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

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6
votes
3answers
194 views

Finding the ideals in a ring of fractions

I am dealing with the ring $$R=\left\{\frac{a}{b} \mid a,b\in\mathbb{Z}\mbox{, $b$ is not divisible by 3}\right\}$$ with addition and multiplication as defined in $\mathbb{Q}$ and I'm trying to find ...
3
votes
1answer
104 views

Commutative $ \mathbb{C} $-algebra with involution.

The following is a problem in the Professor’s lecture notes; I don’t know how to prove it: A complex $ ^{*} $-algebra $ A $ is commutative if and only if the set of self-adjoint elements of $ A $ ...
6
votes
0answers
180 views

Punctured plane algebraic over a finite field?

I've been asked to prove, by my algebraic geometry teacher, that the punctured affine plane $\mathbb{A}_k^2 \backslash \{0,0\}$ is not an algebraic set, i.e. is not the zero set of any set of ...
13
votes
2answers
576 views

Structure of Finite Commutative Rings

Is every finite commutative ring $A$ a direct product of finite algebras over $\mathbb Z/p^n$?
1
vote
1answer
157 views

Boolean ring. Representation as direct product?

I am reading Atiyah, Macdonald's book "Commutative algebra". There is an exercise, which states that that if a ring with identity has idempotent element $\ne 0,1$, then the ring is a direct product ...
5
votes
3answers
95 views

A question about prime ideals

suppose $p$ a prime ideal. If $p=a \cap b$, $a,b$ ideals. Is it true that we have to have $p=a$ or $p=b$?
1
vote
1answer
80 views

All associated primes appear in a series of submodules

This is essentially Ex VI.4.8 of Algebra: Chapter 0. Let $R$ be a commutative ring and $M$ be an $R$-module. Define \begin{equation} \operatorname{Ann}_{R}(m)=\{r\in R:rm=0\} \end{equation} for each ...
7
votes
1answer
151 views

Projective objects in the category of rings

What are the projective objects in the category of rings with identity ? Remarks: The only projectives I could find so far are $\{ 0\}$ and $\mathbb{Z}$. If $R$ is projective and ...
1
vote
1answer
2k views

The (Jacobson) radical of modules over commutative rings

Let $M$ be a module over a commutative ring $R$. Let $\Omega$ be the set of all maximal ideals of $R$. Prove that $\operatorname{Rad}(M)=\bigcap_{\mathfrak m\in \Omega}\mathfrak mM$, where ...
5
votes
3answers
139 views

Lifting isomorphisms of fields to automorphisms of polynomial rings

Let $L$ be a field and $\alpha, \beta$ algebraic over $L$ such that $L(\alpha)\cong L(\beta)$. If $q(t)$ and $p(t)$ are the minimum polynomials of $\alpha$ and $\beta$, respectively, does it follow ...
1
vote
0answers
219 views

An Algorithm to Find the Generators of the Radical of a Monomial Ideal

Working over $R=\mathbb{C}[x_1,...,x_n]$, I'm given a ring homomorphism with $i\in{1,...,n}$ and $t\in \mathbb{C}$. $\phi_{i,t}(x_j)=x_j$ for $j\neq i$ to themselves. From this I've proven that an ...
13
votes
5answers
667 views

Irreducibility of Polynomials in $k[x,y]$

I'm working through some Hartshorne problems and have noticed that in order to do certain problems properly one must prove a given polynomial $f\in k[x,y]$ is irreducible. For example, in problem ...
0
votes
1answer
90 views

Isomorphic completed modules, that were not isomorphic before completion

Let $M$ and $N$ be $R$-modules. Suppose we complete them with respect to an ideal $\frak{m}$ of $R$. If we have $$M^\wedge_\mathfrak{m} \simeq N^\wedge_\mathfrak{m}$$must if be the case that $M ...
7
votes
1answer
178 views

Prime elements in $\mathbb{Q}[[X,Y,Z]]$ whose status as an infinite series is unchanged by arbitrary multiplication

Let's suppose $R$ is the ring $\mathbb{Q}[[X,Y,Z]]$. I'm interested in finding power series $f(x,y,z) \in R \setminus \mathbb{Q}[X,Y,Z]$ which are, first of all, prime elements in $R$, but also ...
4
votes
1answer
335 views

When is the maximal ideal of a zero-dimensional local non-noetherian commutative ring nilpotent?

Let $R$ be a non-Noetherian local commutative ring with identity such that it is of Krull-dimension zero. I am wondering if there are conditions which will force the maximal ideal to be nilpotent.
3
votes
1answer
136 views

Existence of a ring automorphism

Let $A$ be a commutative ring, $B$ a commutative ring that is also an $A$-algebra of finite presentation. Let $f_1$ and $f_2$ be two elements of $B$ such that $(f_1) + (f_2) = B$ and as $A$-modules, ...
0
votes
2answers
114 views

An example of a divisible ideal

Let $R$ be a commutative ring (other than a field) with identity. I am looking for an example of a divisible ideal of $R$.
1
vote
0answers
29 views

Modules who have equal support

What can we say about two R-modules, if we know that there supports are equal? What if we know that this modules are Abelian groups? (Z- modules)
2
votes
1answer
169 views

Tensors in math and physics

I know how tensor product f two modules is defined in communtative algebra. But there is also a concept of tensors used in physics. Are these two concepts related? If yes, can someone explain me ...
2
votes
2answers
169 views

Basic Questions on Tensor Product of Modules

Let $M$, $N$ be $A$-modules where $A$ is a commutative ring with $1$. I am studying $M \otimes_{A} N \simeq A^{(M \times N)}/(4 \text{ generators that makes $\otimes$ bilinear})$, as module ...
1
vote
2answers
100 views

Associated prime ideals of $\mathbb C^3$

Let $$A=\begin{pmatrix} 3&2&0 \\0&1&-1\\1&1&1\end{pmatrix}. $$The $\mathbb C$-vector space $\mathbb C^3$ becomes a $\mathbb C[T]$-module via ...
2
votes
2answers
223 views

Associated prime ideals in F[X,Y]

Let $F$ be a field and $R=F[X,Y]$ be the polynomial ring of two variables over $F$. Let $I\subset R$ be the ideal generated by $X^2$ and $XY$, find the associated prime ideals of $R/I$.I'm really ...
2
votes
1answer
119 views

$X_K$ normal imply $X$ normal

In Algebraic Geometry and Arithmetic Curves of Qing Liu, I have two problems with the lemma 4.1.18 (page 119). The lemma is so: let $\mathcal{O}_K$ a DVR (uniformizing parameter $t$) with residue ...
0
votes
1answer
60 views

The point $(0,1)$ on coordinate axis, and any point on a single line look the same. [whereas $(0,0)$ is not]

I am trying to show that $S_{1}^{-1}[K[X,Y]/(XY)]$ is isomorphic as rings to $S_{2}^{-1}K[X]$, where $K$ is a field, $S_{1}=(k[X,Y]/(XY))\setminus(\bar{X},\bar{Y}-1)$, and ...
4
votes
3answers
153 views

Disjoint Union of Spectra

The following it from Atiyah-Macdonald's Introduction to Commutative Algebra, exercise 1.22. Apparently this should be very easy, my apologies for asking. I have been stuck for almost 1 day and I ...
1
vote
3answers
136 views

The relation between been the quotient ring of a prime ideal and its localization

Let $A$ be a ring and $\mathfrak{p} \subset R$ be a prime ideal. Set $A_\mathfrak{p}=R[U^{-1}]$, where $U= A-\mathfrak{p}$. What is the relation between $A/\mathfrak{p}$ and $A_\mathfrak{p}$? My ...
9
votes
2answers
1k views

Tensor product of domains is a domain

I'm reading Milne's Algebraic Geometry course notes, version 5.22, as a companion to an algebraic geometry course I'm taking now. Proposition 4.15 states: Let $A$ and $B$ be $k$-algebras, which are ...
2
votes
1answer
88 views

Vanishing Ideal of a Linear Subspace

Let $F$ be an infinite field. Let $V$ be a subspace of $F^n$. Let $V^{\perp}$ be the set of all linear functionals $F^n \rightarrow F$ that vanish on $V$. Let $I(V)$ be the vanishing ideal of $V$, ...
9
votes
2answers
2k views

Why is the localization at a prime ideal a local ring?

I would like to know, why $ \mathfrak{p} A_{\mathfrak{p}} $ is the maximal ideal of the local ring $ A_{\mathfrak{p}} $, where $ \mathfrak{p} $ is a prime ideal of $ A $ and $ A_{\mathfrak{p}} $ is ...
5
votes
2answers
397 views

Is every Artinian module over an Artinian ring finitely generated?

I know that if $R$ is Artinian, then a f.g. $R$-module is Artinian. Is f.g. a necessary condition?
4
votes
2answers
368 views

Motivation behind the definition of Zariski tangent space

Intuitively I think of tangent space at a point as the set of all points lying in the tangent plane passing throug that point. Here is the definition of Zariski tangent space Let X be an ...
0
votes
1answer
40 views

Two different definitions of Derivation

Here are two definition of derivations Definition 1 Let $A \rightarrow B$ be a homomorphism of commutative algebras, and $M$ a $B$-module. We define the derivations ...
28
votes
3answers
951 views

Ideals of $\mathbb{Z}[X]$

Is it possible to classify all ideals of $\mathbb{Z}[X]$? By this I mean a preferably short enumerable list which contains every ideal exactly once, preferably specified by generators. The prime ...
0
votes
2answers
28 views

Does the concept of localizing at an extension of a prime ideal make sense?

If $A,B$ are commutative rings with $1$, $p$ is a prime ideal in $A$ and $f:A\rightarrow B$ makes $B$ an $A$-algebra, I want to know if it is possible to define the localization $B_p$ of $B$ at the ...
19
votes
3answers
671 views

What does the topology on $\operatorname{Spec}(R)$ tells us about $R$?

Let $R$ be a commutative ring with a unit. $\newcommand{\spec}{\operatorname{Spec}}\spec(R)$ denotes the set of all prime ideals in $R$, and it can be topologized using the Zariski topology. Last ...
6
votes
4answers
331 views

Show that the ideal of all polynomials of degree at least 5 in $\mathbb Q[x]$ is not prime

Let $I$ be the subset of $\mathbb{Q}[x]$ that consists of all the polynomials whose first five terms are 0. I've proven that $I$ is an ideal (any polynomial multiplied by a polynomial in $I$ ...
-1
votes
1answer
176 views

$\mathbb{Z}[X]$ is noetherian

This follows from Hilbert's basis theorem, which is valid for polynomial rings over any noetherian ring. But is there a more elementary proof, knowing that $\mathbb{Z}$ is a PID (even a Euclidean ...
2
votes
3answers
263 views

Showing an ideal is a projective module via a split exact sequence

Let $R=\mathbb{Z}[\sqrt{-6}]$ and $I=(2,\sqrt{-6})$ the ideal generated by $2$ and $\sqrt{-6}$. I want to show that $I$ is a projective $R$-module by producing a short exact sequence that splits, ...
7
votes
3answers
713 views

Tensor product of a module with an ideal is isomorphic to their standard product

Let $A$ be a commutative ring and $M$ an $A$-module. Let $I$ be any ideal of $A$. We have an epimorphism $M \otimes_A I \rightarrow IM$. It seems to me that this is not in general an isomorphism. ...
3
votes
3answers
150 views

Ring of invariants of Klein Four group

Assume $F$ is a field and assume $f\in F[x_1,\ldots,x_4]$ is a polynomial that is invariant under the Klein Four group $V_4$. How can I show that this polynomial can then be rewritten as a polynomial ...
0
votes
0answers
45 views

Show that $\mathcal{O}^+_K$ contains $\mathcal{O}_K$, and that the discriminant $\Delta(K)$ is the index $[\mathcal{O}^+_K : O_K]$. [duplicate]

Let $K$ be a number field, let $\mathcal{O}_K$ be its ring of integers, and let $B = \{b_1,\ldots,b_d\}$ be a subset of $K$ of cardinality $d$ such that $\mathcal{O}_K = ...
0
votes
1answer
91 views

Is $\mathfrak I^n/ \mathfrak I^{n+1}$ defined when $\mathfrak I = (0)$

Just now I realized that I asked a stupid question.. Please ignore this question. While learning Associated graded modules, I defined associated graded module, which ended in this particular ...
3
votes
1answer
349 views

Use Nakayama's Lemma to show that $I$ is principal, generated by an idempotent. [duplicate]

Let $I$ be a finitely generated ideal in $R$, such that $I^2 = I$. Using the fact there exists $x\in R$ such that $e = 1 - x\in I$ and $xI = 0$, use Nakayama's Lemma to show that $I$ is principal, ...
1
vote
1answer
33 views

$pM \neq M$ $\stackrel{?}{\Rightarrow}$ $p M_p \neq M_p$

Let $A$ be a ring, $M$ an $A$-module and $p$ a prime ideal of $A$ such that $pM \neq M$. According to my intuition, it is not necessarily true that $p M_p \neq M_p$. Any counterexample?
4
votes
3answers
504 views

Does any integral domain contain an irreducible element?

Let $R$ be an integral domain which is not a field. Does $R$ necessarily have an irreducible element? I suspect the answer is no, but I couldn't find an example showing that...
3
votes
1answer
161 views

What has projectiveness to do with Cohen-Macaulay rings?

I read in Jacob Lurie's lecture notes that if $R=k[x_{1},\dots,x_{s}]/p$, and $R'=k[y_{1},\dots,y_{t}]$ injects into $R$ via Noether normalization such that $R$ is finite over $R'$, then $R$ is ...
-2
votes
1answer
148 views

show that M[X] is an R[X]-module.

Let $M$ be an $R$-module and let $M[X]$ be the set of polynomials in $X$ with coefficients in $M$. using the fact that $M[X]$ is an $R$-module with the obvious addition and scalar multiplication. ...
-2
votes
2answers
484 views

Some localization is not finitely-generated as an R-module

Let $R$ be an integral domain with field of fractions $K$, and let $f \in R$ be a non-zero non-unit. Prove that the subring $S=R[1/f]$ of $K$ is not finitely-generated as an $R$-module, using the fact ...
5
votes
1answer
208 views

How many primes can lie over a prime?

Let $R \subset S$ be an integral extension of domains and $\mathfrak p \subset R$ a prime ideal. Can it be the case that there are infinitely many distinct primes ${\cal P} \subset S$ such that ${\cal ...
5
votes
2answers
174 views

Question on an isomorphism in the proof that $k[V \times W] \cong k[V] \otimes_k k[W]$

First I should say that I am aware of the existence of this question here and this question here. My question is a little different from these two because I am asking about a certain detail in the ...