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

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14
votes
2answers
1k views

Inverse Image of Maximal Ideals

Given a map of commutative rings with unit, it is often the case that the inverse image of a maximal ideal is not maximal. For example, consider the inclusion $\mathbb{Z} \subseteq \mathbb{Q}$. ...
4
votes
1answer
349 views

Two definitions of exactness

Given a functor $F:A\to B$ of abelian categories we may say that $F$ is left exact if it maps exact sequences to left exact sequences, and similarily for right. For arbitrary categories, we may say ...
6
votes
2answers
399 views

injective $R$-module homomorphism vs. injective ring homomorphism

The following question has been lingering in my mind for months. Let $R$ be a non-zero commutative ring with $1$. Consider $\phi : R^n \rightarrow R^m$, 1) as an injective $R$-module homomorphism. ...
0
votes
1answer
69 views

Flat ring maps and composition

Suppose $A$, $B$ and $C$ are commutative noetherian rings. Suppose we are given maps $f:A\to B$ and $g:B \to C$. Suppose further that $f$ is flat and that $g \circ f$ is also flat. Does it follows ...
1
vote
5answers
290 views

Maximal ideal of Dedekind domain

Let $\Lambda$ be a Dedekind domain and $\mathcal{m}$ be a maximal ideal of $\Lambda$. Is it possible that $\mathcal{m}=\mathcal{m}^2$? If not, how can I prove it?
4
votes
1answer
618 views

Flat not projective, projective not free [duplicate]

I am looking for examples of a flat but not projective module, and of a projective but not free module.
11
votes
3answers
1k views

Primary ideals of Noetherian rings which are not irreducible

It is known that all prime ideals are irreducible (meaning that they cannot be written as an finite intersection of ideals properly containing them). While for Noetherian rings an irreducible ideal is ...
4
votes
1answer
168 views

Are dualizing modules stable under localization

Let $(R,m,k)$ be a (noetherian) regular local ring of depth=dimension $d$, and let $D$ be a dualizing module for $R$ (say, the injective envelope of $R/m$). Then is $D_p$ dualizing for $R_p$ for ...
5
votes
1answer
99 views

What is the right categorial definition of localisation of a module

Let $A$ be a ring, $S$ be a multiplicative subset, $M$ an $A$-module. let $\iota : A \to S^{-1}A$ be the map $a \mapsto a/1$. $\iota$ can be defined categorically as an initial object in the category ...
2
votes
1answer
172 views

An identity on symmetric polynomial

In the polynomial algebra $\mathbb{C}[X_1, X_2,\ldots, X_n]$, we define a set of symmetric polynomials as follows $h_i(X_k, X_{k+1}, \ldots, X_n)$ = sum of all monomials of total degree $i$ in the set ...
12
votes
4answers
5k views

Example of modules that are projective but not free; torsion-free but not free

Free modules are projective, and projective modules are direct summand of free modules. Is there any example of projective modules that are not free? (I know this is not possible for modules of ...
2
votes
0answers
143 views

A question about localization

In localization of a ring $R$ or a module $M$ over $R$ at a multiplicative subset $S$ of $R$, we define an equivalence relation on $R\times S$ or $M\times S$ and define addition and multiplication on ...
0
votes
1answer
51 views

Applications of the Formal Laurent Lattice

Attach a (Laurent) monomial weight $x_1^{i_1} \cdots x_n^{i_n}$ to each point $(i_1, \dots, i_n)$ of $\mathbb{Z}^{n}$ and call it $\mathbb{Z}^{n}[x_1, x_{1}^{-1}, \dots, x_n, x_n^{-1}]$. Does this ...
2
votes
1answer
548 views

Minimal number of generators of an ideal in a Noetherian local ring

I have seen the phrase "minimal number of generators of an ideal" (in a Noetherian local ring) several times. I am unable to see how this is a well defined. Explicitly, how do we show, if ...
14
votes
3answers
1k views

Complement of maximal multiplicative set is a prime ideal

Let $R$ be a commutative ring with identity. I've been trying to prove the following: If $S \subset R$ is a maximal multiplicative set, then $R \setminus S$ is a prime ideal of $R$. I have spent ...
27
votes
5answers
2k views

Why should I care about adjoint functors

I am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image and ...
6
votes
1answer
177 views

restriction map in a Sheaf of $\mathcal{O}_X$ modules

Let $R$ be a commutative ring with identity and $M$ be an $R$-module. I have trouble understanding the restriction map in the definition of the sheaf of $\mathcal{O}_X$ modules. Explicitly, let ...
2
votes
2answers
133 views

Independence of valuations

I'm reading Matsumura's proof on independence of valuation (p.87, "Commutative Ring Theory"), and I understand most of the proof, except the last bit about principal ideal ring, and if anyone could ...
2
votes
1answer
1k views

Problem on idempotent finitely generated ideal

I have a question. Could you please help me to solve this? Thanks in advance Let $\mathfrak a$ be a finitely generated ideal of $A$, commutative ring with identity, such that $\mathfrak a^2 = ...
4
votes
2answers
124 views

Proving this algebra fact

The Nullstellensatz for $\mathbb{C}[x_1, \ldots, x_n]$ gives a dictionary between radical ideals and varieties, which makes the following assertion obvious: a radical ideal in $\mathbb{C}[x_1, \ldots, ...
4
votes
1answer
313 views

I-adic completion of a ring

Let $R$ be a ring, $I$ an ideal. According to Atiyah-Macdonald, if $R$ is Noetherian, then, we have $\hat{I}=\hat{R}I$ where hat denotes $I$-adic completion of $R$ and (I presume) $\hat{I}$ denotes ...
17
votes
2answers
1k views

Atiyah-Macdonald Exercises 5.16-5.19

I have solutions to Exercises 5.16–5.19 in Atiyah–Macdonald's Introduction to Commutative Algebra, but not in the order desired; I find myself needing later exercises to do earlier ones, ...
2
votes
1answer
125 views

Does $\forall n,d\!\in\!\mathbb{N}$ $\forall$ field $\mathbb{F}$ exist an irreducible $f\!\in\!\mathbb{F}[x_1,\ldots,x_n]$ of degree $d$?

how can one show (hopefully in an elementary manner) that there exist irreducible polynomials of arbitrary degree and number of variables over arbitrary field? thank you P.S. induction? EDIT: ehm, ...
2
votes
1answer
188 views

About Krull domain

If $A$ is a Krull domain in its field of fractions $F$, and if $F'\subset F$ is a subfield, then $A\cap F'$ is a Krull domain. But is it necessarily a Krull domain of $F'$? That is, is ...
7
votes
2answers
388 views

A question related to Krull-Akizuki theorem

Let $(R,m)$ be a D.V.R with field of fraction $K$ and $L$ any finite algebraic field extension of $K$. Suppose $\bar{R}$ is the integral closure of $R$ in $L$. Then it is well known that $\bar{R}$ is ...
4
votes
3answers
226 views

what is the definition of a line in $\mathbb{P}^n(k)$ + how to compute the hilbert polynomial of two intersecting lines?

(1) I have never studied any projective/affine geometry or algebraic curves. I'd like to see a clear definition of a line in the projective space $\mathbb{P}^n(k)$, since I need it for my algebraic ...
2
votes
2answers
94 views

Sub-discrete valuation ring?

Let $A$ be a DVR in its field of fractions $F$, and $F'\subset F$ a subfield. Then is it true that $A\cap F'$ is a DVR in $F'$? I can see that it is a valuation ring of $F'$, but how ...
7
votes
1answer
322 views

Projective module over $R[X]$

Let $(R,m)$ be commutative noetherian local ring with unity. Suppose $P$ is a finitely generated projective module over $R[X]$ of rank $n$ . Is $P$ free? If not, what is the counter example?
17
votes
3answers
917 views

Motivation for Eisenstein Criterion

I have been thinking about this for quite sometime. Eisentein Criterion for Irreducibility: Let $f$ be a primitive polynomial over a commutative unique factorization domain $R$, say $$f(x)=a_0 + ...
1
vote
1answer
84 views

How to show that the projective dimension is infinite

How can we show that the projective dimension of the $\mathbb{Z}/p^2 \mathbb{Z}$-module $\mathbb{Z}/p \mathbb{Z}$ is infinite?
2
votes
3answers
719 views

Is it true that, in a Dedekind domain, all maximal ideals are prime?

Is it true that, in a Dedekind domain, all maximal ideals are prime?
10
votes
2answers
269 views

A commutative group structure on $R\times R$ for a ring $R$

Let $R$ be a commutative ring. The Cartesian square $A=R\times R$ is endowed with the operation $(a_1,b_1)\circ(a_2,b_2)=(a_1+a_2,b_1+b_2+a_1a_2^2+a_1^2a_2)$ which turns $A$ into a commutative ...
3
votes
1answer
382 views

projective dimension

All rings and algebra in this question are commutative and contains unity. Suppose $M$ is an $A$ module and $A$ a $R$ algebra. If $pd_R(M) < \infty$, then will that imply $pd_A(M) < \infty$? ...
4
votes
0answers
235 views

intersection multiplicity and partial derivatives of algebraic curves

this will probably be an easy-to-answer and a not-well-posed question, since I'm a total beginner in the field, but here goes: Let $V(F)$ and $V(G)$ be two projective curves in $\mathbb{P}^2$ ...
2
votes
2answers
158 views

For a ring R with an ideal I, the I-adic topology makes R into a topological ring

Let $R$ be a commutative ring with identity. Let $I$ be an ideal of $R$. Suppose, we give a topology on $R$ where a set is open if and only if it is a union of cosets of powers of $I$. Then, is $R$ a ...
5
votes
1answer
99 views

Locally a domain and connected implies a domain

Let $R$ be a commutative ring with unit. Let $R_p$ be a domain for all $p\in SpecR$ and let $SpecR$ be connected. Is it true that $R$ is a domain or can someone provide a counterexample. Note here ...
7
votes
1answer
211 views

Why is UFD a Krull domain?

Matsumura mentions this as if it is obvious, and I can't find this result anywhere. Am I missing something obvious here?
11
votes
2answers
277 views

Does inclusion of a ring into a polynomial ring induce a closed map on prime spectra?

Let $A$ be a commutative (unital) ring, and $A[x_1,\ldots,x_n]$ a polynomial ring over it in some finite number of variables. The inclusion $i\colon A \hookrightarrow A[x_1,\ldots,x_n]$ induces (by ...
3
votes
1answer
122 views

Find two submodules $E_1$ and $E_2$ such that $\textrm{Ass}(E_1) \cup \textrm{Ass}(E_2) \subsetneq \textrm{Ass}(E_1 + E_2)$

I'm looking for a module $E$ of a commutative ring $A$ with two submodules $E_1$ and $E_2$ such that the associated primes of $E_1 + E_2$ strictly contain the union of the associated primes of $E_1$ ...
2
votes
2answers
148 views

Local complete intersections and the cotangent module

Let $A$ be a (commutative) noetherian ring, and let $I \subseteq A$ be an ideal. It is not hard to show that if $I$ is generated by a length $n$ regular sequence, then the $A/I$-module $I/I^2$ is a ...
2
votes
1answer
120 views

Subsequences of regular sequence

I was trying to answer this question - whether a subsequence of a regular sequence is regular in a Noetherian ring which is not local. In the local case, regular sequences can be permuted and so a ...
10
votes
1answer
267 views

Minimal spectrum of a commutative ring

Can anyone explain to me why the minimal prime ideals of a commutative ring (with the subspace topology inherited from the Zariski topology) form a totally disconnected space, or give a reference? I ...
2
votes
2answers
402 views

Example of an injective module

I can't find an example of a countable injective module over a non-Noetherian ring.
6
votes
3answers
418 views

Why is the prime spectrum of a domain irreducible in the Zariski topology

How does one show that the prime spectrum of a domain is irreducible in the Zariski topology?
2
votes
1answer
373 views

Max Noether's $AF + BG$ theorem

Wikipedia tells me about Max Noether's $AF + BG$ theorem but only gives one reference and one external link. I've had a look at the MathWorld link but it seems to be an entirely geometric formulation ...
10
votes
1answer
372 views

Finitely generated modules over PID

Let $A$, $B$, $C$, and $D$ be finitely generated modules over a PID $R$ such that $A\oplus $ $B$ $\cong$ $C\oplus $ $D$ and $A\oplus $ $D$ $\cong$ $C\oplus $ $B$ . Prove that $A$ $\cong$ $C$ and $B$ ...
2
votes
2answers
157 views

why a reduced ring (commutative with 1) can be embedded into a sum of integral rings?

the question is exactly "why a reduced ring (commutative with 1) can be embedded into a sum of integral rings?" Is this simply because in the normalization process we can have many irreducible ...
5
votes
1answer
92 views

In what generality does the following isomorphism involving tensors and homs hold?

Let $R$ be a CRing and let $M,N$ be $R$-modules. Let $M^*:=Hom_R(M,R)$. I have seen the following isomorphism asserted in the case where $R$ is a field and $M$ and $N$ are f.g. vector spaces: ...
1
vote
1answer
117 views

In what generality does the second argument of Hom distribute over tensor?

Let $R$ be a commutative ring, and let $M,N,P$ be $R$-modules. In what generality can we say that $Hom_R(M,N\otimes_R P)\cong Hom_R(M,N)\otimes_R Hom_R(M,P)$. This is true in a cartesian monoidal ...
2
votes
1answer
268 views

Tensoring a monomorphism of free modules with an identity map

Suppose $R$ is a commutative ring, $f\colon F_1\to F_2$ is a homomorphism of free modules, and $M$ is an $R$-module. If $f$ is a surjective homomorphism, then $f\otimes_R \mathrm{id}_M$ is ...