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

learn more… | top users | synonyms (1)

6
votes
1answer
259 views

Does maximal Cohen-Macaulay modules localize?

Let $A$ be a Noetherian local ring and $M$ a finitely generated $A$-module such that $$\operatorname{depth}M= \dim M=\dim A.$$ I can prove that $$\operatorname{depth}M_{\mathfrak{p}}= \dim ...
2
votes
4answers
131 views

The valuation ring $R$ in $K(T)$, such that $K[T] \subsetneq R \subsetneq K(T)$

$K$ is an algebraically closed field, $K[T]$ is the ring of polynomials of one indeterminate over $K$, and $K(T)$ is its field of fractions. A valuation ring $R$ in $K(T)$ which includes $k[T]$ and ...
11
votes
4answers
4k views

Why are maximal ideals prime?

Could anyone explain to me why maximal ideals are prime? I'm approaching it like this, let $R$ be a commutative ring with $1$ and $A$ be a maximal ideal. Let $a,b\in R:ab\in A$ I'm trying to ...
5
votes
1answer
522 views

A wrong proof about Dedekind domains

I "proved" that a Dedekind domain is a PID, but as we know this is wrong (for example $\mathbb{Z}[\sqrt{-5}]$). I do not know what is wrong in my proof: Suppose $R$ is a Dedekind domain, $I$ is ...
3
votes
1answer
121 views

the reduced locus of a Noetherian ring

Let A be a Noetherian ring, Is the set of prime ideals $\{p\in \operatorname{Spec} A| A_p$ is a reduced ring $\}$ an open subset of $\operatorname{Spec} A$ in Zariski topology?
5
votes
1answer
136 views

Some question ideal of variety

For an affine variety $X=V(x^{2}+y^{2}-1, x-1)$, I found the ideal of $X$, $I(X)=\langle x-1,y\rangle$. But I don't know $I(X)=\langle x^{2}+y^{2}-1, x-1\rangle$.
4
votes
1answer
414 views

Does this “extension property” for polynomial rings satisfy a universal property?

On page 151 of Paolo Aluffi's Algebra: Chapter 0, an important property of the polynomial ring $\mathbb{Z}[x_1, \cdots, x_n]$ is introduced, namely that it's initial in the category of set functions ...
5
votes
2answers
241 views

The fixed subalgebra of a finitely generated algebra

Let $k$ be a field, $A$ a finitely generated $k$-algebra, put $A^{G}:=\{a \in A \mid g(a)=a ~ \mbox{for all}~g \in G\}$, where $G$ is a finite group of automorphisms of $A$. If (1) the order of $G$ ...
3
votes
0answers
209 views

What's the origin of the terminology “Normalization” in commutative algebra?

Since the terminology "normal", "normalized", etc has different meanings in mathematics (some geometric in flavor, like when referring to perpendicularity) and I just read in Eisenbud's book on ...
6
votes
1answer
258 views

Is there a classification of local rings with trivial group of units?

Out of curiosity, is there a classification of all local rings with trivial group of units? I suppose what I'm trying to ask is, if I asked for all local rings $R$ with $R^\times=\{1\}$, what would ...
4
votes
2answers
467 views

Multiplicative monoid of a commutative ring

Is there any good description of the multiplicative monoid of a commutative ring in general? Or in special cases? I understand that in a UFD, it is the result of adjoining a zero to the Cartesian ...
4
votes
1answer
192 views

About $\operatorname{Supp}_A M_{\mathfrak{p}}$

Let $M$ be a finitely generated $A$-module and $\mathfrak{p}\subset A$ a prime ideal. Is it true that $\operatorname{Supp}_A M_{\mathfrak{p}}$ is the closure of $\operatorname{Supp}_{A_{\mathfrak{p}}} ...
2
votes
2answers
675 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 ...
4
votes
1answer
242 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
99 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 ...
7
votes
1answer
387 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
155 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 ...
6
votes
1answer
388 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
134 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
92 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
145 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 ...
3
votes
1answer
523 views

Going down theorem fails

Maybe this exercise comes from some textbook, but I donot 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. Here $k$ is an ...
3
votes
1answer
66 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
164 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
1k 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?
3
votes
1answer
307 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
86 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
436 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 ...
0
votes
1answer
525 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
237 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
500 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
458 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
57 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
73 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
176 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, ...
6
votes
2answers
603 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
317 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
140 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
471 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
238 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
341 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
152 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 ...
7
votes
1answer
358 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
344 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
107 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
305 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
126 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
48 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
384 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 ...
5
votes
1answer
607 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 ...