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

learn more… | top users | synonyms (1)

6
votes
2answers
687 views

Show that a ring with disconnected spectrum is a product of two subrings. [duplicate]

It's an exercise from the book introduction to commutative algebra by Atiyah and Macdonald. If $\operatorname{Spec}(A)$ is disconnected, I'm asked to show that $A$ is a product of two subrings. I ...
1
vote
1answer
80 views

Localization and extension of rings

Is $\mathbb{Z}_{(3)}[i,\sqrt{2}]=(\mathbb{Z}[i,\sqrt{2}])_{(3)}$ (where by subscript $(3)$ we mean localization at the ideal generated by $3$)? Do both of these rings contain elements like $$ ...
5
votes
0answers
84 views

approximating a variety locally by a vector space

Suppose we have $m$ homogeneous equations with integer coefficients in $n$ variables and that $m >> n$. Let $x_0 \in \mathbb{C}^n$. Question 1: is there a way to approximate the variety ...
6
votes
2answers
127 views

Symmetric and exterior powers of a projective (flat) module are projective (flat)

Assume that $R$ is a commutative ring with unity and $P$ a projective (flat) $R$-module. Why $\mathrm{Sym}^n(P)$ and $\Lambda^n(P)$ are projective (flat) for every $n$?
3
votes
1answer
97 views

Height of a prime ideal and number of generators of its localization

This question is very related to this one: generators of a prime ideal in a noetherian ring. Let $\mathfrak{p}$ be a prime ideal in a Noetherian ring and let $k$ be its height. Further suppose that ...
2
votes
1answer
90 views

Finite generation of Hom between cyclic and artinian module

Let $R$ be a Noetherian ring with unit, and $I$ be a nonzero ideal of $R$. Let $M$ be an artinian $R$ module. Is $\operatorname{Hom}(R/I, M)$ finitely generated? Thanks.
2
votes
0answers
214 views

Question on integral scheme

Let $X=\operatorname{Spec}A$ be an affine scheme. In the book of Hartshone, he claimed that $X$ is integral if and only if $A$ is an integral domain. If $X$ is integral then we can deduce easily that ...
2
votes
1answer
343 views

Relation between spectrum of a ring and its quotient ring and localization.

Let $A$ be a commutative ring. $I$ be an ideal of $A$, $S$ be a multiplicative closed subset. We know that : there is 1-1 correspondence between the prime ideals $\mathfrak{p}\in Spec A$ containing ...
7
votes
2answers
194 views

Property of modules via exact sequences

Suppose $A\neq 0$ is a commutative ring with $1$. Let $L, M, N$ be $A$-modules such that the sequence $$0\longrightarrow L\overset{\alpha}{\longrightarrow} M\overset{\beta}{\longrightarrow} ...
5
votes
1answer
409 views

Number of generators of the maximal ideals in polynomial rings over a field

Hi I'm trying to prove the following If $K$ is a field (not necessary algebraically closed) then every maximal ideal of $K[x_{1},\dots,x_{n}]$ is generated by exactly $n$ elements. I know that ...
5
votes
1answer
409 views

A subset of a field that is a subfield

It can be verified that the following assertion is true: a subset $S$ of a field $F$ is a subfield if $S$ contains the additive and multiplicative identities 0 and 1, if $S$ is closed under addition, ...
5
votes
1answer
149 views

Can we have a Primary Avoidance Theorem ?

Prime Avoidance Theorem says: Let $ P_1, P_2,\dots, P_n $ be prime ideals in a commutative ring $R$ and let $I$ be an ideal of $R$ such that $ I \subseteq P_1 \cup P_2 \cup \cdots \cup P_n$. ...
2
votes
1answer
68 views

Maximal element in set of Ann(m) for m in M is prime

Exercise 15.1.32 in Dummit & Foote, along with the included hint, is Suppose that $M$ is a $R$-module and that $P$ is a maximal element in the collection of ideals of the form ...
5
votes
2answers
130 views

Change of variables in $k$-algebras

Suppose $k$ is an algebraically closed field, and let $I$ be a proper ideal of $k[x_1, \dots, x_n]$. Does there exist an ideal $J \subseteq (x_1, \dots, x_n)$ such that $k[x_1, \dots, x_n]/I \cong ...
3
votes
1answer
61 views

Question about completions.

I am reading the book Introduction to commutative algebra by Atiyah and Macdonald. I have some questions about Corollary 10.3 and Corollary 10.4. Why the sequence $$ 0 \to \frac{G'}{G' \cap G_n} ...
1
vote
1answer
191 views

Questions about the intersection of all neighborhoods of $0$ in a topological abelian group.

Let $H$ be the intersection of all neighborhoods of $0$ in a topological abelian group. On page 102 of the book introduction to commutative algebra by Atiyah and Macdonald, the fourth line of the ...
3
votes
1answer
104 views

Atiyah-Macdonald, Proposition 2.12, uniqueness of the tensor product.

The following is a result from Atiyah-Macdonald, defining and showing existence and uniqueness of tensor product of modules over a commutative ring. Proposition 2.12. Let $M, N$ be $A$-modules. ...
2
votes
0answers
45 views

Relation between inverse limits (and direct limits) with limits in calculus. [duplicate]

What is the relation between inverse limit (and direct limit) with limits in calculus? Are there some special cases that an inverse limit (or direct limit) is a limit in calculus (for example, the ...
1
vote
0answers
182 views

Integral extension inside a polynomial ring over a field

Let $K$ be a field and $D = K[X]$. I need to show that if $f\in D$ is non constant, then the extension of rings $K[f]\subset D$ is integral, and if $A$ is a subring of $D$ which contains $K$ and has ...
3
votes
0answers
133 views

associated graded ring is the quotient of a free algebra by a homogeneous ideal

Let $A$ be a semilocal ring with Jacobson radical $m$ and let $I$ be an ideal of definition, i.e. an ideal such that $m^{\nu} \subset I \subset m$. Consider the associated graded ring of $A$, given by ...
1
vote
1answer
109 views

maximal ideal properly contains union of its square with the union of minimal prime ideals

One of the first theorems one encounters in the study of commutative algebra is that if $I$ is an ideal of a ring $A$ not contained in any of the prime ideals $P_1,\cdots,P_n$, then $I$ is not ...
2
votes
1answer
51 views

Question about inverse limits.

I am reading the book Introduction to commutative algebra by Atiyah and Macdonald. On Page 104, I have some questions about the proof that $\{A_n\}$ is surjective implies $d^A$ is surjective. We have ...
4
votes
1answer
81 views

Question about homomorphisms $f_{!}, f^{!}$.

Let $f: A \to B$ be a finite ring homomorphism and $N$ a $B$-module. $N$ can be considered as an $A$-module if we define $A \times N \to N$, $(a, n) \mapsto f(a)n$. Therefore we have a map $f_{!}: ...
3
votes
1answer
85 views

Question about the lying over theorem.

I have some questions about the proof of the Lying over theorem in the book Introduction to commutative algebra by Atiyah and Macdonald. (1) In the proof of Theorem 5.10 of Page 62, is the map ...
5
votes
1answer
79 views

Radical of prime ideal in homogeneous localization is prime

Let $B$ be a graded ring, $B=\oplus_{d\ge 0} B_d$. If $f\in B$ is homogeneous, we let $B_{(f)}$ denote the subring of $B_f$ made up of elements of the form $af^{-N}$, $N>0$, where $a$ is a ...
0
votes
1answer
102 views

Are these prime ideals?

Let $R=\mathbb Z[\sqrt{-5}]$. I want to show $P=3\,R+(1+\sqrt{-5})\,R$ and $Q= 3\,R+(1-\sqrt{-5})\,R$ are prime ideals of $R$.
3
votes
1answer
135 views

Radical of ideals in local one dimensional rings

Let $R$ be a local one dimensional ring. I want to show that for all $ a,b\in R$, $\sqrt{Ra+Rb}$ is equal to $\sqrt{Ry}$ for some $y\in Ra+Rb$ or is equal to $R$.
2
votes
1answer
176 views

Questions about Grothendieck groups.

I have a question of the exercise 26 on page 88 of the book introduction to commutative algebra by Atiyah and Macdonald. In 26(iii), let $A$ be a field. Then finitely generated $A$-modules are finite ...
0
votes
1answer
120 views

The relation between minimal prime ideals and nilpotents

Show that a prime ideal $I$ of a ring $R$ is minimal if and only if for each $x\in I$ there exists $a\in R\setminus I$ such that $ax$ is nilpotent.
0
votes
1answer
109 views

Question about zero-divisors and a quotient of a polynomial ring by an ideal in the book Introduction to commutative algebra by Atiyah and Macdonald.

I am reading the book the book Introduction to commutative algebra by Atiyah and Macdonald. I have two questions On Page 51. On Line 5 of Page 51, it is said that the zero-divisors in ...
0
votes
1answer
38 views

Question about primary decompositions.

I am reading the book Introduction to commutative algebra by Atiyah and Macdonald. On page 50, Line -7, it is said that "if $f: A \to B$ and $\mathfrak{q}$ is a primary ideal in $B$, then ...
3
votes
0answers
126 views

If $M \otimes M \simeq M$ is there anything we can say about $M$? [duplicate]

Over a commutative (and unital) ring, if $M \otimes M \simeq M$ can we say anything about $M$? If we base change to a point, ie tensor with a map from the ring into a field, then $M$ becomes a vector ...
2
votes
2answers
346 views

Questions of the book Introduction to commutative algebra by M. F. Atiyah and I. G. Macdonald.

I have some questions of the book Introduction to commutative algebra by M. F. Atiyah and I. G. Macdonald. On Line 8-9 of Page 42, it is said that $(xs-a)t=0$ for some $t\in S$ iff $xst\in ...
2
votes
1answer
152 views

Question about the book introduction to commutative algebra by M. F. Atiyah and I. G. Macdonald.

On Line 2 of Page 40 of the book introduction to commutative algebra by M. F. Atiyah and I. G. Macdonald, it is said that $m/s =0$ implies $tm=0$ for some $t \in S$. I think that if $m/s=0$, then $m/s ...
3
votes
2answers
87 views

Question about radical of powers of prime ideals.

Let $Q$ be an ideal of a commutative ring $A$ and $$r(Q) = \{x \in A : x^n \in Q \text{ for some } n >0 \},$$ the radical of $Q$. Suppose that $P$ is a prime ideal of $A$. How to show that $r(P^n) ...
2
votes
0answers
161 views

Associated prime ideals of Hom (Bruns and Herzog, exercise 1.2.27)

Let $R$ be a Noetherian ring and $M,N$ finitely generated modules. I want to show that $$\mathrm{Ass}_R(\mathrm{Hom}_R(M,N)) = \mathrm{Ass}_R(N) \cap \mathrm{Supp}(M).$$ I don't understand what ...
2
votes
1answer
73 views

Why is $k \rightarrow A \rightarrow A / I$ and isomorphism of rings if $I \subset A$ is maximal?

Let $k$ be a algebraically closed field, $A$ a finitely generated $k$-Algebra and $I \subset A$ a maximal ideal. Let $\varphi: k \rightarrow A$ be a ring homomorphism. Why is this combination $$k ...
4
votes
2answers
199 views

'homotopy' between morphisms of a 'topological' or 'algebraic' category (Stanley-Reisner ring)

In what follows, a homotopy is a congruence $\simeq$ on a given category. Given such a homotopy, objects $X$ and $Y$ of the given category are homotopy equivalent when there exist morphisms ...
0
votes
1answer
133 views

On projective dimension of quotients of polynomial rings

Let $A$ be a commutative ring, $B=A[X]/(X^2)$, and $C=B/(x)$. (Here $x$ denotes the residue class of $X$ modulo $(X^2)$.) Why the projective dimension of $C$ is infinite ?
0
votes
1answer
80 views

Zariski Topology on Primary Spectrum $P$-$\operatorname{Spec}(R)$

So there's a proposition in Hummadi's journal Primary Spectrum I'd like to ask. It is said if given $R$ principal ideal domain and $a,b \in R$, then $D_p(a) \cap D_p(b) \supseteq D_p(ab)$, and ...
2
votes
1answer
182 views

Extended ideals in power series ring

Let $A$ be a commutative ring with $1$ and consider the ring of formal power series $A[[X]]$. If $I \subseteq A$ is an ideal, let $I[[X]]$ denote the set of power series with coefficients in $I$. This ...
0
votes
2answers
445 views

Every finitely generated algebra over a field is a Jacobson ring

Knowing that the polynomial ring in $n$ variables over a field $k$ is a Jacobson ring, how can we prove from it that every finitely generated $k$-algebra is a Jacobson ring? EDIT: We define a ...
1
vote
1answer
78 views

Why do Artinian rings have dimension 0 and not 1?

One of the properties of an Artinian ring $R$ is that every prime ideal is maximal. So, if $\mathfrak{m}$ is a nonzero prime ideal, $(0)\subseteq \mathfrak{m}$ is a length-$1$ chain of prime ideals, ...
4
votes
1answer
83 views

Explicit generators of syzygies

Consider an $1\times n$ matrix $$ \mathbf{A}=\begin{pmatrix} f_1 &f_2 & \dots & f_n \end{pmatrix} $$ over $R=\mathbb{C}[X_1,\dots,X_r]$. Let $M=\oplus_{i=1}^n R\mathbf{e}_i$ be the ...
1
vote
1answer
191 views

Quotient of a polynomial ring by a polynomial is equal to the direct sum of quotients by the roots

Reading through Claudio Procesi's Lie Groups: An Approach through Invariants and Representations, I came across the following claim, stated without proof during the derivation of some properties of ...
1
vote
1answer
101 views

algebraic-geometric interpretation of the principal ideal theorem

This quote is from Matsumura's Commutative Ring Theory, page 100: "The principal ideal theorem corresponds to the familiar and obvious-looking proposition of geometrical and physical intuition (which ...
-1
votes
1answer
105 views

The h-vector of a simplicial complex

Let $S$ be a polynomial ring over a field. I want to find an ideal $ I\subseteq S$ such that $(1,2,3,1,1,1)$ is the $h$-vector of $S/I$. We have a relation between $f$-vector and $h$-vector and ...
4
votes
1answer
199 views

Finite surjective morphism to normal affine variety is open

We have a finite surjective morphism $\phi: X \to Y$ (it means that $k[X]$ is a finitely generated module over $\phi^*(k[Y])$), $Y$ is normal (it means that $k[Y]$ is normal). Why is it open in ...
3
votes
1answer
85 views

Does this complex remain exact after I restrict the maps?

$R$ is a commutative ring with unity. Assume you have two matrices $A:R^n\rightarrow R^m$ and $B:R^m\rightarrow R^n$ such that they form an exact complex in the obvious way, i.e., $$\cdots\rightarrow ...
5
votes
2answers
287 views

Is every field the field of fractions for some integral domain?

Given an integral domain $R$, one can construct its field of fractions (or quotients) $\operatorname{Quot}(R)$ which is of course a field. Does every field arise in this way? That is: Given a ...