Questions about commutative rings, their ideals, and their modules.
2
votes
3answers
63 views
Why is $\mathbb{Z}[\alpha ]$ not finitely generated as an $\mathbb{Z}$-module?
Assume that $\alpha \in \mathbb{C}$ is an algebraic number which is not an algebraic integer. My question is why $\mathbb{Z}[\alpha]$ is not finitely generated as an $\mathbb{Z}$-module. Clearly there ...
7
votes
1answer
82 views
Names for specific rings
Is there a name for a ring $A$ (commutative with $1 \neq 0$) all of whose ideals are radical?
Is there a name for a ring $A$ (commutative with $1 \neq 0$) such that for each $x \in A$, there is $n ...
1
vote
1answer
35 views
Example of ring such that the nil radical is prime and 0 is not
I was just trying to think about an example of a ring that is not a domain and the nilradical is prime, however I could not find anyone.
Thanks in advance.
2
votes
1answer
43 views
Is there an open mapping theorem for affine morphisms?
Let $A$ and $B$ be rings. If $\varphi : A \longrightarrow B$ is such that $^a\varphi : Spec(B) \longrightarrow Spec(A)$ is bijective, then in what conditions $^a\varphi$ is a homeomorphism? Or, more ...
3
votes
2answers
42 views
An exercise about zerodivisors
If $A$ is a commutative ring with unity, $f\in A$ and $x\in SpecA$, with the notation $f(x)$ I mean the coset $x+f\in A/x$. Now look at this exercise:
Prove that a nonzero element $f\in A$ is a ...
1
vote
0answers
35 views
ideals in rings of algebraic integers are finitely generated
I am trying to write about rings of algebraic integers $\mathcal{O}_K$ in a number field $K$ without introducing to much field theory. I want to show that these rings are Dedekind. First of all I want ...
5
votes
2answers
89 views
Examples of a non-finitely generated $\mathbb Z$-module
I am looking for a couple of examples of a $\mathbb Z$-module $M \neq 0$ with
$$
\mathrm{Hom}_{\mathbb Z}(M, \mathbb Z) = 0 \\
\mathrm{Ext}^1_{\mathbb Z}(M, \mathbb Z) = 0
$$
If $M$ was finitely ...
4
votes
3answers
70 views
Example of a finitely generated flat module which is not free
I couldn't come up with an example of a finitely generated flat module which is not free. I know that over local rings, freeness and flatness are equivalent. So the ring cannot be a local ring.
5
votes
2answers
100 views
A paradox? Or a wrong definition?
Let $A$ be a commutative ring with $1 \neq 0$. Then writing $V(1) = V((1))$, we have $\bigcap_{\mathfrak{p} \in V(1)}\mathfrak{p} = \sqrt{(1)} = (1)$.
But then $\bigcap_{\mathfrak{p} \in ...
2
votes
1answer
27 views
If $M=R/(p^k)$ and $N=p^{k-1}M$ then N is contained in every non-zero submodules of M
I have got this problem where I can do but only the tiny last bit of the proof where I got stuck. So here is the problem:
Let $R$ be a PID, $p\in R$ and irreducible element, $k\ge 1$ and let $M$ ...
2
votes
1answer
42 views
Prime and Primary Ideals in Completion of a ring
Let $(R,\mathfrak m)$ be a local noetherian ring and $\widehat{R}$ its $\mathfrak m$-adic completion.
If $\mathfrak q\in \operatorname{Spec}(\widehat{R})$ then can we find $\mathfrak p\in ...
5
votes
2answers
84 views
a flatness criterion
I'm having trouble with part (b) of Exercise 10.5.25 from Dummit & Foote (the goal of the problem is to prove that $A$ is a flat $R$-module iff $A\otimes_R I\to A\otimes_R R$ is one-to-one for all ...
4
votes
1answer
76 views
+50
a case where contraction of a principal ideal is principal
Let $K$ be a field and $R_1,\cdots,R_n$ DVRs of $K$ with $m_i$ the maximal ideal of $R_i$. Define $A=\cap R_i$. Then $A$ is semilocal with maximal ideals $p_i=m_i \cap A$. Also, $A_{p_i} = R_i$.
...
1
vote
1answer
31 views
deciding if a chain is a composition series (sanity check)
A small sanity check related to Question 2 from here: proof of the Krull-Akizuki theorem (Matsumura)
Let $C$ be an $A$-module, with $A$ commutative ring and suppose that there exists a chain of ...
3
votes
4answers
102 views
Proof of the uniqueness of maximal ideal
Let $R$ be a commutative ring with $1$. Let $M$ be a maximal ideal of $R$ such that $M^2 = 0$. Prove that $M$ is the only maximal ideal of $R$.
1
vote
0answers
20 views
Relation between $H^i_I(-)$ and $H^i_J(-)$ when $I\subset J$
What is the relation between $H^i_I(-)$ and $H^i_J(-)$ (cohomological functors) when $I\subset J$ are ideals of a (local) noetherian ring?
4
votes
1answer
61 views
Irreducible ideal implies prime ideal in Dedekind Domains?
An ideal is irreducible if it can not be written as the finite intersection of strictly larger ideals. In a Noetherian ring every irreducible ideal is primary, but the converse doesn't hold. I wonder ...
2
votes
1answer
57 views
Determinants of free modules
This should be easy, but I'm stuck.
Let $A$ be a DVR and $B$ a finite algebra over $A$ that is free as an $A$-module. For $b \in B$ one can define the norm of $b$ as the determinant of ...
2
votes
1answer
50 views
$\mathrm{Hom}(R/I, R/J\otimes M)\cong ?$
Let $R$ be a Noetherian commutative ring, $I,J$ two ideal of $R$ and $M$ an $R-$module. Does anyone see the isomorphism $\mathrm{Hom}(R/I, R/J\otimes M)\cong \ldots$?
Thanks.
5
votes
0answers
99 views
Question about the nullstellensatz for projective schemes
Assume that $ G $ is a graded ring. Assume that $A$ is a relevant homogeneous ideal (that is, it does not contain the irrelevant ideal $ \oplus_{n > 0}G_n$). I am having trouble proving the ...
11
votes
1answer
74 views
Uniformly solvable families of polynomials
It is a famous theorem that there is no "quintic formula", i.e. there is no formula which expresses the roots of a quintic polynomial $x^5+a_4x^4+\cdots+a_0$ in terms of $a_4,\ldots,a_0$ and rational ...
4
votes
3answers
63 views
example of a flat but not faithfully flat ring extension
I am learning commutative algebra and there is a definition about faithfully flat modules or ring extensions. I can't think of an example of a flat but not faithfully flat ring extension or module. ...
2
votes
0answers
26 views
Support of a direct sum of local cohomology modules
Let $R$ be a Noetherian ring with unit, $I$ be an ideal of $R$. Let $M$ be a finitely generated $R$ module. How can we show the following:
$$\operatorname{Supp}(\bigoplus_{j\ge ...
2
votes
1answer
37 views
special case of Nagata's Lemma (Matsumura p.86)
Let $K$ be a field and $R$ a valuation ring of $K$ with maximal ideal $m_R$. Let $a \in R$ such that $1-a \in m_R$.
Statement: For any $s$ that is not a multiple of the characteristic of $R/m_R$, ...
3
votes
1answer
42 views
All local cohomology modules being zero
Let $R$ be a Noetherian ring with unit, $I$ be an ideal of $R$ and let $M$ be a finitely generated $R$-module. Suppose $H_{I}^j(M)=0$ for all $j$, then how can one show that $M=IM$?
The converse of ...
8
votes
2answers
208 views
Vanishing of a certain Tor
I am reading about the construction of the Affine Grassmannian in Dennis Gaitsgory's seminar notes
and there are some commutative algebra facts that I am not able to figure out by myself apparently, ...
4
votes
0answers
36 views
Lattices as invertible module
Let $E$ be an etale algebra over $\mathbb{Q}$. In other words, $E$ is a finite sum of number fields. Let $L$ be a lattice in $E$, and $R$ the order associated to $L$. More explicitly, $$R=\{ e\in ...
6
votes
1answer
74 views
is the dual of a finitely generated module finitely generated?
I recently thought of this and have no idea whether over a general commutative ring the dual of a finitely generated module is finitely generated. This must be known.
2
votes
2answers
61 views
Some question on localization of polynomial ring
Let $S=A[x_1,\dots, x_r](r \geq 2)$ be a polynomial ring where $A$ is a commutative ring.
Then is it true that $S=\bigcap_{i=1}^r S_{x_i}$? If $S$ is $A$-algebra and $x_i$ are not zero divisors, then ...
3
votes
1answer
43 views
Finite Projective Dimension implies non vanishing Ext
Suppose the projective dimension of a module $M$ is $n < \infty$. Does there exist a free $R$-module $F$ such that $\operatorname{Ext}^n(M, F) \not = 0$?
Can't we write the free module as a direct ...
3
votes
1answer
61 views
$H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$
Let $R$ be a Noetherian ring and $I$ an ideal of $R$. If $n$ is the cohomological dimension of $I$, then why is the following isomorphism true:
$$H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$$
The ...
7
votes
2answers
71 views
Show field of fractions is finite extension of $\mathbb{Q}$
Let $A$ be a ring which is also a finitely generated $\mathbb{Z}$-module. If $A$ is an integral domain and $K$ is its field of fractions and $K$ has characteristic zero, then why is $K$ a finite ...
3
votes
1answer
44 views
Discrete Valuation Ring and Subring of the Fractions Field
Let $R$ be a Discrete Valuation Ring, and $K$ its fractions field. Now if $B\subseteq K$ is a subring with $R\subseteq B$ then we have $$B=R \text{ or } B=K.$$
Now this seems to be a very basic ...
2
votes
1answer
97 views
+50
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 ...
1
vote
1answer
26 views
Integral closure of k-algebra
Let $k$ be a field and $A$ a finitely generated algebra over $k$ that doesn't have zero divisors. Why is the integral closure of $A$ a finitely generated module over $A$ ?
(edited)
8
votes
2answers
62 views
If $R$ is a commutative ring with unity and $R$ has only one maximal ideal then show that the equation $x^2=x$ has only two solutions
If $R$ is a commutative ring with unity and $R$ has only one maximal ideal then show that the equation $x^2=x$ has only two solutions.
I know that $0$ and $1$ are the solutions, but I can't proceed ...
2
votes
0answers
63 views
Descent Theorem Problem
If $A$ is a finitely generated $K$-algebra which is a domain, and $p_0 \subsetneqq \cdots \subsetneqq p_t$ a chain of prime ideals, how I can show that if $F$ is the field of fractions of $A$ and ...
1
vote
0answers
54 views
Reduction of ideals in a commutative ring
Is it possible to have an infinitely generated reduction of a finitely generated ideal in a commutative ring with identity ? If yes, why ? If no, an example to this effect will be helpful.
Thank ...
3
votes
1answer
51 views
Automorphism of $A[t]/(t^m)$
Let $A$ be a commutative ring and $t$ an indeterminate over $A$. If $f$ is an automorphism of the ring $A[t]/(t^m)$ satisfying $f(x)\equiv x\pmod{(t)}$ for each $x\in A[t]/(t^m)$ with $m$ a positive ...
4
votes
0answers
55 views
Artinian rings are perfect
Is there a simple way to prove that an Artinian ring is perfect? (in the commutative case)
3
votes
1answer
60 views
Property of Hom-functor
How to prove $$\operatorname{Hom}_{R}(A,\operatorname{Hom}_{\mathbb{Z}}(R,B))\cong \operatorname{Hom}_{\mathbb{Z}}(A,B)$$ where $R$ is a commutative ring, $A$ an $R$-module and $B$ an abelian group?
...
8
votes
2answers
57 views
Why over $\mathbb{Z}/n\mathbb{Z}$ projectivity, injectivity and flatness coincide for cyclic modules?
Assume $R=\mathbb{Z}/n\mathbb{Z}$ ($n\neq0$) and let $M$ be a cyclic $R$-module. Could you tell me how to prove that $M$ is projective if and only if it is injective if and only if it is flat? And ...
5
votes
1answer
37 views
How to show that differential operator can be defined in terms of certain commutator operators
Let $U$ be any open subset of $\mathbb{R}^n$ (or, more general, of some smooth manifold). Define $\mathcal{D}_{-1}(U):=\{0\}$. For any two linear operators $A$ and $B$, the commutator operator $[A,B]$ ...
3
votes
2answers
64 views
A noetherian ring whose ideals are idempotent is artinian
I have to prove the folowing:
If $R$ is a Noetherian ring, and for every ideal $I$ of $R$ we have $I = I^{2}$, then $R$ is Artinian.
My first thought was to try to prove that the nilradical of ...
6
votes
0answers
94 views
An example of a commutative ring in which every primary ideal is prime
It is clear that every prime ideal in a commutative ring is primary. The converse is false; for example, in the ring $\mathbb{Z}$ the ideal $(p^2)$ is an example of a primary ideal that is not prime ...
4
votes
1answer
49 views
studying the topology of a real algebraic set
Let $f_1,\ldots,f_n \in \mathbb{R}[x_1,\ldots,x_m]$ be polynomials with real coefficients and let $I$ be the ideal that they generate. Denote by $V_{\mathbb{R}}(I)$ the corresponding real variety, ...
1
vote
2answers
49 views
radical of sum of two ideals
$I$ and $J$ are ideals in $k[x_1,\cdots,x_n]$.
Show that $\sqrt{I+J}=\sqrt{\sqrt{I}+\sqrt{J}}$.
I have no idea how to prove it. Can someone help?
2
votes
0answers
43 views
Flatness over Jacobson ring
I need either a reference or a counter-example to the following statement. Let $A$ be a noetherian Jacobson ring (i.e. a noetherian ring where every prime ideal $\mathfrak{p} \subset A$ is an ...
5
votes
1answer
50 views
What can be said about $p\in Spec(R)$ when $R_p$ is a field?
What can be said about $p\in Spec(R)$ when $R_p$ is a field? Especially when $R$ is local noetherian
-6
votes
0answers
70 views
Are there two or three unique paths for tonal movement in music? [closed]
In music theory we learn there are 2 unique tone value class pathways that include in sequence every note in an octave: the chromatic path and the circle of 5ths. This makes a toroidal manifold. We ...
