{M}_r$ be a $\mathbb{K}$-vector ...
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0answers
31 views

is there a relationship between $\ell (R/I^n)$ and $\ell (R/I)$

$(R,m)$ is local neotherian cohen-macaulay ring of dimension $d$, and $I$ is an $m$-primary ideal of $R$. since $I$ is an $m$-primary, $\dim R /I=\dim R/I^n =0$. so $\ell(R/I^n)$ and $\ell (R/I)$ are ...
0
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0answers
17 views

Weighted degree property

Let $f\in \mathbb{C}[x,y]$ be a polynomial and $a,b$ be positive relatively prime integers. We introduce a new polyomial degree the following way: the monomial $x^i y^j$ has degree $ai+bj$ and we ...
5
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2answers
93 views

How to show $\mathbb{Z}[\sqrt{-5}]_2$ is a UFD?

I would like to know how to show $\mathbb{Z}[\sqrt{-5}]_2$ is a UFD. I am actually given hints that $\mathbb{Z}[\sqrt{-5}]$ has class group $\mathbb{Z}/2$ and that $(1 + \sqrt{-5},2)$ is not ...
0
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0answers
40 views

Cocommutative k-Hopf algebra finite as k-vector space represents a constant group functor

I am working through some of my first exercises regarding Hopf algebras and I am kind of stuck with this one: Given an algebraically closed field $k$ and a cocommutative $k$-Hopf algebra $A$ finite ...
2
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1answer
42 views

How to find a non-surjective and non-injective tensor products of the formal completion?

Let $A$ be a commutative ring with unit endowed with $I$-adic topology where $I$ is the ideal of $A$. Let $\hat A$ be the formal completion of $A$ for the $I$-adic topology, and $M$ an $A$-module. Let ...
0
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1answer
69 views

maximal ideal not containing some expression

Does there exists a domain $R$ with fraction field $K$, and $x \in K \setminus R$, such that for any maximal ideal $\mathfrak{m} \leq R[x]$, there exists $a \in R$ such that $x-a \in \mathfrak{m}$.
1
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1answer
54 views

The localization of a ring at a maximal ideal

I am working on the following problem: If $R$ is a local ring whose maximal ideal is denoted $\mathfrak{p}$ then show that $R \cong R_\mathfrak{p}$. $R_\mathfrak{p} := \{\frac{r}{u} : r\in R, u\in ...
1
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0answers
67 views

Prime ideals in multivariate polynomial rings over $\mathbb R$ and in their quotients

1) Which are the prime ideals of $\mathbb{R}[X_1,\dots,X_n]$? 2) Which are the prime ideals of $\mathbb{R}[x,y]/\langle x^2+y^2-1\rangle$? About the question 2), I know that ...
1
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1answer
40 views

Proving if $ \Gamma_{2}(R)\smallsetminus J(R) $ is a forest then it is either totally disconnected or a star graph

These days I am reading the research paper Graphs associated to co-maximal ideals of commutative rings by Hsin-Ju Wang. In this paper, $ R $ denotes a commutative ring with the identity element. $ ...
0
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1answer
34 views

Hopkins-Levitzki Theorem

Let $R$ be a an artinian ring and $M$ a f.g. $R-$module. By Hopkins-Levitzki Theorem, $M$ is an artinian module. I am looking for an example such that according to the conditions above, $M$ is not ...
1
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0answers
39 views

Direct Sum of Simple Modules

Let $A$ be a commutative ring and $M$ a module of finite length. Under what conditions on $A$ is it true that $M$ is a direct sum of simple modules? Is it true if we assume $A$ to be a Dedekind ...
0
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1answer
34 views

An algebra is a vector space

Let $F$ be a field. If I have $A$ is a finite-dimensional $F$-algebra, then can I conclude that $A$ is a finite-dimensional over $F$?
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1answer
29 views

Factoring a homogeneous element in graded ring

Let $k$ be a field, and $A = k[w,x,y,x] / (wz-xy)$, which is an integral domain. I would like to show that if $h$ is a homogeneous element in $A$, not irreducible, then it factors into a product of ...
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3answers
25 views

Ring of germs and induced isomorphism

Let $A$ be the ring of germs of real analytic functions in $0\in\mathbb R$. Let $x\in A$ be the identity map on $\mathbb R$. How can I show that the map $f\mapsto \sum_n(f^{(n)}(0)/n!)T^n$ from $A$ to ...
1
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1answer
65 views

Does this definition of the “roots” of an element of an arbitary $R$-algebra make sense? If so, where can I learn more?

(All my rings and $R$-algbras are commutative and unital.) Question. I think it makes sense to speak of the "roots" of an element of an arbitary $R$-algebra; a definition is given below. Does it ...
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0answers
39 views

$IM=mM$. can we say that $I$ is a reduction ideal of $m$

Definition. Let $R$ be a Noetherian ring􀀀, $I$ a proper ideal,􀀀 and $M$ a finite $R$-module. An ideal $J\subset I$ is called a reduction ideal of $I$ with respect to $M$ if $JI^nM = I^{n+1}􀀀M$ for ...
2
votes
1answer
51 views

Localization of Minimal free Resolution

Let $(R,m)$ be a local ring and $p \in \operatorname{Spec}(R)$. Let $$\cdots \longrightarrow F_n \longrightarrow F_{n-1}\longrightarrow\dots\longrightarrow F_1\longrightarrow F_0 \longrightarrow ...
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2answers
79 views

In a finitely generated $k$-algebra, the nilradical is $0$ iff the Jacobson radical is $0$.

I was solving an exercise in Vakil's notes Foundations of Algebraic Geometry 3.6.K, and eventually proved the following statement: Let $\mathscr{A}$ be a finitely generated $k$-algebra, where $k$ ...
0
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1answer
49 views

Let K be a field, and $I=(XY,(X-Y)Z)⊆K[X,Y,Z]$. Prove that $√I=(XY,XZ,YZ)$.

Let $K$ be a field, and let $I=(XY,(X-Y)Z) \subset K[X,Y,Z]$. Prove that $\sqrt{I}=(XY,XZ,YZ)$. I have no idea how to start with this question, can anybody give me some hint? Thanks a lot.
3
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1answer
123 views

Grothendieck Group

Let $A=\mathbb{Z}$ be a ring, $K=\mathbb{Q}$ its field of fractions, $L$ a number field, and $B = \mathcal{O}_L$, the integral closure of $A$ in $L$. Define the category $C_A$ of $A$-modules of finite ...
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2answers
99 views

Showing $\operatorname{Spec} k[x,y,z]/(x^2 + y^2+z^2)$ is normal [closed]

Let $k$ be a field of characteristic not $2$ and algebraically closed. I would like to show that $\operatorname{Spec} k[x,y,z]/(x^2 + y^2+z^2)$ is normal. I would appreciate any help/hint. Thank ...
0
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0answers
31 views

Is there always a unit in topological rings?

I'm reading Qing Liu's Algebraic Geometry and Arithmetic Curves, http://176.58.104.245/ALGANT/TONG/Liu-1-4.pdf . On the first page he wrote that "Unless otherwise specified, all rings is this book ...
1
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1answer
33 views

Frobenius powers of an ideal does not depend on the choice of a system of generators

Let $I$ = $(x_1 , . . . , x_n )$ be an ideal of a ring $R$ of characteristic $p$. For each nonnegative integer $e$ we set $I^{[p^e]}$=$(x_1^{p^e},...x_n^{p^e}$)$R$. These ideals are called the ...
1
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1answer
27 views

Tensorproduct of $R$-module and $R$-algebra

let $M$ be an $R$-module and let $S$ be an $R$-algebra through the ring homomorphism $\phi$. I can make $M\otimes S$ into a $R$-module in several different ways. Either by defining $r. (m\otimes ...
0
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0answers
32 views

intuitive interpretation of the multiplicity

Although logically I can understand and use multiplicity (for defi􀀀nition see 4.1.5 of Bruns_Herzog), yet, the concept of multiplicity of a module is not completely clear for me. Is there an ...
2
votes
1answer
73 views

Is $\operatorname{Hom}_R(R/m,R/(x_1,…,x_d))$ isomorphic to $R/m$?

Let $(R,m)$ be a local ring. Let $x_1,...,x_d$ be a maximal $R$-sequence. Is $\operatorname{Hom}_R(R/m,R/(x_1,...,x_d))$ isomorphic to $R/m$?
2
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1answer
35 views

Question on homogeneous ideal in a graded ring

In an $\Bbb{N}$-graded ring $R=\bigoplus_nR_n$, an element is called homogenous (of degree $n$) if it is contained in $R_n$. An ideal is called homogenous if it is generated by homogenous elements. ...
0
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0answers
27 views

Algebraic Ideal & affine varieties equality

Let $I=\langle x^2-y-4\rangle$ and $G= \langle x^2+y-4\rangle$ be two ideals on polynomial ring $\mathbb{C}[x,y]$. As obviously $V(I)=V(G)$, what can we conclude for $I$ and $G$? How they are related ...
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0answers
28 views

a f.g., projective, non free $R-$module [duplicate]

I know that if $R$ is a PID ring, then a projective $R-$module is free. Now, i want an example of a f.g., projective, non free $R-$module where $R$ is a non PID ring.
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0answers
59 views

Morphisms of Affine Sets and Morphisms of Corresponding Coordinate Rings.

I stumbled across something that I really couldn't really figure out. So suppose you have a morphism of affine algebraic sets: $f: X \rightarrow Y$ and the corresponding coordinate ring morphisms: ...
1
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3answers
65 views

The generic fiber of a morphism of schemes

Let $m$ be a non-zero integer and $f: \operatorname{Spec}\mathbb{Z}[T_1,T_2]/\langle T_1T_2^2-m\rangle \to \operatorname{Spec}\mathbb{Z}$. Why is the generic fiber $\operatorname{Spec} ...
2
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1answer
100 views

Normalization or integral closure of ring over $\mathbb Z_p$

Let $p$ be a prime larger than four. And denote the $p$ adic integers by $\mathbb Z_p$. Consider the ring $A=\mathbb Z_p[x]$ and its field of fractions $K=\mathbb Q_p(x)$. Now let's extend $K$ to a ...
5
votes
1answer
57 views

Example of morphism of ringed spaces not induced by homomorphism of rings

What is the example of non-local morphism of ringed spaces $\phi:\text{Spec}(B)\to\text{Spec}(A)$, which is not induced by the ring homomorphism $A\to B$? Thank you.
1
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1answer
38 views

If there exists a vertex of $ \Gamma_{2}(R)\setminus J(R) $ which is adjacent to every other vertex then $ R \cong \mathbb{Z}_{2}\times F$

I am reading the research paper Comaximal Graph of Commutative Rings by H.R. Maimani, M. Salimki, A. Sattari, S. Yassemi. In this paper, $ R $ denotes a commutative ring with the identity element. $ ...
2
votes
0answers
55 views

Algebraic characterization of commutative rings with Krull dimension=1,2, or 3

A commutative ring $R$ (with $1$) is $0$-dimensional if and only if $R/\sqrt 0$ is von Neumann regular. Besides this result, there is a wealth of information about zero-dimensional rings. I could not ...
1
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1answer
38 views

Maximal submodules and maximal ideals

Let $R$ be a commutative ring and $M$ be an $R$-module. We know that $Rad(M)$ is the intersection of all maximal submodules of $M$. If $K$ is a maximal ideal of $R$, is it true that $Rad(M)$ is ...
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0answers
76 views

How can every divisor be reached by a sequence of blow-ups?

The following is a result of Zariski [Lemma 2.45 of Birational Geometry of Algebraic Varieties]. $X$ : an algebraic variety over a field $k$. $(R,m)$ : a DVR of the quotient field $K(X)$ of ...
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0answers
50 views

Inverse of the product of fractional ideals

Let $R$ be a integral domain, $K$ its field of fractions and $\mathfrak M,\mathfrak N$ fractional ideals, i.e. non-zero finitely generated $R$-submodules of $K$. $$\mathfrak M^{-1}=\{x\in K: ...
1
vote
1answer
26 views

A submodule filtration that does not define the structure of topological module.

I was reading the following from Liu's Algebraic Geometry and Arithmetic Curves on page 18. In this book all rings are commutative and with unit. Let $A$ be a ring endowed with the $I$-adic topology. ...
12
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4answers
236 views

Is $\mathbb Z[[X]]\otimes \mathbb Q$ isomorphic to $\mathbb Q[[X]]$?

Is $\mathbb Z[[X]]\otimes \mathbb Q$ isomorphic to $\mathbb Q[[X]]$? Here tensor product is over the ring $\mathbb Z$ and $\mathbb Z[[X]] $ denotes formal power series over $\mathbb Z$. I think ...
-1
votes
1answer
54 views

Canonical map is injective

Let $A$ and $B$ be commutative rings, and let $f:A\to B$ be a faithfully flat ring homomorphism. How can I show that for any $A$-module $M$, the canonical map $M\to M\otimes_AB$ is injective? I was ...
2
votes
1answer
62 views

Local cohomology killed by a power of I

Notations:: $H^i_I(M)$ is $i^{th}$ local cohomology of $M$ with support in $I$ and $H^i_I(M)=R^i\Gamma_I(M)$ where $R^i\Gamma_I(M)$ is the right derived functor of a covariant left exact functor, ...
3
votes
1answer
157 views

Is the converse of Proposition 3.5.4 (c) of Bruns_Herzog true?

Question 1. Is the converse of Proposition $3.5.4 (c)$ of Bruns_Herzog true? I can see that $R$ is cohen-macaulay. so if one can prove that $r(R)=1$ , $R$ will be Gorenstein. ...
0
votes
1answer
30 views

Cartesian product of projective system

Let $(M_i,\mu_i^j,I)$ and $(N_i,\nu_i^j,I)$ be two projective system of $R$-module ($R$ a commutative ring) How to prove that : $$\varprojlim_{i\in I}(M_i\times N_i)\cong \varprojlim_{i\in ...