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

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8
votes
1answer
120 views

Infinite linear independent family in a finitely generated $A$-module

So I'm stuck with this problem. Let $A$ be a commutative ring (with unit). I have several questions that are really close to each other. 1) Let $M$ be a finitely generated module over $A$. Can we ...
8
votes
3answers
173 views

Is Orzech's generalization of the surjective-endomorphism-is-injective theorem correct?

In math.stackexchange answer #239445, Makoto Kato quoted a statement from the paper Morris Orzech, Onto Endomorphisms are Isomorphisms, Amer. Math. Monthly 78 (1971), 357--362. The statement ...
1
vote
1answer
28 views

Is every field a Krull domain?

Background: A Krull domain is an integral domain $A$ with a family $(v_i)$ of valuations on the field of fractions $K$ for $A$ satisfying the following conditions: Each $v_i$ is discrete. The ...
0
votes
1answer
93 views

Trying to use the Zariski topology in a problem without knowing scheme theory.

I don't know scheme theory, and I am doing a problem and the solution involves making conclusions based on the Zariski topology, and I want to make sure that I am "intuiting" things correctly when ...
0
votes
1answer
25 views

$SL(2)$ invariant polynomials are generated by determinant?

Suppose $SL_2(\mathbf{C})$ acts on the space of quadratics $aX^2+2bXY+cY^2$ by $X\to \alpha X+\beta Y, Y\to\gamma X+\delta Y$, where $\alpha,\beta,\gamma,\delta$ consists a matrix in ...
2
votes
2answers
32 views

Flat extension of noetherian rings and formal power series

Let $A \to B$ be a flat homomorphism of Noetherian rings. Is it true that it induces a flat homomorphism of formal power series $A[[x]] \to B[[x]]$?
2
votes
2answers
90 views

When does the regularity of $A$ implies the regularity of $A[w]$?

Let $A$ be a commutative noetherian ring (I do not mind to assume that $A$ is a UFD), and assume that $A$ is regular. Recall that a commutative noetherian ring is called regular if all its ...
8
votes
0answers
126 views

Subring of $\mathcal O(\mathbb C)$

Let $\mathfrak A \subset \mathcal O(\mathbb C)$ be the subring generated by the nowhere zero analytic functions $f: \mathbb C \to \mathbb C$. Does we have a precise description of $\mathfrak A$ ? Is ...
1
vote
1answer
23 views

Two dimensional valuation domain

Let $R$ be a two-dimensional valuation domain with prime ideals $0 \subset P \subset M$ and value group $G=\Bbb Z \oplus \Bbb Q$. Then $M^2=M$ and $P^2 \neq P$. Why $M^2=M$ and $P^2\neq P$? Can we ...
3
votes
2answers
50 views

How to choose a left-add$(X)$-approximation with a certain property

Let $A$ be an artin algebra and $X,Y$ in mod-$A$. Suppose $0\rightarrow Y \stackrel{\alpha}{\rightarrow} X^n\stackrel{\beta}{\rightarrow} X^m$ is exact. Set $C:=Coker(\alpha)$ (as module) and ...
3
votes
1answer
85 views

If ring $B$ is integral over $A$, then an element of $A$ which is a unit in $B$ is also a unit in $A$.

Let $A$ be a subring of ring $B$, with $B$ integral over $A$. If $x\in A$ is a unit in $B$, then it is a unit in $A$. I know that $f(t) = t - x$ is in $A[t]$ with $f(x) = 0$, and that there ...
0
votes
1answer
43 views

Element invertible in integral extension of ring implies invertible in ring [duplicate]

Please excuse some minor hiccups in terminology, I am primarily reading this in Swedish so feel free to correct any. Let $A\subseteq B$ be an integral extension and $\alpha\in A$ an invertible ...
1
vote
1answer
65 views

$R/I$ satisfies $S_2$ conditions

Let $R=k[x_1,...,x_{n},y_1,...,y_n]$ be a ring over $k$ and $I=\langle \{x_iy_j|$ for some $i,j \in\{1,...,n\}\}\rangle$ be ideal of $R$ and there are $r,s\in\{1,...,n\}$ such that $x_ry_s\notin I$. ...
3
votes
2answers
101 views

One-dimensional Noetherian UFD is a PID

I am looking for a reference which has a self-contained (elementary, that is, at the "undergraduate algebra level") proof of the the fact that any one-dimensional Noetherian UFD is a PID. Does anyone ...
1
vote
1answer
22 views

Find the integral closure of an integral domain in its field of fractions [duplicate]

Let $k$ be a field and let $R = k[x,y]/(x^2-y^2+y^3)$. Note that $R$ is an integral domain. Let $F$ be the field of fractions of $R$. How to determine the integral closure of $R$ in $F$? I have ...
6
votes
1answer
276 views

Normalization of a quotient ring of polynomial rings (Reid, Exercise 4.6)

I solved all parts of Exercise 4.6 of the book Undergraduate Commutative Algebra of Miles Reid except the last one. Let $A=k[X]$ and $f\in A$ has a square factor but it is not a square polynomial ...
5
votes
1answer
86 views

What is the kernel of $K[x^2,x^3][T] \to K[x]$, defined by: $T \mapsto x$?

Consider $K[x^2,x^3] \subset K[x]$, where $x$ is an indeterminate over a (zero characteristic) field $K$. Clearly, $x$ vanishes the following polynomials $\in K[x^2,x^3][T]$: $f(T)=x^2T-x^3$, ...
1
vote
1answer
74 views

$\operatorname{Hom}_R(\mathfrak{a},M)$ is isomorphic to $\mathfrak{a}^{-1}M$ if $R$ is a Dedekind domain

I want to prove Lemma 2.5.1 of Silverman's Advanced Topics in The Arithmetic of Elliptic Curves (whose proof is left to the reader): Let $R$ be a Dedekind domain, let $\mathfrak{a}$ be a ...
1
vote
1answer
14 views

Radical of the annihilator of an element of a Noetherian module

Assume $M$ is a commutative Noetherian $R$-module and $m\in M$ is such that $P=\sqrt{\operatorname{Ann}(m)}$ is a prime ideal in $R$. Is it true that $P$ is an associated prime of $M$, i.e. there is ...
3
votes
1answer
63 views

Reconciling two different definitions of constructible sets

This question is really about sets and topology, but it is motivated from commutative algebra, hence the tag. Setup: Let $X$ be a set and let $\{U_\lambda\}_{\lambda\in\Lambda}\subset 2^X$ be a ...
7
votes
2answers
244 views

Category of binomial rings

A binomial ring is a commutative ring $R$ such that (1) the additive group of $R$ is torsionfree and (2) $n!$ divides $x(x-1)\dotsc(x-n+1)$ for all $n \in \mathbb{N}$ and $x \in R$. We may then define ...
0
votes
1answer
28 views

Associated prime of $M/Q$ where $Q$ is $\mathfrak{p}$-primary

I need check if my statement is true and proof check (for some reason I couldn't find this anywhere): Let $Q$ be a $\mathfrak{p}$-primary submodule of $A$-module $M$. Then $\mathfrak{p}$ is the ...
4
votes
0answers
79 views

When flatness of $B$ over $A$ implies flatness of $B$ over $C$, where $A \subseteq C \subseteq B$?

Assume $A \subseteq C \subseteq B$ are integral domains, with $B$ flat over $A$. Generally, $B$ is not necessarily flat over $C$. For example, see van den Essen's book "Polynomial Automorphisms and ...
4
votes
0answers
63 views

Example of $A$-module but not $A$-algebra. [duplicate]

If $A$, $B$ are commutative rings, and if $B$ is an $A$-algebra then it is also an $A$-module. I am looking for an example that shows that the converse is not true. That is, I am looking for ...
4
votes
1answer
136 views
+50

What is $\operatorname{Hom}_R(P,R)$ isomorphic to when $P$ is projective?

Let $R$ be a (possibly noncommutative) ring with $1$. Now, quite clearly we have $$\operatorname{Hom}_R(R^n,R)\cong R^n.$$ I am wondering if there is any similar result for ...
8
votes
1answer
71 views

$A$ regular, $k'/k$ transcendental. How to prove that $A \otimes_k k'$ is regular?

Let $k$ be a field and $k'$ a purely transcendental extension of $k$. Let now $A$ be an integral finitely generated $k$-algebra. How to prove that if $A$ is regular then $A \otimes_k k'$ is also ...
2
votes
2answers
46 views

Conductor of a ring

An easy (possibly trivial) question from Neukirch's Algebraic Number Theory, p.47. Let $A$ be a Dedekind domain, $K$ its fraction field, $L$ a finite separable extension of $K$ and $B$ the integral ...
2
votes
1answer
43 views

Flatness of closure of subring

Assume we are given Noetherian local rings $(A,\mathfrak{m})$ and $(B,\mathfrak{n})$ such that: $A \subset B$ and $\mathfrak{m} = A \cap \mathfrak{n}$, $B$ is a finite $A$-module. It is known ...
2
votes
0answers
36 views

Generalization of the Going up Theorem to arbitrary chains of prime ideals

Let $S$ and $R$ be commutative rings with $1$. This is the usual form of the Going up theorem that one encounters in commutative algebra texts: Let $S$ be integral over $R$, and suppose that we have ...
0
votes
1answer
52 views

Theorem 31.7 of Matsumura, Commutative Ring Theory

Theorem: If A is a Noetherian local ring and A[x] catenary, then A is formally catenary. In the proof, it is assumed that A is integral domain and A* (the completion of A) is not equidimensional and ...
1
vote
0answers
49 views

Projectivity of a (prime) ideal in a noetherian integral domain

Assume $R$ is a noetherian integral domain (and assume $R \neq k[x_1,\ldots,x_n]$), $I$ is a non-zero ideal of $R$ ($I$ is finitely generated, since $R$ is noetherian), and $I$ is not necessarily ...
0
votes
1answer
116 views

Projective dimension of all principal ideals is finite. Is R an integral domain?

$R$ is a noetherian ring in which projective dimension of all principal ideals is finite. Is $R$ an integral domain? What condition can be added on it to be a regular ring? thanks for any help. ...
3
votes
1answer
96 views

If $R$ is a noetherian local ring, then every 2-generated ideal has finite projective dimension iff $R$ is a UFD

This question is about zcn's comment on the answer to this question. It's a good point. So I ask it for use of everybody: if $R$ is a noetherian local ring, then every 2-generated ideal has ...
2
votes
0answers
49 views

When a two-generated ideal of a noetherian integral domain have a finite projective resolution?

Let $R$ be a noetherian integral domain, and $I$ a non-zero ideal of $R$ which can be generated by two elements. (We do not know if $I$, considered as an $R$-module, is $R$-projective; maybe yes maybe ...
2
votes
2answers
53 views

Can one decompose any ring into a (possibly infinite) product of indecomposable rings?

Let $R$ be a ring. Do there exist indecomposable rings $R_i$, $i\in I$, such that $R={\prod}_{i\in I} R_i$?
5
votes
0answers
73 views

Weak Global Dimension and Global Dimension

Let $R$ be a commutative ring (not necessarily Noetherian) with unit. Is there an example such that weak global dimension of $R$ is finite but the global dimension of $R$ is infinite? Can we find such ...
2
votes
1answer
123 views

Global dimension.

What is the global dimension of $\mathbb{Z}_{(p)}$ and $\mathbb{Z}_{(p)}/t\mathbb{Z}_{(p)}$, where $\mathbb{Z}_{(p)}$ is the local ring, $p$ prime and $p \mid t$? What is the global dimension of ...
2
votes
1answer
39 views

Global dimension of an intermediate ring

Assume $A \subseteq B \subseteq C$ are noetherian integral domains, where $A$ and $C$ have the same finite global dimension $n$. Also assume that $C$ is a finitely generated $B$-algebra and $B$ is a ...
0
votes
0answers
38 views

Prime ideal in indecomposable commutative ring [closed]

Let $R$ be a commutative indecomposable ring with Jacobson radical $J$. When can we find a prime ideal contained in $J$?
3
votes
2answers
76 views

Semilocal commutative ring with two or three maximal ideals

Is there any equivalence condition for a commutative ring to have exactly two or three maximal ideals?
12
votes
2answers
483 views

Vanishing of a certain Tor

I am reading about the construction of the Affine Grassmannian in Dennis Gaitsgory's seminar notes and there are some commutative algebra facts that I am not able to figure out by myself apparently, ...
2
votes
1answer
122 views

Radical of an ideal using Macaulay2 software.

What is the radical ideal of $(u^2v-a^3,uv^2-b^3,uv-ab)$ in $\mathbb{C}[u,v,a,b]?$ Above all, to learn how to fish, what would be code that I can use to get the radical? I have not worked with ...
3
votes
2answers
228 views

Radical ideal computation (Macaulay2)

Is there a way to find the radical ideal of $I_i=(a^n-u^{n-i+1}v^{n-i}, b^n-u^{i-1}v^i, uv-ab)$ for $1\leq i \leq n$ in $\mathbb{C}[u,v,a,b]?$ This is the generalization of my question here where ...
0
votes
0answers
49 views

Transitivity of discriminant for flat algebras

Let $A$ be an finite flat $R$-algebra and $A'$ be an finite flat $A$-algebra such that it is also finite flat as an $R$-algebra. Then we have a notion of discriminant ideals ...
1
vote
1answer
56 views

Prove that $f$ is a nonzerodivisor on $R[x_1,\dots,x_r]/IR[x_1,\dots,x_r]$ for every ideal $I$ in $R$

Let $R$ be a Noetherian commutative ring with unity, and $S=R[x_1,\dots,x_r]$. Let $f\in S$ be a nonzerodivisor of $S$. Suppose that the ideal generated by the coefficients of $f$ is $R$. How to ...
2
votes
1answer
133 views

Regular subrings of a polynomial ring

Let $R=\mathbb{C}[x,y]$. I have the following situation: $\mathbb{C} \subseteq D \subseteq R$ is affine (= finitely generated as a $\mathbb{C}$-algebra), noetherian, has field of fractions ...
2
votes
1answer
50 views

Galois group and traslations by rational numbers.

Is a well known result that, for every $n \in \mathbb{N}$, there exist an irreducible polynomial $p \in \mathbb{Q}[x]$ such that the Galois Group of its splitting field is $S_n$. Now my question: ...
10
votes
0answers
112 views

Example of a commutative, local, dual ring with nilradical $N$ such that $ann(N)\nsubseteq N$

For an ideal $I\lhd R$ in a commutative ring $R$, let $ann(I)$ denote the annihilator of $\{x\in R\mid xI=\{0\}\}$. A commutative ring $R$ is said to be a dual ring if for every ideal $I$ of $R$, ...
2
votes
1answer
97 views

$R$ is normal. Are $R[x]$ and $R[[x]]$ normal?

Studying about normalizations I've bumped in the following theorem: Theorem. Let $R$ be a normal (integrally closed) domain, then $R[x]$ is a normal domain. How to prove (elegantly, if possible) ...
6
votes
1answer
120 views

For a group-algebra $k[G]$ ($G$ finite), why is a $k[G]$-module the same as a $k$-representation of $G$?

I'm reading the Atiyah-MacDonald book on Commutative Algebra. At the beginning of the module chapter on page 17, they make an example which I don't understand. Example 5) is: $G$ = finite group, ...