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

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2
votes
2answers
755 views

Grading of the quotient module $M/N$

Let $S$ be a graded ring, $M$ a graded $S$-module, and $N$ a graded submodule of $M$. I'm trying to convince myself (of the well known fact) that $M/N$ is graded by $$M/N=\oplus_{i\geq0} (M_i/N\cap ...
5
votes
1answer
282 views

“Instructive” proof of “If I is maximal among ideals not …, then I is prime”

In this question all rings are commutative with identity. Consider the following well-known statement: (*) Let $R$ be a ring and $S$ a multiplicatively closed subset of $R$. Suppose $I$ is an ...
4
votes
1answer
106 views

A nonreflexive module isomorphic to its double dual

I know that the definition of reflexive module is that the $R$-module $M$ should be isomomorphic to its double dual $M^{**}$ via the canonical map $M\rightarrow M^{**}$. I'd like to know an ...
8
votes
1answer
399 views

Question about Zariski topology

Here is the question: Let $A$ be a commutative ring with unit, $X=\mathrm{Spec}A$, $U_i$s be quasi-compact open sets of $X$ such that $\emptyset=\cap_{i\in I}U_i$, then there is a finite subset ...
4
votes
2answers
157 views

Is $\operatorname{Hom}_A(M,N)$ a set without axiom of choice?

Let $M$ and $N$ be $A$-modules, $\operatorname{Hom}_A(M,N)$ the set of all $A$-module homomorphisms $M\rightarrow N$. $\operatorname{Hom}_A(M,N)$ can be viewed as a subset of the cartesian product ...
7
votes
1answer
449 views

Question about a proof on Atiyah Macdonald

I have a question about a step of a proof in Atiyah Macdonald. It's the proposition 2.4. Let M be a finitely generate A-module, let a be an ideal of A, and let $ \phi $ be an A-module endomorphism of ...
0
votes
2answers
137 views

Problem about Assasin of module

I found this problem and I don't understand the solution. I will appreciate your help. Let $A = \mathbb{Q}[X_1,...,X_n,...], a = (X_1^2,...,X_n^2,...)$ and $ M = A/a$. Show that $Ass_A (M) = ...
3
votes
1answer
97 views

Appropriate notion of localization of a Galois ring extension

Earlier, I had a asked a question for a notion of Galois ring extension. I was particularly interested in Peter Patzt's answer. So, given an integral domain $R$ with field of fractions $F$ and a ...
3
votes
1answer
161 views

Proving projective equivalence of Auslander Transpose

Let $$P_1\overset{\partial}{\rightarrow} P_0\rightarrow M\rightarrow 0$$ be an exact sequence of $A$-modules with $P_0$, $P_1$ finitely generated and projective. The transpose $T(M)$ is defined as ...
5
votes
1answer
638 views

Going down theorem fails

Maybe this exercise comes from some textbook, but I do not know. It said that this ring extension $k[x(x-1),x^2(x-1),z]\subset k[x,z]$ does not have the Going-Down property. I observe that ...
3
votes
1answer
69 views

About depth$(I,M)$ when $IM=M$

Suppose $A$ is a Noetherian ring, $I\subset A$ an ideal, and $M$ a finitely generated $A$-module. If $IM\neq M$, then the length of a maximal $M$-sequence inside $I$ is fixed by the number ...
0
votes
1answer
168 views

faithful, finitely generated module over a local ring

Let $A$ be a commutative local ring, with unique maximal ideal $\mathfrak{m}$, and residue field $k:=A/\mathfrak{m}$. Let $M$ be a faithful, finitely generated $A$-module. If $M/\mathfrak{m}M$ is ...
3
votes
3answers
2k views

Intersection maximal ideals of a polynomial ring

Let $k$ be a field and let $k[x,y]$ be the polynomial ring in two variables. Why this ring has trivial Jacobson radical?
4
votes
1answer
356 views

Extensions and contractions of prime ideals under integral extensions

Let $R\subseteq S$ be an integral extension of commutative rings with identity. Let $P$ be a prime ideal in $R$ and $Q$ a prime ideal in $S$. If $Q=PS$ and $P=Q\cap R$ what can we say about $Q^n\cap ...
1
vote
1answer
96 views

direct summands of modules

Let $R$ be a commutative DVR, and let $M$ be the free $R$-module of finite rank $k\ge 2$. Let $N$ be a submodule of $M$ isomorphic to $R$. Is it true that $N$ is a direct summand of $M$? Thanks in ...
5
votes
2answers
471 views

Localizing and taking degree zero commutes with tensor product

Let $S$ be a graded ring ($S_n=0$ for $n<0$), $f\in S$ a homogeneous element, and $M, N$ two graded $S$-modules. I'm trying to prove that $$(M\otimes_S N)_{(f)}\simeq ...
1
vote
1answer
614 views

localizations of a direct sum module

Consider the $\mathbb{Z}$-module $M=\bigoplus{\mathbb{Z}/p\mathbb{Z}}$, where the direct sum is taken over the set of all prime numbers. How do I show that the localizations $M_\mathfrak{p}$ are ...
4
votes
1answer
248 views

Chain of prime ideals of maximal length

Consider the domain $R=\mathbb{C}[x,y]/(y^2-x^3)$. What would be an example of a chain of prime ideals of $R$ of maximal length?
14
votes
3answers
534 views

Question about UFD

I want to know some examples with the following properies. Let $R$ be a domain such that every non unit element $x$ is a product of finite irreducible elements,but $R$ is not a UFD, and there is ...
8
votes
2answers
486 views

Is this ring Noetherian?

The subring of $\mathbb{C}[x,y]$ consisting of all polynomials $f(x,y)$ whose gradient vanishes at the point $x=y=0$. Is this ring Noetherian?
1
vote
1answer
58 views

Prescribing linear projection

Let R be a commutative pid, and let M be the free R-module of finite rank k. Given a non-zero proper submodule N of M, does there always exist a projection P such that ker(P)=N? If so, how can we ...
3
votes
1answer
76 views

a submodule of $R^n$

Let $R$ be a commutative ring of positive characteristic p. If $M$ is a submodule of $R^n$, let $M^{[p]}$ be the submodule of $R^n$ generated by $(a_1^p,\cdots,a_n^p)$ where $(a_1,\cdots,a_n)\in M$. ...
4
votes
2answers
189 views

Subrings of formal series rings

Let $k$ be a field and $A = k[[x_1, \dots, x_n ]]$ be the ring of formal series in $n$ variables. Consider $g_1, \dots, g_m \in A$ such that $g_1(0) = \cdots = g_m(0) = 0$. For every $f \in k[[t_1, ...
7
votes
2answers
694 views

Does every Noetherian domain have finitely many height 1 prime ideals?

Let $A$ be a Noetherian domain. Is the set $\{P\subset A \mid P \mbox{ prime ideal, } \dim A_P=1\}$ always finite? I can prove for $f \neq 0, f\in A$, the set $\{P\subset A \mid \dim A_P=1, f\in ...
2
votes
2answers
320 views

I can prove a Contradiction - Where's my mistake?

I am trying to understand a certain scenario and to do so, I sat down and calculated an explicit example. While doing so, I was able to "prove" two statements that directly contradict each other, ...
2
votes
0answers
142 views

isomorphism of $p$-adic groups

Let $(A_n)<$ be a projective and inductive system of $\mathbb{Z}/p^n$-modules. Is then $\operatorname{Hom}{(\projlim A_n, \mathbb{Z}_p)}$ isomorphic to ...
5
votes
1answer
509 views

Proof of Hensel's lemma

I am reading up the proof of Hensel's lemma here. On page 2, after equation 2, the author concludes that the degree of $\delta h_k$ is less than $n$ since the degree of $\Delta$ and $\epsilon g_k$ is ...
1
vote
2answers
247 views

Little question about Nakayama's Lemma

Let $M$ and $N$ be finitely generated modules over a local ring $A$ with residue field $k$ and $f:M\rightarrow N$ a A-homomorphism, such that the induced morphism $M\otimes_{A}k \rightarrow ...
1
vote
1answer
387 views

Socle is the intersection of essential submodules?

Let $M$ be an $A$-module. How do I show that Soc$(M)$ is the intersection $Q$ of all essential submodules of $M$? One direction is easy enough (Soc$(M)\subset Q$), but I can't seem to show the other ...
4
votes
1answer
158 views

Henselian rings with the same quotient field

I was recently reading these notes, where it is proved (a theorem of Kaplansky-Schilling) that a field that admits two distinct valuations with respect to which it is henselian is separably closed. A ...
8
votes
1answer
376 views

Kernel of map between polynomial rings that takes monomials to monomials

Let $k$ be a field (say of characteristic $0$). Let $z_1,\ldots,z_n \in k[y_1,\ldots,y_m]$ be monomials, and consider the ring homomorphism $\phi : k[x_1,\ldots,x_n] \rightarrow k[y_1,\ldots,y_m]$ ...
6
votes
1answer
361 views

Integral closure of p-adic integers in maximal unramified extension

Let $\mathbb Q_p$ be the field of p-adic numbers, and let $\mathbb Q_p^{\text{unr}}$ be maximal unramified extension in some algebraic closure of $\mathbb Q_p$. My understanding is that $\mathbb ...
3
votes
1answer
109 views

Example showing why Macaulay's lemma doesn't work for inhomogeneous ideals

Macaulay's lemma states: Let R be a polynomial ring and I a homogeneous ideal. Then the Hilbert function of I is the same as the Hilbert function of in(I). (Schenck, Computational Algebraic ...
1
vote
1answer
367 views

Scalar operators and commutators

Given a scalar operator $S$ and vector operators $V_1, V_2$, show that the commutator $[S,V_1\times V_2]= [S,V_1]\times V_2+V_1\times [S,V_2]$. I don't quite understand what a scalar operator is. But ...
1
vote
1answer
129 views

Question about derived functors

Let $F,G, H: Mod \to Mod$ be three left exact functors such that $R^iF(-)\cong R^iG(-)$ for all $i\in\mathbb{N}$. We consider the exact sequence $$\cdots\to R^iF(M)\to R^iG(M)\to R^iH(M)\to ...
0
votes
1answer
49 views

Roots of Units in Complete $\mathbb{C}$-Algebras

$\newcommand{\cc}{\mathbb C}$ Let $R$ be a finitely generated $\cc$-algebra and ${\frak m}\subset R$ a maximal ideal. Denote by $\hat R$ the completion of $R$ with respect to $\frak m$. Assume that ...
11
votes
3answers
401 views

When to use Zorn's Lemma

I was looking at an exercise this morning which I was able to reduce to showing that the nilradical is the the intersection of the prime ideals in a ring -- a fact I remembered was true, but which I ...
6
votes
1answer
668 views

Irreducible quadratics in polynomial ring of two variables over algebraically closed field

I'm currently stuck at problem 1.1 c) in Hartshorne's algebraic geometry book. I just can't let it go. Setting is as title says (field $k$, variables $x$ and $y$). Problem 1.1. a) and b) concerns ...
3
votes
4answers
631 views

Are there any nontrivial, finite subrings of an infinite ring?

For example, $S\subset\mathbb{R}$ where $S=\{0\}$ is the trivial subring which is finite. Is there a nontrivial subring of an infinite ring (i.e. of $\mathbb{R}$ or not) that is non-infinite? This ...
0
votes
1answer
118 views

Some question of polynomial ring

I saw the following statement "If polynomial ring $k[X_1,\cdots,X_n]$ is field then this implies that $n=0$" but I can't understand this statement. I want to completed proof of this statement.
1
vote
0answers
83 views

Use the degree of Hilbert polynomial to define the dimension of a ring

Let $R$ be a local ring , $\mathfrak{m}$ the maximal ideal, $q$ is $\mathfrak m$-primary . Then we can prove that there exists a polynomial $F_q(t)\in \mathbb{Q}(t)$ such that ...
7
votes
1answer
274 views

Associated primes and integral closure

Let $A$ be an integral domain which is finitely generated as a $k$-algebra and let $I\subset A$ be an ideal. Let $B$ be its integral closure (in the fraction field $\mathrm{Frac}\ A$) - in this case ...
5
votes
2answers
178 views

Why is this module reflexive?

Let $A\subset B$ be two integrally closed Noetherian domains with $B$ finitely generated as an $A$-module. Then $B$ is reflexive. Could you explain me why, please? Reflexive means that the ...
10
votes
2answers
2k views

Prerequisites for Atiyah Macdonald

I am currently doing a one semester course on groups and rings where we have learned about (so far): Definitions of groups, subgroups, cyclic and normal subgroups, the symmetric group, homomorphisms, ...
9
votes
2answers
648 views

Derived functors of torsion functor

Let $A$ be a domain. For every $A$-module $M$ consider its torsion submodule $M^{tor}$ made up of elements of $M$ which are annihilated by a non zero-element of $A$. If $f \colon M \to N$ is a ...
0
votes
1answer
1k views

Zero divisors, nilpotents and units in the ring of functions $\mathbb{R} \to \mathbb{R}$

Let $R$ be the set of all real valued functions defined for all real numbers under function addition and multiplication. i have to show that all the zero divisors of $R$ all nilpotent elements of ...
2
votes
2answers
235 views

Non-zero divisor in an integral domain

Let $R$ be an integral domain and $P$ a prime ideal. Let $x$ be an element such that $xP^{m-1}=P^m$ for some $m>0$. Is $P$ generated by $x$?
2
votes
2answers
101 views

Sum of localization maps

In Eisenbud's Commutative algebra with a view..., he shows that if an $A$-module $M$ has a finite length, then the sum of localization maps at maximal ideals is an isomorphism: ...
2
votes
1answer
222 views

Finding the Hilbert Function for a certain ring

Right now I'm trying to find the Hilbert Function , and the corresponding Hilbert Polynomial for the ring $M=k[x,y,z,w]/(x,y) \cap (z,w)$. I just finished reading the first chapter of Eisenbud, so I ...
5
votes
1answer
204 views

When is the ring of continuous functions absolutely flat?

This question was created in a discussion. Let $X$ be a topological space. Denote by $C(X; \mathbb{R})$ the ring of real-valued continuous functions defined on $X.$ Characterize those compact ...