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

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4
votes
1answer
347 views

Flat algebras and tensor product

All rings are commutative. Suppose $B$ is a flat $A$-algebra, and that $M$ and $N$ are flat $B$-modules. Is there a way to compare the two $A$-modules $M \otimes_A N$ and $M \otimes_B N$? Thanks
4
votes
4answers
368 views

Every set of $n$ generators is a basis for $A^{n}$

This is problem $15$ from Chapter $3$ of Atiyah's and Macdonald's book. Let $A$ be a ring let $F$ be the $A$-module $A^{n}$. Show that every set of $n$ generators is a basis of $F$. Here's the hint: ...
4
votes
1answer
292 views

Generating a regular sequence out of two

Here is the last problem of my final exam in "Commutative algebra" which I think, no one has solved it completely, today! Let $R$ be a commutative Noetherian ring. Let $a_1,\dots,a_n$ and ...
3
votes
2answers
47 views

If $M_*$ and $N_*$ are graded modules over the *graded* ring $R_*$, what is the definition of $M_* \otimes_{R_*} N_*$?

Quick question (hopefully): What is the correct definition of a tensor product of two graded $R_*$-modules and/or graded $R_*$-algebras $M_*$ and $N_*$ over the graded ring $R_*$? $M_* \otimes_{R_*} ...
3
votes
1answer
112 views

Is $\mathbb{Z}/p^\mathbb{N} \mathbb{Z}$ widely studied, does it have an accepted name/notation, and where can I learn more about it?

Fix a positive integer $p$, possibly prime. For each natural number $n$, there is a ring $\mathbb{Z}/p^n \mathbb{Z}$ together with a distinguished ring homomorphism $$\pi_n:\mathbb{Z} \rightarrow ...
3
votes
2answers
110 views

Direct sum of non-zero ideals over an integral domain

Let $R$ be an integral domain. Let $I$ and $J$ be non-zero ideals of $R$. Is this statement always true: $$R\oplus(I\cap J)\cong I\oplus J\ ?$$ I regarded the short exact sequence $0\to I\cap ...
3
votes
1answer
124 views

When a holomorphy ring is a PID?

I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes. Let $F$ be a function field over a finite field $\mathbb F_q$, $S$ a non empty set of places (possibly ...
3
votes
1answer
281 views

Other proof for tensor product equal to zero (without use Nakayama lemma and Jacobson radical)

Let $R$ is commutative local ring, $M$ is $R$-module, $N$ is $R$-module. If $M,N$ are finitely generated, how to prove: $M \otimes_R N=0$ if and only if $M=0$ or $N=0$. If we delete the ...
3
votes
1answer
105 views

Is Serre's $S_1$ condition equivalent to having no embedded primes?

Today I tried to prove that if a Noetherian ring $A$ satisfies Serre's $R_0$ and $S_1$ conditions, then $A$ is reduced. Now we recall that $R_0$ means the localization at any minimal prime is a field ...
3
votes
2answers
223 views

Scheme: Countable union of affine lines

Let $X$ be a countable union of $A_n$ ($n \in \Bbb{N}$), where $A_i$ are affine lines, i.e., $A_i=\operatorname{Spec}k[x]$, with $k$ algebraically closed field, such that $A_i$ meets $A_{i+1}$ in the ...
3
votes
1answer
73 views

Criterion for locally free modules of rank $1$

Let $R$ be a commutative ring and let $M$ be a finitely generated $R$-module such that the $R_{\mathfrak{p}}$-module $M_{\mathfrak{p}}$ is free of rank $1$ for every prime ideal $\mathfrak{p}$. Can we ...
3
votes
1answer
386 views

proof of the Krull-Akizuki theorem (Matsumura)

This set of questions refers to the proof of the Krull-Akizuki theorem given in Matsumura's Commutative Ring Theory, pages 84-85. For those who don't have the text, i will provide the details. The ...
3
votes
1answer
91 views

$I|J \iff I \supseteq J$ using localisation?

Let $R$ be a Dedekind domain. We know that for ideals $I$ and $J$ of $R$ that $I|J \iff I \supseteq J$. This fact is used for example in Marcus' Number Fields to show that we have unique factorisation ...
3
votes
1answer
143 views

Integral extension (Exercise 4.9, M. Reid, Undergraduate Commutative Algebra)

Let $k$ be any field and let $A = k[X,Y,Z]/(X^2 - Y^3 - 1, XZ - 1)$. How can I find $\alpha, \beta \in k$ such that $A$ is integral over $B = k[X + \alpha Y + \beta Z]$? For these values of ...
3
votes
2answers
215 views

a ring of fractions which has finitely many maximal ideals

Let $R$ be a commutative ring and $P_1,\ldots ,P_n$ be prime ideals of $R$. If $S=\bigcap_{i=1}^n (R\setminus P_i)$ then show that the ring of fractions $S^{-1}R$ has only finitely many maximal ...
3
votes
1answer
1k views

A surjective homomorphism between finite free modules of the same rank

I know a proof of the following theorem using determinants. For some reason, I'd like to know a proof without using them. Theorem Let $A$ be a commutative ring. Let $E$ and $F$ be finite free modules ...
3
votes
1answer
232 views

Example of a module whose support is not closed? [duplicate]

Possible Duplicate: The support of a module is closed? Is there a simple example of a module $M$ of a Noetherian commutative ring $R$ such that ...
2
votes
1answer
162 views

Is every ideal of $R$ a sum of a nilpotent ideal and an idempotent ideal?

I have a question concerning the following local ring: $$R=K[X_1,...X_n,...]/(X_1,X_2^2-X_1,...,X^2_{n+1}-X_n,...).$$ Is every ideal of $R$ a sum of a nilpotent ideal and an idempotent ideal? ...
2
votes
1answer
69 views

prove the inverse image of a maximal ideal is also a maximal ideal

The following is the problem: Let $K$ be a field, and $A$ is a commutative ring containing $K$. $\phi:A \rightarrow K[X]$ is a ring homomorphism which is the identity on $K$. If $M$ is a maximal ...
2
votes
2answers
545 views

Krull dimension of polynomial rings over noetherian rings

I want to prove the following theorem concerning Krull dimension: Theorem If $A$ is a noetherian ring then $$\dim(A[x_1,x_2, \dots , x_n]) = \dim(A) + n$$ where $\dim$ stands for the Krull ...
2
votes
1answer
60 views

An equivalent condition for zero dimensional Noetherian local rings

Let $(A,m)$ be a Noetherian local ring. Why $A$ is zero dimensional if and only if a power of $m$ is $\{0\}$ ?
2
votes
5answers
534 views

A finite dimensional algebra over a field has only finitely many prime ideals and all of them are maximal [duplicate]

Let $K$ be a field and let $R$ be a $K$-algebra with unity which is finite dimensional as a $K$-vector space. Prove that $R$ has only finitely many prime ideals all of which are maximal. (Hint: ...
2
votes
1answer
113 views

Is the integral closure of local domain a local ring?

Suppose $A$ is a local domain, with field of fractions $K$, let $A'$ be the integral closure of $A$ in $K$, is $A'$ a local ring?
2
votes
1answer
919 views

Is any UFD also a PID?

Is there any counterexample that will disprove that every unique factorization domain (UFD) is also a principal ideal domain (PID)? I mean, any PID is a UFD, does the converse hold? Thanks in ...
2
votes
2answers
884 views

What is a lift?

What exactly is a lift? I wanted to prove that for appropriately chosen topological groups $G$ we can show that the completion of $\widehat{G}$ is isomorphic to the inverse limit ...
2
votes
2answers
792 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 ...
2
votes
1answer
620 views

Minimal generating sets of free modules, and endomorphisms of free modules

I know that it seems very loose as a title but I hope this post will be beneficial to all the forum members. One thing I like about free modules is that they help one define maps directly as we do in ...
1
vote
2answers
138 views

Is a specific ring extension $B$ of $K[x,y]$ integrally closed? separable?

Let $A=K[x,y] \subset K[x,y][w]=B$, $K$ is a field of characteristic zero, $w$ is integral over $A$ (so $B$ is a f.g. $A$-module), but $w$ is not in the field of fractions of $A$, and $B$ is an ...
1
vote
1answer
427 views

Homogeneous and maximal ideal in a $\mathbb Z$-graded ring

Is Exercise 2.8 from Marley's notes on "GRADED RINGS AND MODULES" true? Exercise 2.8: Let $R$ be a graded ring and $M$ a homogeneous maximal ideal of $R$. Prove that $M =…⊕R_{-1}⨁m_0⨁R_1⨁…$, ...
1
vote
1answer
484 views

calculating minimal prime ideals

Is there a "general approach" to determine the minimal prime ideals over an ideal $J$? I checked some books and didn't find a general approach. Maybe the theory of Gröbner bases is related to these ...
1
vote
1answer
59 views

$\mathfrak{a}(M/N) = (\alpha M + N) / N$

I want to prove $\mathfrak{a}(M/N) = (\mathfrak{a}M + N) / N$, where $M$ is an $A$-module and $\mathfrak{a}$ is an ideal of $A$. There will be many ways, for example, define a map ...
0
votes
1answer
42 views

A be an affine K-algebra and f be a non-zero divisor of A then can one say that dim A=dim A_f

Let $A$ be an affine $K$-algebra and $f$ be a non-zero divisor of $A$ then can one say that $\dim A=\dim A_f $ ? What I proved that if $A$ is an affine domain and $f$ is a non-zero element in ...
12
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 ...
10
votes
1answer
461 views

Extending Herstein's Challenging Exercise to Modules

Anybody who has worked through Herstein's Topics in Algebra might remember Exercise 26 of Section 2.5 (in second edition): If $G$ is an abelian group containing subgroups of order $m$ and $n$, ...
10
votes
1answer
323 views

Conditions for $\sqrt{\mathfrak{a + b}} = \sqrt{\mathfrak{a}} + \sqrt{\mathfrak{b}}$

Let $A$ be a commutative ring with identity and, $\mathfrak{a}$ and $\mathfrak{b}$ ideals.I'm trying to find sufficient and necessary conditions for $\sqrt{\mathfrak{a + b}} = \sqrt{\mathfrak{a}} + ...
10
votes
1answer
444 views

Primes in a Power series ring

Let $\mathbb Z$ be the ring of rational integers. Consider the power series ring $\mathbb Z[[x]]$. It is known that $\mathbb Z[[x]]$ is unique factorization domain. What are the primes in $\mathbb ...
10
votes
2answers
197 views

Can an ideal in a commutative integral domain be its own square?

If $I$ is a non-zero proper ideal of a commutative integral domain, is it possible for $I$ to be its own square?
10
votes
2answers
956 views

When is the integral closure of a local ring also a local ring?

Suppose $A$ is a normal local domain contained in a field $K$. Suppose $B$ is the integral closure of $A$ in $K$. Under what conditions on $A$ is $B$ local?
9
votes
1answer
161 views

Chinese remainder theorem as sheaf condition?

The chinese remainder theorem in its usual version says that for a finite set of pairwise comaximal ideals $R/\bigcap _jI_j\cong \prod _j R/I_j$. In the binary case, the following general statement ...
9
votes
1answer
214 views

Weil does not imply Cartier on variety $X$.

Show that the divisor $D$ defined by $a = b = 0$ in the variety $X \subset \mathbb{A}^4$ defined by $ad - bc = 0$ $($the cone on a smooth quadric surface$)$ is not locally principal. My attempt ...
9
votes
2answers
339 views

$K[x_1, x_2,\dots ]$ is a UFD

I wonder about how to conclude that $R=K[x_1, x_2,\dots ]$ is a UFD for $K$ a field. If $f\in R$ then $f$ is a polynomial in only finitely many variables, how do I prove that any factorization ...
9
votes
2answers
666 views

Nilradical of polynomial ring

Let $R$ be a commutative ring. The nilradical $\text{nil}(R)$ is the set of all nilpotent elements, and it is the intersection of all the prime ideals of $R$. Is the following true in the polynomial ...
9
votes
2answers
454 views

Why is a variety over a non-alg. closed field a hypersurface?

Exercise $3$ on page $8$ of Kunz's Introduction to Commutative Algebra and Algebraic Geometry is as follows: If the field $K$ is not algebraically closed, then any $K$-variety $V \subset A^n(K)$ can ...
8
votes
1answer
2k views

Finitely generated modules over a Noetherian ring are Noetherian

I'm trying to prove that if the ring $R$ is Noetherian then every finitely generated $R$-module is Noetherian. First of all, it is known that every module is a homomorphic image of a free module, ...
8
votes
4answers
867 views

Are bimodules over a commutative ring always modules?

Let $R$ be a commutative ring. It is true that every module over $R$ is an $(R,R)$-bimodule. Is the converse true? In other words is it possible that there is an $R$-module where left multiplication ...
8
votes
2answers
546 views

Approximation Lemma in Serre's Local Fields

Let $A$ be a Dedekind domain, and let $K$ be its field of fractions. In Serre's Local Fields, the following Lemma is stated. Approximation Lemma Let $k$ be a positive integer. For every $i$, ...
8
votes
3answers
250 views

Proof that a certain derivation is well defined

I have spent several hours on this, apparently straightforward issue. This is with reference to page 17 in the following notes http://www.math.lsa.umich.edu/~hochster/615W10/615.pdf Suppose, $R$ is ...
8
votes
2answers
931 views

Connectedness in Zariski topology

How to show that, for example, the Zariski topology of a cyclic group ring (thus $\mathbb{Z}[\rho]/(\rho^n-1)$) is connected? Does this still hold for an Abelian group? Or in general, how do we ...
7
votes
1answer
162 views

Can an element in a Noetherian ring have arbitrarily long factorizations?

Suppose $R$ is a Noetherian ring. Is it possible that an element $r\in R$ have arbitrarily long factorizations? That is, for all $n>0$, is there a factorization $r=a_{1n}a_{2n}\cdots a_{nn}$ such ...
7
votes
3answers
360 views

Isomorphism of quotients of powers of maximal ideals

Let $R$ be an integral domain, and $\mathfrak{m}$ a maximal ideal of $R$. Let $R_\mathfrak{m}$ denote the ring localized at $\mathfrak{m}$, and let $\mathfrak{m}_\mathfrak{m} = ...