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

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5
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1answer
594 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
159 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
72 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
135 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
64 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
279 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
145 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
49 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
213 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
166 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
126 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
55 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
86 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
90 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
122 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
139 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
238 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
147 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
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1answer
120 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
41 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
128 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
419 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
174 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
142 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) ...
3
votes
1answer
200 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
74 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
227 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
182 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
84 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
202 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
653 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
81 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
96 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
243 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
120 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
111 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
246 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
364 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 ...
4
votes
1answer
53 views

The identifications of $R$ in its ring of fractions $S^{-1}R$

If $R$ is an integral domain, we can identify the elements $r\in R$ as elements $rs/s$ of the ring of fractions $S^{-1}R$. In this way, we can identify $r\in R$ as $r/1_R$. I've seen in somewhere that ...
4
votes
1answer
68 views

Does the triangle inequality follow from the rest of the properties of a subfield-valued absolute value?

(This is a much more specific version of my earlier question from over a year ago.) Let $F$ be a field, let $E$ be an ordered subfield of $F$, and let $\;\; |\hspace{-0.03 in}\cdot\hspace{-0.03 in}| ...
32
votes
4answers
1k views

$\mathbb C[X]/(X^2)$ is isomorphic to $\mathbb R[Y]/((Y^2+1)^2)$

This question led me to the following: Prove that $\mathbb C[X]/(X^2)$ is isomorphic to $\mathbb R[Y]/((Y^2+1)^2)$.
13
votes
1answer
173 views

Modules over Completion

I have the following question. Let $R$ be a commutative ring with unit, and let $\hat{R}$ denote its completion (w.r.t. any ideal $I$). Let $M$ be an $\hat{R}$-module. Is $M= N\otimes_R \hat{R}$ for ...
1
vote
1answer
67 views

Ring of integers

I know that $\mathbb{Z} [\sqrt{3}, \sqrt{7} ]$ is not the ring of integers in $\mathbb{Q} [\sqrt{3}, \sqrt{7} ]$. But, I don't know how to explain. Can someone help in this. Thnx
1
vote
0answers
35 views

Description of certain invariant polynomials (not a group action)

Working on a recent question led me to the following invariant-computation problem : let $$ A=\bigg\lbrace P \in {\mathbb Q}[X_1,X_2,X_3,X_4] \ \bigg| \\ \quad\ P(X_1X_3+X_2X_4+X_1X_4,\ X_2X_3,\ ...
2
votes
2answers
867 views

What is the kernel of the evaluation homomorphism?

I'm studying Sharp's Steps in Commutative Algebra, and I need a hint how to proceed with this exercise in the page 26: First of all, I didn't understand even the notation, what did the author mean ...
3
votes
4answers
104 views

Waring’s problem in commutative rings

Let $k\geq 2$ be a fixed integer. If $R$ is a commutative, integral, unital ring, the Waring height of an element $r\in R$ is the smallest number of $k$-ths powers whose sum is $r$ (this height can ...
8
votes
2answers
1k views

Product of two primitive polynomials

I'm having troubles with one of the problems in the book Introduction to Commutative Algebra by Atiyah and MacDonald. It's on page 11, and is the last part of the second question. Given $R$ a ...
11
votes
2answers
249 views

Is $k[x,y,z]/(x^2+y^2-z^2)$ a UFD?

Let $k$ be an algebraically-closed field of characteristic not two. Then is the ring $$k[x,y,z]/(x^2+y^2-z^2)$$ a UFD? I admit that $k[x,y,z]/(xy-z^2)$ is not a UFD.