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

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2
votes
1answer
169 views

direct limit of finitely generated submodule

if $A$ is a module,then the family fin($A$) of all the finitely generated submodules of $A$ is a directed set and direct limit of$M_i$ is isomorphic to$A$. for prove this needed to define to injection ...
3
votes
0answers
94 views

Complete ring and unique continuous homomorphism

Let $n\geq 2$ be an integer, $D=\mathbb Z[1/n]$, and $A$ a complete commutative ring with unit for the $I$-adic topology, where $I$ is an ideal of $A$. Suppose that $n$ is invertible in $A$. Let $x\in ...
2
votes
1answer
38 views

Direct limit of a subdirected system

Let $(M_{i},\phi_{ij})$ be a direct system over a directed set $I$ and let $M$ be its direct limit. Suppose we remove a finite/infinite number of terms from the direct system $(M_{i},\phi_{ij})$. Does ...
0
votes
1answer
46 views

$n$th root of a polynomial

Let $n\geq 2$ be an integer, and $D=\mathbb Z[1/n]$. Consider the polynomial $S=(1+T)^n-1\in D[T]$. How can I show that $D[[S]]=D[[T]]$ and there exists an $f(S)\in SD[[S]]$ such that $1+S=(1+f(S))^n$?...
0
votes
2answers
296 views

Finding Radical of an Ideal [duplicate]

Given the ideal $J^\prime=\langle xy,xz-yz\rangle$, find it's radical. I know that the ideal $\langle xy,yz,zx\rangle$ is radical ideal but that's not the case. How can I compute the radical here?
0
votes
0answers
101 views

How to show that $\text{Spec}(\mathbb{Z}[x_1,x_2,\ldots, x_n]) \cong (\mathbb{G}_m)^n$?

How to show that $\text{Spec}(\mathbb{Z}[x_1,x_2,\ldots, x_n]) \cong (\mathbb{G}_m)^n$? Here $\mathbb{G}_m$ is the multiplicative group. Thank you very much. Edit: $\mathbb{G}_m = k - \{0\}$ is ...
1
vote
1answer
142 views

Showing an ideal of $k[x,y]/\langle xy \rangle$ is prime

I am currently trying to show that the ideal $I = \langle x, y-1 \rangle$ is a prime ideal in $R = k[x,y]/\langle xy \rangle$ (for some field $k$). My first thought was to rewrite the ideal as ...
2
votes
1answer
68 views

Commutative ring product of elements in terms of sum of powers of sums

The following fact is stated (but not proved) in Chapter 0 of Noll's Finite Dimensional Spaces (it's 07.21): If $R$ is a commutative ring with unity and $r_1, ..., r_n$ elements of $R$ then $$ n!...
1
vote
1answer
105 views

If $\varphi(I)$ is an Ideal $\forall I $ ideal of $A$, is $\varphi$ surjective?

Today I heared some young students talking about the fact that if an homomorphism of rings (commutative with identity) $\varphi:A \rightarrow B$ is surjective then the image of any ideal of $A$ is an ...
1
vote
2answers
48 views

Existence of a canonical isomorphism of completions

How one can do the problem 1.3.8 from Qing Liu's Algebraic Geometry and Arithmetical Curves. Namely, Let $A$ be a Noetherian ring, and $I,J$ ideals of $A$. Let $\widehat{A}$ be the $I$-adic ...
3
votes
0answers
100 views

Localization of tensor products

Assume $A$ and $B$ are finite type $k$-algebras over a field. Consider the tensor product $C = A\otimes_k B$. There are maps $A\to C$ and $B\to C$. Given a prime ideal $p \subseteq C$, I can ...
2
votes
2answers
146 views

Linear Algebra Basis Trick

Let $V$ be a finite dimensional vector space over a field $F$. More generally, we can consider a free module $M$ over a commutative ring $A$, with rank $n$. Let $(m_1,...,m_n)$ be a basis for $M/A$ ...
2
votes
1answer
88 views

$\operatorname{Spec}\mathbb{K}[x,y,z]/\langle x^2-yz \rangle$ is normal and singular

Let $X=\operatorname{Spec}\mathbb{K}[x,y,z]/\langle x^2-yz\rangle$ an affine scheme. It is singular because only at the rational point $0$ corresponding to the ideal $\langle x,y,z\rangle$, the ...
2
votes
1answer
43 views

Example of filtration which is not stable

The following proposition was given in Liu's Algebraic Geometry and Arithmetic Curves: Let $A$ be a Noetherian ring, $I$ an ideal of $A$, and $M$ a finitely generated $A$-module endowed with stable $...
0
votes
1answer
48 views

Valuation rings are conjugate

Let $F/K$ be a finite Galois extension where $(K,v)$ is a valued field (i.e. $v$ is a valuation on $K$). Let $w_1,w_2$ be extensions of $v$ to $F$. Then, we have associated valuation rings $O_{w_1}$ ...
2
votes
1answer
31 views

$I=(I:s)\cap (I, s)$

Somewhere I've read the following: Theorem Let $I \subset A$ an ideal of a domain $A$. Let $S$ a multiplicatively closed set and let be $I^e$ the image of $I$ in $S^{-1}A$. Let $s \in S$ be such ...
5
votes
1answer
95 views

What is the injective hull of a graded polynomial ring (Graded category of modules)?

Let $R=K[x]$ be usual graded ring (where $K$ is a field). What is the injective hull of $R$?
3
votes
1answer
140 views

Spectrum of a ring of continuous functions

The set $A=\{f:[0,1] \rightarrow \mathbb{R} : f$ is continuous$\}$ is a ring with the standard function addition and multiplication. Which are the prime ideals in $A$? The only thing I've managed ...
1
vote
1answer
230 views

Bruns Herzog Theorem 9.2.1

In Theorem 9.2.1 of Bruns-Herzog Cohen-Macaulay rings it is proven that if $x_1,...,x_n$ is a regular sequence on an $R$-module $M$, and $t \ge 0$, then $x_1^t x_2^t \dotsm x_n^t \notin (x_1^{t+1},......
0
votes
1answer
49 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, $m^t=0$...
0
votes
1answer
31 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
vote
2answers
194 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 R[1/f]...
0
votes
1answer
38 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
vote
1answer
66 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
vote
0answers
20 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 ...
0
votes
0answers
37 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 ...
7
votes
2answers
128 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 ...
2
votes
1answer
69 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
votes
1answer
78 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 $x-a \in \mathfrak{m}$.
3
votes
1answer
127 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
vote
1answer
43 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
votes
1answer
74 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
vote
0answers
55 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 domain?...
0
votes
1answer
39 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 $F$?
0
votes
1answer
39 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 ...
1
vote
3answers
55 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
vote
2answers
82 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 ...
0
votes
0answers
49 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 ...
3
votes
1answer
74 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 M\...
1
vote
2answers
182 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
votes
1answer
86 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
votes
1answer
166 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 ...
2
votes
2answers
114 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 ...
2
votes
1answer
62 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
vote
1answer
33 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 s)...
4
votes
1answer
88 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
votes
1answer
93 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
votes
1answer
54 views

Algebraic Ideal and affine varieties equality

Let $I=\langle x^2-y-4\rangle$ and $G= \langle x^2+y-4\rangle$ be two ideals in the polynomial ring $\mathbb{C}[x,y]$. As obviously $V(I)=V(G)$, what can we conclude for $I$ and $G$? How they are ...
1
vote
0answers
34 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.
2
votes
1answer
85 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: $f'...