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

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2
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1answer
89 views

Ascending chain “stabilizing temporarily” infinitely many times

A commutative ring $R$ is called Noetherian if every ascending chain of ideals in $R$ stabilizes, that is, $$ I_1\subseteq I_2\subseteq I_3\subseteq\cdots $$ implies the existence of $n\in\mathbb{N}$ ...
4
votes
2answers
284 views

definition of convergence of a spectral sequence

In Lecture Notes in Algebraic Topology by Davis & Kirk, on page 241, there is written: What do $E^\infty$ and $\lim_{r\to\infty}E^r_{p,q}$ mean? If a spectral sequence is not first-quadrant and ...
12
votes
2answers
425 views

If every ascending chain of primary ideals in $R$ stabilizes, is $R$ a Noetherian ring?

A commutative ring $R$ is called Noetherian if every ascending chain of ideals in $R$ stabilizes, that is, $$ I_1\subseteq I_2\subseteq I_3\subseteq\cdots $$ implies the existence of $n\in\mathbb{N}$ ...
3
votes
1answer
114 views

Equivalence/Similarity of matrices over domains

Let $K$ be a field, $R$ some domain with $R\subsetneq K$. Take some square-matrix $A \in \operatorname{Mat}_n(R)$ with $n>1$, and suppose the matrix $B \in \operatorname{Mat}_n(R)$ is equivalent to ...
4
votes
1answer
96 views

Reducing the dimension of a finitely generated $k$-algebra by $1$

Let $k$ be a field and $Q$ an ideal of $k[x]=k[x_1,\dots,x_s]$ such that $R=k[x]/Q$ has Krull dimension equal to $d>0$. Define $V=x_1 k + \cdots +x_s k$ to be the vector space of linear forms over ...
3
votes
1answer
102 views

filtered modules (LNAT, Davis & Kirk)

In Lecture Notes in Algebraic Topology by Davis & Kirk, on page 240, there is written: Q1: Is convergence of the filtration assumed in the first underline? Otherwise $\forall p: F_p=A$ is a ...
5
votes
1answer
76 views

Is there a simple way to state continuity for $I$-adic topology?

Let $R$ be a commutative ring with the $I$-adic topology defined by an ideal $I$, and let $S$ be a commutative ring with the $J$-adic topology for an ideal $J$. How would you translate saying that a ...
1
vote
1answer
49 views

Integral dependence of an algebraic element

Let $A$ be a UFD, $K$ its field of fractions, and $L$ an extension of $K$. Then, let $\alpha \in L$ and let $f_\alpha \in K[x]$ be its minimal polynomial over $K$. Is it true that $\alpha$ is ...
7
votes
1answer
198 views

Zariski topology of $k^2$

I've found this "proof left to the reader" in some lecture notes: Let $k$ be an arbitrary field. In the Zariski-Topology of $k^2$ every closed set of $k^2$ is either finite or the zero set of a ...
3
votes
1answer
776 views

Can a non-square matrix be called “invertible”?

Let $R$ be a commutative ring with $1 \neq 0$. It is known that $R^{n} \hookrightarrow R^{m}$ implies $n \leq m$ and $R^{n} \twoheadrightarrow R^{m}$ implies $n \geq m$, and I might use this without ...
1
vote
1answer
109 views

The depth sensitivity of the Koszul complex

Assume $R$ is a local ring and $M$ a finitely generated $R$-module. Call $K^R$ the Koszul complex of $R$ (over a minimal set of generators of the maximal ideal) and call $K^M$ the complex $K^R\otimes ...
1
vote
1answer
108 views

The minimal free resolution of a module with a direct summand is not periodic

Assume that $R$ is a local ring and that $M$ is a finitely generated $R$-module with a free direct summand, then why its minimal free resolution cannot be periodic?
1
vote
0answers
112 views

Depth of ideals in a commutative ring

Let $I,J\subset S=k[x_1,\dots,x_n]$ be two monomial ideals and $k$ a field. If every regular element on $S/J$ is also a regular element on $S/I$ is it true that $\operatorname{depth}_S S/I\ge ...
0
votes
2answers
96 views

Isomorphism of power series

Let $R$ be a commutative ring and let $Q$ be a prime ideal of $R$. Show that $R ([[x]])/(Q[[x]])$ is isomorphic to $R/Q [[x]].$
3
votes
1answer
73 views

a question on the minimal prime divisors of an ideal

This question is motivated by the second part of Step 1 in the proof of Theorem 14.14 in Matsumura's Commutative Ring Theory, p. 112. Let $k$ be an infinite field and $Q$ a homogeneous ideal of ...
5
votes
1answer
472 views

Primary decomposition example

I want to find the primary decomposition of $(x^2, xy^2)$ as an ideal of $k[x,y,z]$ where $k$ is some field. My guess is $(x^2, xy^2) = (x) \cap (x^2, y^2)$ however I am not 100% certain if $(x^2, ...
1
vote
1answer
230 views

Is the polynomial ring over a discrete valuation ring local?

Let $R$ be a discrete valuation ring. Is the polynomial ring $R[X_1,\dots, X_n]$ a local ring?
3
votes
2answers
210 views

Local rank and direct sum decomposition

Let $A$ be a non zero commutative ring with unit. Let $n_1, n_2,\cdots , n_r$ be the distinct local ranks of the finitely generated projective $A$ module $M$. Could somebody help me to show that $A$ ...
2
votes
0answers
65 views

an argument regarding the dimension of a real algebra

Let $S$ be an algebraic subset of $\mathbb{R}^d$ with vanishing ideal $I_S=(f_1(x),\cdots,f_m(x))$. Suppose that $S$ has infinite cardinality (countable or uncountable, does not matter). I want to ...
6
votes
2answers
85 views

Finite extension of integrally closed ring again integrally closed

Let $S\subset R$ be a finite ring extension, i.e. $R$ is finitely generated as an $S$-module. Assume that $S$ is integrally closed. Does this imply that also $R$ is integrally closed (in its quotient ...
5
votes
0answers
203 views

generic regularity of affine varieties

Suppose that $V\subset {\mathbb C}^n$ is an affine subvariety of codimension $p$. How does one prove that $V$ is regular (i.e., is a smooth manifold) at its generic points? In view of the Jacobian ...
3
votes
1answer
87 views

Map between two direct limits

Let $\{ M_i, ϕ_j^i\}_{i\in I}$ be a direct system of $R$-modules over a direct index set $I$. Show that there exists a direct system $\{P_i,\psi_j^i\}_{i\in I}$ of projective $R$-modules and a ...
6
votes
1answer
186 views

Atiyah and Macdonald Exercise 1.27

Please do not ruin the fun by telling me why $\mu$ is surjective! I am having trouble understanding the idea of the coordinate functions on the affine algebraic variety $X$. I am trying to understand ...
0
votes
1answer
151 views

The clique number of zero-divisor graphs

If $R$ is finite commutative ring with exactly $8$ elements, show that the clique number of the zero-divisor graph is $2$. Edit. Let $R$ be a commutative ring and $Z(R)$ be the set of all ...
2
votes
1answer
70 views

In $\mathbb{C}[x]$ is it true that $F_{a,b}=\{p\in\mathbb{C}[x] : p(a)=p(b)\}$ for $a\neq b$ is a maximal subring?

The problem is in the title. It is clear that $F_{a,b}$ is a ring, but it is not so clear to me that it is maximal in $\mathbb{C}[x]$. I tried to consider it as a vector space and show that it has ...
-1
votes
1answer
135 views

Show that if an ideal is free as a module then it is principal.

Here $\mathfrak a$ is an ideal of a commutative ring $A$. Show that $\mathfrak a$ is principal if it is free as an $A$-module.
14
votes
5answers
1k views

Favourite applications of the Nakayama Lemma

Inspired by a recent question on the nilradical of an absolutely flat ring, what are some of your favourite applications of the Nakayama Lemma? It would be good if you outlined a proof for the result ...
2
votes
0answers
79 views

Infinitely many primary decompositions of an ideal

Given a noetherian ring $R$ and an ideal $I$, we know that if the associated primes of $I$ coincide with its minimal primes (i.e. $\text{Min}(I)=Ass(I)$ ) , then there is a unique irredundant primary ...
2
votes
0answers
134 views

(Finite) continued fractions over a general domain

I am looking for some literature (articles or books) where finite continued fractions over a general integral domains (that is, in a fraction field of that domain, but the "coefficients" are from the ...
2
votes
3answers
166 views

Confusion about Spec of quotient ring

Consider the ring $A := \dfrac{\mathbb C[x]}{(x(x-1)(x-2))}$. According to some sources (cf. Vakil) $sp(Spec (A))$ should be just the three points $\{0,1,2\}.$ It seems right, because $A$ is the ring ...
3
votes
1answer
138 views

Does this morphism necessarily give rise to a finite extension of residue fields?

Let $f:X\rightarrow Y$ be a morphism of finite type of locally Notherian schemes. Let $x\in X$ and $y=f(x)$. Recall that $f$ is said to be unramified if the map of stalks $g:\mathcal O_{Y,y} ...
6
votes
2answers
434 views

Nilradical of absolutely flat ring

Suppose $A$ is an absolutely flat ring (i.e. every $A$-module is flat). Is it true that nilradical of $A$ is trivial, i.e. $\mathfrak{N}(A)=\{0\}$? I believe the answer is yes. Here is my attempted ...
1
vote
1answer
161 views

Determining generators for vanishing ideal of projective closure

This question is along the lines of 2.9 from Hartshorne. Notation: Let $S:=k[x_0,\ldots,x_n]$ be the coordinate ring of $\mathbb{P}^n$, and let $A:=k[y_1,\ldots,y_n]$ be the coordinate ring of ...
3
votes
1answer
156 views

An affine open neighborhood of a nonsingular point

Let $X$ be an algebraic variety over an algebraically closed field $k$. Here a variety is an integral separated scheme of finite type over $k$ as in Hartshorne's book. Let $x \in X$ be a closed point. ...
3
votes
1answer
134 views

degree of the Hilbert polynomial of a quotient

Let $A=\bigoplus_{n \ge0} A_n$ be a Noetherian graded ring with $A_0$ Artinian. Suppose that $A=A_0[a_1,\dotsc,a_d]$ with $a_i$ having degree $1$. Let $M$ be a finitely-generated graded $A$-module. ...
3
votes
2answers
208 views

Does product distribute with respect to intersection for ideals in a ring.

Let $I,\, J$ and $K$ be ideals in a commutative ring $R$. Could you please give an example such that $(I\cap J)K = IK\cap JK$ is not true?
2
votes
0answers
80 views

A criterion for finite generation of subalgebras of a polynomial ring

In her 1926 paper on invariant theory, Emmy Noether uses a certain "finiteness criterion" which I wish to translate and maybe find a more modern reference to. The original wording is: Ein ...
4
votes
1answer
118 views

Flat closed immersion into a Noetherian scheme is open

Let $X$ be an irreducible Noetherian scheme. Consider some flat closed immersion into it. I want to show that it is also open, so that the morphism is surjective. I have a few thoughts, but I can't ...
1
vote
1answer
36 views

Help with this definition of $(G:_M I)$

I didn't understand why in this definition $I$ has to be an ideal to make sense. REMARK This is from Steps in Commutative Algebra, page 107. Thanks a lot
6
votes
1answer
286 views

Finite set of zero-divisors implies finite ring

Show that any commutative ring $R$ having only $n$ non-zero zero divisors ($n\geq 1$) is finite and doesn't contains more than $(n+1)^2$ elements.
0
votes
0answers
47 views

Rees algebra of a monomial ideal [duplicate]

Let $R=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$ and $I=(f_1,\ldots,f_q)$ a monomial ideal of $R$. If $f_i$ is homogeneous of degree $d\geq 1$ for all $i$, then prove that $$ ...
-1
votes
1answer
73 views

Intersection of radical primal ideal

Let $A$ a noetherian ring, $a_{1},...,a_{n}$ primary ideals, with $rad(a_{i})=m_{i} $ maximal ideal and $m_{i}\neq m_{j}$ si $i\neq j$. How can I prove that $a_{1}\cap a_{2}\cap \ldots\cap ...
7
votes
0answers
458 views

Regular Noetherian local rings are integral domains - questions about the proof

I am reading a proof that if $(A,\mathfrak m)$ is a regular local ring, then $A$ is an integral domain. I put the major questions I'm worried about in bold, but there are a lot of little things I'm ...
5
votes
2answers
349 views

Commutative rings whose non-trivial ideals are maximal

It is well known that a local ring is a ring containing only one maximal ideal. I was wondering if there is a characterization (or any information) of the commutative rings such that all their ...
1
vote
1answer
115 views

Question on normal Noetherian local rings

Consider a normal Noetherian local ring $(A,\mathfrak m)$ of dimension $1$. I am working through a proof that such a ring is a principal ideal domain. Consider $x\in \mathfrak m \backslash \mathfrak ...
1
vote
1answer
80 views

Normal at every localization implies normal

I'm having some trouble with basic ring theory. Let $A$ be an integral domain and $\alpha$ an element of its fraction field integral over $A$. I am trying to understand a proof that $\alpha\in A$ ...
1
vote
0answers
42 views

Factoring maps between noetherian rings

Let $A,B$ be commutative noetherian rings, and let $f:A\to B$ be a ring map. Can one always factor $f$ as $A\to C\to B$ where $C$ is a noetherian ring, $A\to C$ is flat, and $C \to B$ is surjective?
1
vote
1answer
238 views

Integral closure of $\mathbb{Z}$ in $\mathbb{Q}[i]$

I am trying to compute the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[i].$ I have managed to show that $\mathbb{Z}[i]$ is inside the integral closure, and I suspect it is the entire thing. Can ...
2
votes
0answers
47 views

Homocyclic primary module over PID

Let $R$ be a PID, $M$ be an $R$-module. If $M$ is isomorphic to $r$ copies of cyclic primary module $R/\langle p^s\rangle$ where $p$ is a prime element of $R$, then does $M$ possess the following ...
2
votes
1answer
148 views

Finitely Generated Modules over Quotient of a DVR.

Let $(R,t)$ be a DVR with uniformizing parameter $t$ and let $M$ be a finitely generated module over $R/(t^n)$. Then $M\cong \bigoplus_r R/(t^r)$. Question: How many summands of type $R/(t^r)$, where ...