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

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3
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3answers
1k views

irreducibility of a polynomial in several variables over ANY field

The irreeducibility of a polynomial $f\!\in\!K[x_1,\ldots,x_n]$ in general depends on what the field $K$ is (for example, if $K=\mathbb{R}$, then $f=x_1^2+1$ is irreducible, but if $K=\mathbb{C}$, it ...
7
votes
1answer
184 views

Integral closure in the total ring of fractions

My question is linked with normalization of reduced algebraic curves that are not necessarily irreducible. Let $(A,\mathfrak{m})$ be a local reduced noetherian ring with Krull dimension $1$, let ...
1
vote
1answer
149 views

Queries on proof that every PID is a factorisation domain

I'm reading a proof from C. Musili's Rings and Modules that every PID is a factorisation domain. The author defines a factorisation domain as a commutative integral domain $R$ with a unit such that ...
2
votes
1answer
303 views

Finitely presented modules

I know that one can compute Fitting ideals of a finitely presented module (over a commutative ring with identity). However, are they the only invariants of such a module? In other words, my question ...
5
votes
1answer
314 views

definition of Krull dimension of a module

Let $R$ be a commutative ring with $1$. We know that the Krull dimension of $R$ is by definition the length of the longest chain of prime ideals of $R$. Now if $M$ is a $R$-module, the Krull ...
3
votes
2answers
407 views

$\operatorname{MaxSpec}(A)$ closed

If $A$ is an arbitrary commutative ring, is $\operatorname{MaxSpec}(A)$ closed as a subset of $\operatorname{Spec}(A)$? I wanted to think of a counterexample, but so far without success. I tried to ...
11
votes
1answer
459 views

Are all subrings of the rationals Euclidean domains?

This is a purely recreational question -- I came up with it when setting an undergraduate example sheet. Let's go with Wikipedia's definition of a Euclidean domain. So an ID $R$ is a Euclidean domain ...
3
votes
3answers
824 views

Height and minimal number of generators of an ideal

How can I determine the height and the least number of generators of the ideal $ I=(xz-y^2,x^3-yz,z^2-x^2y) \subset K[x,y,z] $? I tried to calculate the dimension of the vector space ...
0
votes
2answers
140 views

integral cohomology groups of an aspherical manifold are isomorphic to the integral cohomology groups of it's fundemental group

(of corresponding dimensions). how can I prove this? I think my main stumbling block is my general ignorance of group cohomology.
0
votes
1answer
466 views

Quotient of a local regular ring

How can I prove this: Let $A$ be a local regular ring with maximal ideal $\mathfrak m$ and $x \in \mathfrak m-\mathfrak m^2$. Then $A/(x)$ is a regular ring. Prove also that if $x\in\mathfrak ...
5
votes
2answers
178 views

Computing contractions of ideals in Macaulay2

Does Macaulay2 compute contractions of ideals under ring homomorphisms. Specifically, if $R\subseteq S$ is a ring extension (say polynomial rings over $\mathbb{Q}$ which can be specified in M2) and ...
6
votes
2answers
872 views

$\mathbb{Q}[x,y]/\langle x^2+y^2-1 \rangle$ is an integral domain

How can I prove that $\mathbb{Q}[x,y]/\langle x^2+y^2-1 \rangle$ is an integral domain? Also, I need to prove that its field of fractions is isomorphic to the field of rational functions ...
6
votes
1answer
364 views

Localizing a quotient ring $A/ \mathfrak{p}$

Let's assume $A$ is a commutative ring with $1$ and $\mathfrak{p} \subset A$ is a prime ideal. We shall consider $A/ \mathfrak{p}$ as an $A$-module, so there is a localization $(A/ ...
1
vote
1answer
167 views

Prime ideals in coordinate rings

Is there a way to characterise prime ideals in affine coordinate rings (i.e. quotients of polynomial rings). To be more specific, how can I say if principal ideals in such rings are prime or not in an ...
6
votes
2answers
622 views

Finite length modules over local rings

Let $A$ be a noetherian local ring and $M$ be an artinian and noetherian module over $A$. Does one know a priori anything about the structure of $M$? Furthermore: if one knows that the length of $M$ ...
2
votes
1answer
386 views

concrete isomorphisms of polynomial rings

Question 1: In A Singular Introduction to Commutative Algebra, page 222, there is written: How can I check that this isomorphism actually holds? I would really prefer a computational proof (using a ...
7
votes
5answers
947 views

Commutative rings without assuming identity

I was going through Exercises in Dummit&Foote, which does not assume identity in the definition of a ring, and reached the following exercise: Prove that in a Boolean ring ($a^2 = a$ for all ...
4
votes
1answer
468 views

The ideal $(x,y)$ is not a free $K[x,y]$-module

Given a field $K$ we have the polynomial ring $K[x,y]$ in $2$ variables, which is also a left module (over itself). How can we prove that the ideal $(x,y)$ is not a free module?
1
vote
2answers
71 views

a simple question about a local ring, and modules.

I put the paragraph to clarify because it is a vector space. I have a question with the proposition, I don´t know why he concludes the red line assertion, only knowing that there exist a surjective ...
12
votes
1answer
218 views

An inverse limit

Let $k$ be a field. Consider the inverse limit $\varprojlim k[x,y]/(y\cdot x^n)$. I wonder if there is a nice description of this ring? Geometrically, we look at the union of the line $y=0$ ...
0
votes
4answers
112 views

there exist an extension such that this element is a zerodivisor?

Everyone knows that if in a ring A a unit a $\in$ A can´t be a zerodivisor. But could also be possible that "a" not be a zero divisor ( i.e does not exist a nonzero x $\in$ A , such that $ax=0$) but ...
10
votes
2answers
2k views

Methods to check if an ideal of a polynomial ring is prime or at least radical

I am looking for methods to check whether a given ideal in $K[x_0,\dots,x_n]$ is prime. I mean something you can effectively use in some concrete non-trivial example. To be more explicit, I am working ...
2
votes
2answers
531 views

Artinian local $k$-algebra

I encountered the term "Artinian local $k$-algebra," where $k$ is a field. I think the author meant an Artinian local ring which is a $k$-algebra, but is it by any chance equivalent to a local ring ...
4
votes
1answer
575 views

Minimal and minimal associated prime ideals

Let $R$ be a Noetherian commutative ring and fix an ideal $I$. It is a classical result that $I$ admits a decomposition as the intersection of primary ideals, all of whose radicals are different and ...
2
votes
2answers
85 views

a basic question about the locus of a polynomial

First a little introduction and notation. I have a question with a words that the book says. Let k be a field. Let F $\in $ $k[x_1,...,x_n]$ We define the locus $V(F)$ of F , by $$ V\left( F \right) ...
4
votes
2answers
399 views

Every ideal of $K[x_1,\ldots,x_n]$ has $\leq n$ generators?

Is this true: Every ideal of $K[x_1,\ldots,x_n]$ is generated by some subset with $\leq n$ elements? It is true when $n=1$, since $K[x]$ is a PID. I'm trying to prove it is not true for $n\geq2$, ...
9
votes
3answers
2k views

If a ring is Noetherian, then every subring is finitely generated?

Let $R$ be a commutative ring with $1$, and let $K$ be a field. We know that $R$ is Noetherian iff every ideal of $R$ is finitely generated as an ideal. Question 1: If $R$ is Noetherian, is every ...
1
vote
2answers
243 views

Geometric view of rings

Intuitively, a map is determined on a region by its values on the points of a region. In the language of affine schemes, this suggests that $f(\mathfrak p_x)$ is determined by $\{f(\mathfrak m)\colon ...
3
votes
1answer
316 views

Computing with ideals: over $K$ or over $\mathbb{Q}\subseteq K$? does it matter? pt.2

Definitions: Let $R$ be any commutative ring with $1$, and let $S\subseteq R$ and $I\unlhd R$. The ideal generated by $S$ is $\langle S\rangle_R:=\bigcap\{J; S\subseteq J\unlhd R\}$, i.e. the ...
5
votes
2answers
588 views

Invertible modules are locally free of rank 1

The original context of the following question is something about coherent sheaves over noetherian schemes, but the question itself is purely (commutatively-)algebraic. The definition of an ...
0
votes
1answer
763 views

Dimension of the vector space of homogeneous polynomials.

Let $R$ be a polynomial ring with $n_k$ variables of degree $k$, for $1\leq k\leq m$. Is there a writeable formula to express the dimension of the vector space $R_l$ of degree $l$ homogeneous ...
4
votes
1answer
161 views

A step in showing that $\oplus_{i\in\mathbb Z}\mathbb Z$ is reflexive

I'm working on an assignment, in which in the end I'm trying to show that the countable direct product $\prod_{i\in \mathbb Z}\mathbb Z$ is not reflexive. I've already made some progress on the ...
7
votes
1answer
700 views

Is each power of a prime ideal a primary ideal?

I want to show that each power of a prime ideal is a primary ideal or I have to think about a counterexample?
5
votes
1answer
121 views

Finitely generated modules over a PID, and normed polynomials

I am Arthur from Belgium student in 2nd year of mathematics and I am repeating the exercises for Algebra I, but this one extra exercise I just can't solve: 14.$\text{}$ i) If $R$ is a PID, ...
6
votes
1answer
114 views

Showing a degree formula $\dim_{\mathbb{C}} R^{2} / L$

If $a,b,c,d$ are in $R=\mathbb{C}[t]$ and $ad-bc \ne 0$, $L= R(a,b)+R(c,d)$ in $R^{2}$. I want to show that $\dim_{\mathbb{C}}R^{2}/L = \deg(ad-bc)$. In a previous theorem it was shown that : ...
5
votes
1answer
545 views

Relation between primary ideal and prime ideal

We know that every prime ideal is primary ideal. But can we say, every primary ideal is a power of prime ideal? if it is not correct a counterexample. Thanks.
15
votes
2answers
2k views

What does a zero tensor product imply?

I'm trying to prove that for two finitely generated $A$-modules $M,N$ ($A$ being any cmmutative ring), the tensor product $M\otimes_A N$ is zero iff $\operatorname{Ann}(M)+\operatorname{Ann}(N)=A$. ...
1
vote
0answers
315 views

Does intersection equal product for infinitely many comaximal ideals?

If $I_1,...,I_n$ are comaximal ideals in a commutative ring, then $I_1\cdots I_n=I_1\cap \cdots \cap I_n$. Does this extend to infinitely many comaximal ideals? The proof I have seen uses induction, ...
6
votes
1answer
151 views

Principal ideals in completions of Zariski rings

Let $A$ be a Noetherian ring and $\mathfrak{a}$ some ideal contained in the Jacobson radical of $A$. Now $A$ is endowed with the $\mathfrak{a}$-adic topology, i.e. $A$ is a Zariski ring. If ...
5
votes
2answers
565 views

When does a polynomial generate a radical ideal?

A polynomial in a polynomial ring in one variable over a field generates a radical ideal iff it has no multiple roots. Is there a sufficient condition for a polynomial in several variables to ...
2
votes
1answer
114 views

$\mathfrak{a}$-adic topology on a submodule equivalent to the induced topology

Suppose $A$ is a Noetherian ring, $M$ a finite $A$-module and $N \subset M$ a submodule. Furthermore, $\mathfrak{a} \subset A$ is an ideal. Consider the $\mathfrak{a}$-adic topology on M, i.e. the ...
2
votes
2answers
566 views

radical of an ideal: $\sqrt{IJ}$ $=$ $\sqrt{I}\sqrt{J}$ $=$ $\sqrt{I}\cap\sqrt{J}$ $=$ $\sqrt{I\cap J}$?

Let $R$ be a commutative ring with $1$ and let $I,J\!\unlhd\!R$. The radical of an ideal $I$ is defined as $$\mathrm{rad}(I):=\sqrt{I}:=\{r\!\in\!R;\;\exists n\!\in\!\mathbb{N}: ...
1
vote
1answer
363 views

Torsion subgroup

Prove that in a finitely generated abelian group $G$ the torsion subgroup is a direct summand (from Scott, Group Theory). Clearly, the torsion subgroup is normal because $G$ is abelian, so we have to ...
1
vote
1answer
106 views

Surjective homomorphism on Laurent polynomial ring, part II

This question is similar to the question link, with a stronger hypothesis. Let $A= \mathbb C [t^2,t^{-2}]$ and $B= \mathbb C [t,t^{-1}]$. Consider $f\in B$ with the form ...
1
vote
1answer
121 views

Surjective homomorphism in Laurent polynomial ring.

Let $A= \mathbb C [t^2,t^{-2}]$ and $B= \mathbb C [t,t^{-1}]$. Consider $f\in B$ with the form $f=(t-a_1)(t-a_2)\cdots(t-a_k)$ where $a_i\in \mathbb C\setminus \{0\}$ and let $I$ be the ideal ...
9
votes
1answer
1k views

Ring of Polynomials is a Principal Ideal Ring implies Coefficient Ring is a Field?

I read this proof that if $D$ is an integral domain and $D[X]$ is a principal ideal domain, then $D$ is a field. My question is if the requirements can be relaxed a bit, namely: Is it true that ...
3
votes
1answer
101 views

Sum involving units of a ring.

Let $A$ be a commutative ring of characteristic zero. Let $a_1,a_2,a_3,a_4 \in A$ be units such that $a_i^k\ne a_j^k$ for $i\ne j$ and $k=1,2$. How to show that $a_1 a_2 a_3 + a_1 a_2 a_4 + a_1 a_3 ...
1
vote
2answers
142 views

Is an ideal the intersection of contractions of expansions to localizations at its minimal primes

Let $I$ be an ideal in a Noetherian ring $R$. Is $I=\cap(IR_P\cap R)$ where the intersection is taken over all minimal primes of $I$? If not, is it true if we assume $I$ has no embedded ...
3
votes
1answer
217 views

What does it mean to say a polynomial has an isolated singularity

In algebraic geometry, what does it mean when people say a polynomial $f$ has an isolated singularity at the origin?
4
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1answer
233 views

Question on proof from Eisenbud's Commutative Algebra.

I don't understand the second part of the proof of Corollary 4.8 (Nakayama's Lemma) in Eisenbud's Commutative Algebra. Let $I$ be an ideal contained in the Jacobson radical of a ring $R$, and let ...