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

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3
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2answers
36 views

Requirements on ring for injective-projectiveness

What requirements could be asked (minimal) of a ring R, so that any module M on R which is injective must also be projective? Is this possible?
3
votes
1answer
224 views

Jacobson radical of rings

I have some question about the Jacobson radical of rings. What is $J(R)$ when $R$ is a Principal Ideal Domain but not a field? e.g. I know that $\mathbb Z$ is a PID and why is $J(\mathbb Z)=0$ but ...
4
votes
1answer
306 views

Inducing homomorphisms on localizations of rings/modules

I'm trying to work out Exercise 2.6 in Commutative Algebra by Eisenbud, which asks to prove the Chinese Remainder Theorem for commutative rings. Exercise: Let $R$ be a commutative ring, and let ...
7
votes
2answers
100 views

Ideals in a non-dedekind domain that cannot be factored into product of primes

For a domain $R$ to be a Dedekind domain it need to satisfy 3 conditions: one-dimensional, Noetherian, integrally closed. I have got three domains satisfying all but one of those three: 1) ...
7
votes
1answer
72 views

Is an abelian group characterized by its localizations?

Let $G$ and $H$ be countable abelian groups. Assume that for every prime number $p$ there is an isomorphism $G\otimes_{\mathbb Z} \mathbb Z[\frac{1}{p}]\cong H\otimes_{\mathbb Z} \mathbb ...
1
vote
1answer
57 views

Subring of a field with a discrete valuation is a euclidean domain

I am confused by the following problem in Aluffi's Algebra Chapter Zero... A discrete valuation on a field $k$ is a surjective group homomorphism $v : k^* \to (\Bbb Z,+)$ such that $v(a + b) \geq ...
3
votes
1answer
55 views

Help in this question in Fulton's algebraic curves

I'm trying to solve this question: In item (a) I used the fact $O_a(V)$ is a Noetherian local ring and the only maximal ideal is $(x-a)$. First note that the non-units of $O_a(V)$ are the elements ...
2
votes
1answer
93 views

Rees ring associated to an ideal

I am reading Atiyah-Macdonald, the chapter on completions. Let $A$ be a ring (not graded), and let $\mathfrak{a}$ be an ideal of $A$. Then we can form a graded ring ...
1
vote
1answer
30 views

Given ring $A$, ideal $I$, and $A$-module $M$, show that $A/I \otimes_A M$ is isomorphic to $M/IM$.

The question is stated as in the title; the hint I am given is to "tensor the exact sequence" $0 \rightarrow I \rightarrow A \rightarrow A/I \rightarrow 0$, which I take to mean using that sequence ...
3
votes
0answers
83 views

Mapping cones and resolutions

Let me preface my question by acknowledging the vagueness of it. I am hoping to find some information in the form of references as opposed to a hard and fast solution. Suppose that $S=k[x_1, \dots, ...
1
vote
0answers
55 views

Construction of projective space of a graded ring

Let $S=k[x_0,x_1,x_2]$, $k$ an algebraically closed field and $ S_d=k[x_0^d,x_0^{d-1}x_1,\ldots,x_2^{d-1}x_1, x_2^d]$. $\mathbb{P}S_d $ is identified with the projective space $\mathbb{P}^{N_d} $, ...
1
vote
1answer
33 views

Unique Maximal ideal in a ring containing $\Bbb C$ is the nilradical.

This is a question from Algebra (Artin) ex.10.8.8 $R$ is a ring containing $\Bbb C$ as a subring. Assume $R$ is a finite dimensional vector space over $\Bbb C$ and that $R$ contains exactly one ...
5
votes
1answer
123 views

Ideal contained in the union of two ideals and a prime

Taken from Miles Reid "Undergraduate Commutative Algebra" p.35 ex. 1.12 b) Let $I,J_1,J_2 \subset A$ be ideals of a commutative ring $A$. Let $P$ be a prime ideal, then if $I \subset J_1 \cup J_2 ...
3
votes
1answer
33 views

Terminology regarding property of ideals

Is there a name for a property that only needs to be checked for either prime or maximal ideals in order to show that it holds for all ideals? An example would be being a principal ideal for which ...
1
vote
1answer
87 views

Hartshorne II Prop. 6.9

Prop. 6.9: Let $X \to Y$ be a finite morphism of non-singular curves, then for any divisor $D$ on $Y$ we have $\deg f^*D=\deg f\deg D$. I can not understand two points in the proof: (1) (Line 9) ...
0
votes
1answer
128 views

Noetherianess of a locally noetherian affine scheme without axiom of choice

I use the definition of a noetherian ring given by Qiaochu in this: A commutative ring is noetherian if, for any nonempty collection of ideals $\mathcal{I}$, there is some $I \in \mathcal{I}$ which is ...
7
votes
2answers
103 views

An integral domain that is Noetherian, integrally closed, but not one-dimensional

A Dededind domain is defined as an integral domain which is integrally closed, one-dimensional, and Noetherian. Also I know an equivalent characterization that a domain is Dedekidn if and only if ...
3
votes
0answers
71 views

Structure of the completion of Z[x] with respect to a maximal ideal

Can we explicitely describe the completion $R$ of $\mathbb{Z}[x]$ with respect to a maximal ideal $\mathfrak{m}\subset \mathbb{Z}[x]$? $(R,n)$ is a complete regular local ring of dimension two with ...
3
votes
1answer
92 views

On one-dimensional socles

Let $(R,m,k)$ be a regular local ring of dimension $n$. Let $b_1,\dots,b_n$ be a maximal $R$-sequence and define $J=(b_1,\dots,b_n)$. Let $y_1,\dots,y_n$ be a regular system of parameters of $R$ and ...
1
vote
1answer
66 views

Radical of a quotient ring

Let $F$ be a field. Suppose that $f\in F[X]$ can be written as $f=g^m$ for some separable polynomial $g\in F[X]$. Assume that $f$ has degree $\geq 1$ and $m\geq 1$. Define the quotient ring ...
7
votes
1answer
147 views

How to compute Ext over an exterior algebra

I found this question in several places (even on mathoverflow and mathstackexchange), but I never found a satisfying answer. Let $k$ be a field and $V$ a finite dimensional $k$-vectorspace. I ...
0
votes
1answer
49 views

Help in this proof on DVRs

I'm trying to understand this proof: Anyone could clarify the converse please? I really need help. Thanks a lot
1
vote
0answers
84 views

'Finitely generated modules' versus 'Finitely generated algebra'

I'm reading Commutative Algebra by Bourbaki, and I'm on chapter III of the book. I really think that the author made a typo there, but I'm not so sure. It's Proposition 1, and its Corollary on page ...
4
votes
1answer
167 views

A question on valuation overrings of a PID

Let $A$ be a PID and let $K$ be its quotient field. Let $V$ be a valuation ring of $K$ containing $A$ and assume $V\neq K$. Show that $V$ is a local ring $A_{(p)}$ for some prime element $p$. I ...
0
votes
1answer
115 views

Preimage of Maximal ideals

It is a common fact that when F is a surjective homomorphism from a commutative ring A to another B (with 1), we have inverse images of the maximals in B maximal in A. Could anyone be so kind to me ...
4
votes
0answers
226 views

Computing toric ideals via saturation

I have recently got interested in toric varieties and I have a question concerning their ideals. Let $A \in \mathbb{Z}^{m \times n}$ and $\ker A = \{ u \in \mathbb{Z}^n \; | \; Au = 0 \}$. For any $u ...
4
votes
2answers
98 views

Stalk of the quotient presheaf

Consider a sheaf of abelian groupS $\mathscr F$ over a topological space $X$. If $\mathscr F'$ is a subsheaf of $\mathscr F$ (over $X$), then we can construct the quotient presheaf $\mathscr ...
6
votes
1answer
279 views

Normalization of a quotient ring of polynomial rings (Reid, Exercise 4.6)

I solved all parts of Exercise 4.6 of the book Undergraduate Commutative Algebra of Miles Reid except the last one. Let $A=k[X]$ and $f\in A$ has a square factor but it is not a square polynomial ...
2
votes
1answer
173 views

Maximal ideals of rings which are finite dimensional vector spaces over $\mathbb C$.

If $K$ is a commutative ring which is a finite dimensional vector space over $\mathbb C$ what can we say about the maximal ideals of $K$? What can we say if instead of $\mathbb C$ we have some ...
1
vote
2answers
51 views

What are some concrete examples of ideals and modules where $I M = M$?

I'm trying to get more of intuition for the cloud of ideas surrounding the abstract Cayley-Hamilton theorem, Nakayama's lemma, etc., so I'd like to see some concrete examples. The problem is that the ...
1
vote
1answer
82 views

K-algebra isomorphic to a polynomial ring

I am trying to understand why this is true: Let K be a field, and let $K[a_1,\ldots,a_r]$ be a finitely generate $K$-algebra. If $a_1,\ldots, a_r$ are algebraically independent, then ...
0
votes
1answer
36 views

Can one find in this specific setting an extension of a given ring map?

All rings in this question are unitary and commutative and all maps are homomorphisms of commutative rings sending $1$ to $1$. Let $R$ and $S$ be regular local rings and let $$ ...
2
votes
1answer
100 views

Does the natural bijection between the set of prime ideals in A disjoint from S and Spec$(S^{-1}A)$ restrict to maximals?

I was studying rings of fractions, and I was wondering about the problem of restricting the canonical bijection (induced by retraction and extension of ideals) $\{p\in \text{Spec}(A) \mid p\cap ...
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votes
1answer
118 views

Finite product of algebras of finite type

Let $A$ be a commutative ring with unity. Let $B, C$ be $A$-algebras of finite type. Is $B \times C$ an $A$-algebra of finite type?
3
votes
2answers
113 views

If the ring map $f: A\rightarrow B$ is integral, fibres of $f^*$ are finite, then $f$ is finite?

If the ring map $f: A\rightarrow B$ is integral, i.e. $B$ is integral over the subring $f(A)$, and each fibre of the induced map $f^*: Spec(B)\rightarrow Spec(A)$ is a finite set, then should $f$ be ...
1
vote
1answer
52 views

Gelfand transform explicity

Let $T$ be a bounded normal operator. Let $A$ be the algebra generated by $T$ and $T^*$. What is the explicit Gelfand transform $G:A\to C(\sigma(T))$? My book says the image of $T$ is the ...
11
votes
1answer
141 views

$B\otimes_A A[x]=B[x]$

Let $A\rightarrow B$ be a homomorphism of commutative rings. Then $B\otimes_A A[x]\cong B[x]$ as $B$-algebras. How can one demonstrate this nicely, i.e. using universal properties alone and the Yoneda ...
2
votes
1answer
56 views

$K(A)\cong \mathbb Z$ for a PID $A$

In Atiyah and Macdonald, chapter 7, exercise 26, iii), it's required to show the Grothendieck group $K(A)\cong \mathbb Z$ for a PID $A$. By ii) of this problem, it's easy to show that $K(A)$ is ...
3
votes
1answer
65 views

Exact sequence induces exact sequences for free parts and torsion parts?

Let $A$ be a PID and consider the exact sequence of finitely generately modules over$A$: $$0\longrightarrow M' \overset{f}{\longrightarrow}M\overset{g}{\longrightarrow}M''\longrightarrow 0 \tag{1}.$$ ...
3
votes
2answers
130 views

Surjection from a Noetherian ring induces open map on spectra?

Let $A$ be a Noetherian ring, $f: A\rightarrow B$ a surjective ring map, then should the induced map on spectra $f^*: Spec(B)\rightarrow Spec(A)$ be an open map? In Atiyah and Macdonald, Chapter 1, ...
1
vote
0answers
121 views

About the “going-down property”

In Atiyah and Macdonald, Chapter 5, Exercise 10, there defines the so called "going-down property" (GDP). Then in Chapter 7, Exercise 24, the hint says, the ring map $f: A\rightarrow B$ has GDP ...
3
votes
0answers
68 views

understanding the language of an argument in Bruns and Herzog Cor. 2.3.10

The following question is heavily related to the context of theorems 2.3.9 and 2.3.10 in the text Cohen-Macaulay rings by Bruns and Herzog. My intention is to make this question as self-contained as ...
2
votes
2answers
84 views

If $N$ is an $R/I$-submodule of $M$ can we view $N$ as an $R$-submodule of $M$?

Let $R$ be an integral domain, $M$ an $R$-module and $I\subseteq \mathrm{Ann}(M)$ an ideal of $R$. $N$ is an $R/I$-submodule of $M$ (as $R/I$-module). Can we view $N$ as an $R$-submodule of $M$ ...
4
votes
2answers
142 views

Module of $R$-valued functions on an infinite set is not countably generated

Let $R$ be an integral domain and $X$ be an infinite set. Let $R^X$ be the set of all functions $f: X \rightarrow R$, viewed as an $R$-module in the usual manner: for $\alpha \in R$, $\alpha f: x \in ...
5
votes
1answer
161 views

Endomorphism of a local $k$-algebra inducing an automorphism modulo $m^2$ is an automorphism

The following is exercise 4.1 of Hartshorne's Deformation Theory, used in the proof given there of the sufficiency of the infinitesimal lifting criterion of smoothness: Let $(A,m)$ be a local ...
7
votes
2answers
114 views

If $k\subset R\subset k[x]$, then $R$ is Noetherian?

Is there a way to prove that any subring $R$ of the polynomial over a field $k$ such that $k\subset R$ is Notherian without appealing to integral extensions, Eakin-Nagata, etc.? The reason I ask is ...
2
votes
0answers
50 views

question on $\mathbb{Q} \otimes R[G]$ his maximal ideals, the action of a Galois group on it

Reasoning on a question a friend posed me, i've found a question in the following setup: Suppose you have a finite group G, now you can pass to the algebra $\mathbb{Q} \otimes R[G]$ where the second ...
3
votes
1answer
99 views

Prove $A_\mathfrak{p} \otimes_A B_\mathfrak{q} = B_\mathfrak{q}$, where $\mathfrak{q}$ prime in $B$

$\require{AMScd}$ Hi, I think I have the answer for this question, but I'm not sure if it's correct. So I would be very glad if someone could have a quick look through it. Let $A$, $B$ be ...
1
vote
0answers
43 views

refining a presentation of a quotient ring

Suppose that we have a commutative ring $R$ which i) is local ii) is the quotient of a regular ring and iii) it is a $k$-algebra, where $k$ is a field. I am trying to prove that in that case we can ...
2
votes
2answers
226 views

Evaluation morphisms of formal power series and nilpotent elements

Given a commutative ring $A$, and a finitely presented (associative) $A$-algebra $B$, show that a morphism of $A$-algebras $A[[x]] \longrightarrow B$ is given by evaluation at an nilpotent element $ ...