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

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2
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1answer
91 views

If M is free with a finite basis then every basis of M over R is finite and has the same number of elements.

Stuck on a proof in my lecture notes. Proposition: Let $R$ be a commutative ring and let $M$ be an $R$-module. If $M$ is free with a finite basis then every basis of $M$ over $R$ is finite and has ...
4
votes
2answers
91 views

$k[X]$ is integral over $k[X^{2}]$

I am trying to show that $k[X]$ is integral over $k[X^2]$, where $k$ is a field. Taking an element $b=b_nx^n+b_{n-1}x^{n-1}+...b_1x+b_0 \in K[X]$ we want to find $a_i \in K[X^2]$ such that ...
0
votes
1answer
34 views

$(x_1,\ldots x_n)=(1)\implies (x_1^{k_1},\ldots, x^{k_n})=(1)$ [duplicate]

Let $R$ be a commutative ring with unit, I'm trying to prove why in this ring $$(x_1,\ldots x_n)=(1)\implies (x_1^{k_1},\ldots, x^{k_n})=(1)$$ It seems an easy question, but I couldn't prove it, I ...
0
votes
0answers
119 views

Jacobson Radical

Let $R$ be a local ring with unity then when can we say that the radical of Jacobson of $R, J$, is a $R/J$ module. By local I meant it has a unique ideal maximal. And when is $R$ is isomorphic to ...
1
vote
1answer
42 views

$f:M\rightarrow N$ 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 ...
-1
votes
1answer
81 views

Depth of infinite direct sum

Let $R$ is a local ring, from the depth lemma, we can get $\operatorname{depth}(R\oplus\dotsb\oplus R)=\operatorname{depth}(R)$, here the direct sum is finite, how about the infinite case? By the ...
3
votes
1answer
62 views

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 ...
3
votes
0answers
167 views

Infinitesimal thickening of a smooth closed subscheme

Let $A$ be a noetherian ring (if it is useful I can assume that $A$ is an algebra of essentially finite type over a field) and $I \subset A$ is an ideal s.t. $A/I$ is smooth. Is it true that extension ...
3
votes
1answer
243 views

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 ...
3
votes
1answer
58 views

Computing the closed subschemes of the projective line over a field

(Specifically, this is III-15 in E&H, but I feel like I've hit a brick wall in actually applying the definitions they've given to this example.) In Chapter I of The Geometry of Schemes, E&H ...
2
votes
1answer
36 views

Question about proof that every f.g. projective module over a local ring is free.

I'm reading the proof here. I'm at the line where they say $$ \psi\pi(f)=\psi(f+FR)=\varphi(f)+PR.$$ Since $\psi\pi$ is surjective, it should follow that $\{\varphi(f)+PR:f\in F\}=P/PR$. I don't ...
9
votes
2answers
184 views

Is $\Bbb Q/\Bbb Z$ artinian as a $\Bbb Z$-module?

I'm confused. Is $\Bbb Q/\Bbb Z$ artinian as a $\Bbb Z$-module? We know that $\Bbb Z_{p^{\infty}} \subset \Bbb Q/\Bbb Z$ is artinian. The following argument is true or not ? $\mathbb Q / ...
1
vote
2answers
127 views

Existence of module of finite injective dimension

At p. 107 of the book Cohen-Macaulay Rings by Bruns and Herzog, the authors write "any module of finite projective dimension (over a Gorenstein ring $R$) has finite injective dimension as well, ...
3
votes
1answer
143 views

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 ...
3
votes
1answer
76 views

Galois cover an affine scheme

Let $X = \operatorname{Spec}(A)$ be an affine scheme, with $A$ noetherian (and normal if this is useful). We suppose that $X$ is a finite étale covering of $Y = \operatorname{Spec}(B)$, Galois with ...
7
votes
2answers
405 views

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 ...
2
votes
1answer
594 views

Is any UFD also a PID?

Is there any counterexample that will disprove that every unique factorization domain (UFD) is also a principal ideal domain (PID)? I mean, any PID is a UFD, does the converse hold? Thanks in ...
1
vote
1answer
222 views

How many ways are there to represent a monomial order, defined by $>$, by term order via matrices?

During the lecture, my professor brought up the list of project ideas to work on. One of the ideas I am interested and currently working on is term order via matrices. That is: I need to find the ...
1
vote
1answer
56 views

Map induced by localization on categories

I have been doing some reading in Hartshorne's Algebraic Geometry on derived functors and subsequent results in cohomology. Given $A$ an abelian category of groups, I have seen that the map ...
2
votes
0answers
46 views

Under what conditions are the resolutions of two modules subcomplexes of the resolution of the tensor product?

I have that $S=k[x_1, \dots, x_n]$, $I$ is a lattice ideal, and $J$ is a monomial ideal. I am interested in the resolution of $S/(I+J)\cong S/I\otimes S/J$. In particular, I am interested in knowing ...
3
votes
1answer
94 views

Wikipedia definition of an order (ring theory)

Wikipedia defines an order $\mathcal O$ of a finite type $\Bbb Q$-algebra $A$ to be a subring of $A$ satisfying the following properties. Here, by finite type $\Bbb Q$-algebra, I mean that $A=\Bbb ...
5
votes
0answers
126 views

Direct image of an ideal sheaf along a blow-up

Suppose that $I\subseteq\mathbb{C}[x_0,\ldots,x_n]$ is a saturated homogeneous ideal. Let $\mathcal{I}\subseteq\mathcal{O}_{\mathbb{P}^n}$ denote the corresponding coherent ideal sheaf, and then let ...
5
votes
1answer
151 views

Why do people look into modules over Dedekind domains?

It is said in this blog that: The reason this turns out to be useful is that many examples in algebraic/arithmetic geometry require you to look no further than understanding modules over Dedekind ...
2
votes
0answers
63 views

Tor dimension in polynomial rings over Artin rings

I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go ...
6
votes
1answer
155 views

Local parameter of curves in affine n-space

I'm looking for a double answer to this question: a mathematical one (say, if the statement is correct or not) and a philosophical one (say, why we do expect this to be true, or not). Let $k$ be a ...
4
votes
0answers
87 views

Regular monomorphisms of commutative rings

What are the regular monomorphisms of $\mathsf{CRing}$? Is there a purely algebraic characterization? Since regular monomorphisms coincide here with effective monomorphisms (see Prop. 1. here), the ...
3
votes
1answer
85 views

tensor, symmetric, exterior power of a module over a PID

Let $R$ be a PID and $M\cong R^r\!\oplus\bigoplus_{i=1}^s\!R/Ra_i$. Denote the tensor, symmetric, exterior power of $M$ by $T^nM=\bigotimes_{k=1}^nM$ and $S^nM= T^nM/\langle ...
3
votes
2answers
583 views

A quotient $\mathcal{O}/\mathfrak{a}$ of a Dedekind domain is principal (Neukirch exer 1.3.5)

The exercise states: The quotient ring $\mathcal{O}/\mathfrak{a}$ of a Dedekind domain by an ideal $\mathfrak{a}\ne 0$ is a principal ideal domain. The proof by localization ...
3
votes
2answers
121 views

Orthogonal idempotents from disjoint union in $\text{Spec}(A)$

Let $A$ be a commutative ring with unity. Suppose that $X=\text{Spec}(A)$ is a disjoint union $X_1\cup X_2$ of topological spaces. Show that $A$ has a pair of orthogonal idempotents $e_1,e_2$ ...
5
votes
1answer
140 views

Proof that $K\otimes_F L$ is not noetherian

Let $F$ be a field and $K$ and $L$ be extension fields of $F$ such that $\mathrm{tr.deg}_F(K) = \infty$ and $\mathrm{tr.deg}_F(L) = \infty$. It seems to be proved that $K\otimes_F L$ is not ...
0
votes
1answer
80 views

Associated prime preserved under the quotient

Let $(R,m,k)$ be a complete local Noetherian ring and let $E$ be an $R$-module such that $\operatorname{Ass}E=\left\{m\right\}$. Let $N$ be a proper submodule of $E$. Question: Is it true that ...
5
votes
0answers
149 views

Existence of finite projective resolution

The situation I'm considering is quite involved. All rings are noetherian commutative with $1$. All modules are finitely generated. First of all we fix a non reduced local ring $A$ where all zero ...
2
votes
1answer
180 views

Annihilators and exact sequences

Let $R$ be a commutative ring. Let $M_1$, $M_2$ and $M_3$ be $R$-modules. Let the following sequence be exact: $$0\longrightarrow M_1 ...
1
vote
1answer
95 views

how to prove a=r(a)<=>a is a intersection of prime ideals.if a is an ideal≠(1) in commutative ring A.

how to prove a=r(a)<=>a is a intersection of prime ideals.if a is an ideal≠(1) in commutative ring A.r mean radical. I can't prove it,here is what I did. a is a intersection of prime ideals mean ...
2
votes
2answers
72 views

Can a chain of irreducible subvarieties always be extended to one of maximal length?

I'm interested in computing the dimension of a variety $X$. I can get a lower bound by exhibiting some strictly increasing chain of irreducible subvarieties $$\varnothing =Z_{-1}\subset Z_0\subset ...
8
votes
3answers
189 views

Why is the topology on $\operatorname{Proj} B$ induced from that on $\operatorname{Spec}(B)?$

In the proof of Lemma $3.36$ in Algebraic Geometry and Arithmetic Curves, it is stated that, if $B=\oplus_{d\ge0}B_d$ is a graded algebra over a ring $A,$ and if $I$ is an ideal of $B,$ then ...
0
votes
1answer
579 views

Difference between Matsumura's Commutative Algebra and Commutative Ring Theory

I am a beginner in more advanced algebra and my question is very simple, I would like to know the difference between these books of the same author, Hideyuki Matsumura Commutative Ring Theory ...
5
votes
2answers
132 views

$\mathbb{Q}_p\otimes_{\mathbb{Q}} \mathbb{Q}_q$ and $\mathbb{Z}_p\otimes_{\mathbb{Z}} \mathbb{Z}_q$

Let $p, q$ be prime numbers which may or may not be distinct. Let $\mathbb{Q}_p$ be the field of $p$-adic numbers. Let $\mathbb{Z}_p$ be the ring of $p$-adic integers. We define similarly ...
3
votes
1answer
119 views

Characterizing Galois field extensions via tensor product

Let $K\subset L$ be a finite field extension of degree $n$. Prove that it is Galois if and only if $L\otimes_{K} L\simeq L^{n}$, as $L$-algebras, considering on $L \otimes_{K}L=L^{n}$ the left ...
5
votes
2answers
265 views

Quotient of a local ring at a point is a finite dimensional vector space

$f,g\in \mathbb{C}[x,y]$ are irreducible polynomials, and the varieties $V_1=V(f)$ and $V_2=V(g)$ are not equal. Is the ring $\mathcal{O}_p/(f,g)$ a finite dimensional vector space over ...
0
votes
2answers
87 views

Nonprincipal prime ideals contain two relatively prime elements

Let $R$ be a principal ideal domain and let $P$ be a nonprincipal prime ideal of $R[x]$. I'm having trouble seeing why $P$ must contain two elements with no common divisor. Can anyone help me? ...
1
vote
1answer
97 views

local Noetherian of zero depth implies Artinian?

Let $(R,m,k)$ be a local Noetherian ring such that $\operatorname{depth}R=0$. Question: Is it true that $R$ is Artinian? PS: If it is true then please only say so, as i am still attempting to ...
2
votes
1answer
127 views

Localization at a maximal ideal and quotients.

If we have a commutative ring $R$ and a maximal ideal $m$, then is $m/m^2$ isomorphic to $m_m/m^2_m$? Thx.
5
votes
1answer
247 views

Exercise 4.5.E a) in Ravi Vakil's Foundations of Algebraic Geometry.

Hi! I am following the hint given in Exercise 4.5.E in Vakil's Foundations of Algebraic Geometry, but I am stuck trying to prove that if $a_1,a_2 \in Q_i$, then $a_1^2 + 2a_1 a_2 + a_2^2 \in Q_{2i}$. ...
1
vote
0answers
55 views

Name of a certain type of rings

What is the name given to (if there exists any) commutative rings $R$ with identity such that $R/(a)$ is finite for every non-zero $a\in R$ Thanks a lot
0
votes
1answer
112 views

combinatorial commutative algebra

Is there anyone who can help me with this problem? Any hint to the solution would be appreciated! Let $\Delta$ be a $(d-1)$-dimensional simplicial complex. Show that the h- and f-vectors of $\Delta$ ...
1
vote
1answer
60 views

Are there homogeneous elements with two distinct grades?

In a graded ring $B=\bigoplus_{d\ge 0} B_d$, the element $0$ is homogeneous with grade $d$ for every $d\ge 0$, in fact since every $B_d$ is an additive subgroup of $B$, then it must contain $0$. Can ...
2
votes
1answer
86 views

Question about some details of a proof of Chinese Remainder Theorem

In the proof of 3rd proposition I can prove the intersection of all ideals is the kernel of the map, but why does it imply this proposition is true?
1
vote
1answer
93 views

Is the completion of $(k[x,y]/f)_\mathfrak{m}$ isomorphic to $k[[x]][y]/f$?

Let $k$ be an algebraically closed field and let $f\in k[x,y]$ be an irreducible polynomial with no constant term that is not a polynomial in $x$ alone. Is it the case that the completion of the ...
1
vote
1answer
39 views

A question about a detail of proof

proposition: x∈The Jacobson radical <=> 1-xy is a unit in commutative ring A for all y∈A I have proved (=>) I don't figure out a detail of the proof of (<=). Here is the proof on book: ...