Tagged Questions
9
votes
1answer
55 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
0answers
33 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 ...
2
votes
2answers
59 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
38 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 ...
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 ...
3
votes
1answer
50 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 ...
5
votes
0answers
84 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 ...
1
vote
2answers
45 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?
12
votes
0answers
85 views
A ring isomorphic to its finite polynomial rings but not to its infinite one.
I was messing with the ring $k[x_1,\dots,x_n,\dots]$ of polynomials in numerable many variables in order to solve an exercise of Atiyah, and the following question came to me and made me curious:
...
2
votes
1answer
52 views
Monomials not in an ideal
Let $R=\mathbb{R}[x,y]$ denote the comutative ring of polynomials in two variables $x,y$ with real coeficients. Show that for each $k \in \mathbb{N}$ there exists a monomial of degree $k$ not ...
1
vote
1answer
30 views
Isomorphism of polynomial rings implying isomorphism of the coefficient rings [duplicate]
Let $R$ and $S$ be commutative rings. Let $x, y$ be indeterminates, and assume that one has an isomorphism $R[x] \rightarrow S[y]$ (not necessarily mapping $x$ to $y$ of course). Does this imply $R ...
2
votes
1answer
41 views
Is localization of a prime ideal still a prime ideal?
Im still new to the topic so this question might seem trivial. But I hope if someone can help explaining to me if a prime ideal $P$ of a domain $A$ is still a prime ideal $P_s$ in the localization ...
0
votes
0answers
31 views
Ideal membership problem for monomial ideals
Hi guys. I'd really appreciate help on understanding the proof for this Lemma above. I'm not sure how we got: "we see that every term on the right side of the equation is divisible by some x^{a(i)}. ...
2
votes
1answer
26 views
degree lexicographic monomial ordering
With respect to deglex X>Y, what would the leading monomials of these polynomials be?
$f_1=XY^3-X^2$ and $f_2=-X^3Y^3-4X^2Y^3+3X^2Y$
My understanding is that you prioritise X over Y based on their ...
7
votes
1answer
47 views
When does “second annihilator” of a (principal) ideal equal the ideal itself
Suppose that $R$ is a (local) ring and $r\in R$. When do the equations $\operatorname{Ann}_R(\operatorname{Ann}_R(r))=Rr$ or $\sqrt{\operatorname{Ann}_R(\operatorname{Ann}_R(r))}=\sqrt{Rr}$ hold?
I ...
1
vote
0answers
62 views
A counterexample for $\operatorname{Ass}(M_1+M_2)$ [duplicate]
$\newcommand{\Ass}{\operatorname{Ass}}$
Let $A$ be a Noetherian ring and let $M$ be an $A$-module. Suppose $M=M_{1}+M_{2}$, then we have $\Ass(M)\supset \Ass(M_{1})\cup \Ass(M_{2})$. What is an ...
3
votes
1answer
61 views
All the Associated Primes are minimal.
Let $R$ be a commutative Noetherian ring with unit and let $I$ be a fixed ideal. I am sorry if the following turns out to be a very silly question.
1) Suppose $\operatorname{Ass}(R/I)$ are all ...
7
votes
0answers
102 views
Find all maximal subrings of $\mathbb{C}[x]$
Definition: A maximal subring $S$ of $R$ is a subring such that if $S \subseteq T \subseteq R$ then $T=S$ or $T=R$.
Find all maximal subrings of $\mathbb{C}[x]$.
Clearly $\mathbb{C}[x^2,x^3]$ ...
1
vote
1answer
60 views
Localization of $K[x,y|x^2-y^3]$ and $K[x,y|xy]$ at $\langle x,y\rangle$ and $\{\text{non-zero-divisors}\}$ (exercise in SICA)
In Greuel & Pfister's A Singular Introduction to Commutative Algebra, p. 38, there is written:
So we have rings
$$\begin{array}{l l}
R_1:= K[x,y|x^2\!-\!y^3], & R_4:= K[x,y|xy],\\
R_2:= ...
3
votes
1answer
43 views
Difficulty Understanding Primary Modules
I have read that any irreducible sub-module $I$ of a Noetherian module $M$ is primary. However if we let $M = \mathbb{Z}_8$ and $I = \mathbb{4Z}_8$ this isn't true, because $I$ is irreducible, and ...
5
votes
2answers
67 views
Intuition behind Direct limits
Let $R$ be a commutative ring and $x\in R$ be a nonzero divisor. Then i know that the direct limit of $R\mapsto R\mapsto R\mapsto\cdots $, where each map is multiplication by $x$ is $R_x$, the ...
3
votes
0answers
38 views
Injective dimension is locally finite but not globally
Let $R$ be a commutative ring. Could someone provide me an example where $\operatorname{id}_{A_{\mathfrak p}}(M_{\mathfrak p})$ is finite for all $\mathfrak p\in \operatorname{Spec}(R)$, but ...
2
votes
0answers
29 views
constructing a sum of squares modulo an ideal
This question refers to the proof of Theorem 7.3, p. 98 of the pdf http://math.berkeley.edu/~bernd/cbms.pdf.
The statement of the theorem and its proof do not depend on what precedes them.
Let $I$ be ...
3
votes
1answer
95 views
About Artinian Rings
I'm studing commutative algebra by the text of Atiyah and Macdonald, and a doubt come at me and I can not prove neither find a counterexample, the problem is:
If a ring (commutative with identity) ...
1
vote
1answer
84 views
Artinian ring and faithful module of finite length
Let $A$ be a ring. How can I prove that:
$A$ is an Artinian ring $\Leftrightarrow \exists$ a faithful $A$-module which is of finite length.
I know that if a ring has a faithful $A$-module which ...
1
vote
0answers
43 views
For $R$-modules $M,N$, what are sufficient conditions for $\operatorname{Supp}(M\otimes_R N)\subseteq \operatorname{Supp}(\operatorname{Hom}_R(M,N))$?
Let $R$ be a commutative ring, $M$ and $N$ be finitely generated $R$-modules. What additional conditions will ensure $\operatorname{Supp}(M\otimes_R N)\subseteq ...
4
votes
1answer
84 views
Field of fractions of $\mathbb{Q}[x,y]/\langle x^2+y^2-1\rangle$ [duplicate]
This problem goes as follows:
Prove that $\mathbb{Q}[x,y]/\langle x^2+y^2-1\rangle$ is an integral domain and that its field of fractions is isomorphic to the ring of rational functions ...
4
votes
0answers
52 views
A question on an answer on Math Overflow about Artin approximation
I have a question on an answer of this Math Overflow question.
Let $(A,I)$ be a commutative excellent normal local domain. The completion
$$
\hat A=\underset{\longleftarrow}{\operatorname{lim}} ...
5
votes
0answers
63 views
Description of $\mathrm{Ext}^1(R/I,R/J)$
Let $R$ be a commutative ring with unit and $I$ and $J$ are nonzero ideals of $R$. Do we have a nice description for $\mathrm{Ext}^1_R(R/I,R/J)$?
What do I mean by a nice description? For example ...
2
votes
1answer
53 views
Noetherian and Artinian modules over subrings
I have a question about whether Noetherian-ness and Artinian-ness of modules are preserved under changes of the base ring. More precisely:
Let $R$ be a commutative ring and $S \subseteq R$ a ...
4
votes
1answer
51 views
A graded abelian group and a graded map
I have an elementary question about a graded abelian group and a graded map. Here is the situation.
Let $A$ be a free abelian group of rank $2$ spanned by $1$ and $x$. Let us make $A$ be a graded ...
2
votes
0answers
44 views
Spectrum of a Laurent polynomial ring
I suspect this question is either easy/known or far too general to answer, but I'm finding it difficult to google for so I'd appreciate directions to good resources on the subject.
Can we describe ...
3
votes
0answers
55 views
Hilbert’s zeros theorem, an application. (The algebraic variation)
Theorem: (Hilbert) If $k$ is a field, $A$ is a finitely generated $k$-algebra, and $M$ is a maximal ideal in $A$, then the factor $A/M$ is a finite extension of $k$. In particular if $k$ is ...
0
votes
1answer
58 views
Seidenberg's Lemma
I have a problem with the proof of this
Lemma. Let $K$ be a field, $R=K[X_1,\dots,X_n]$, and $I\subset R$ a zero-dimensional ideal. For every $i$ there exists $g_i\in I\cap K[X_i]$, $g_i\neq 0$, ...
3
votes
1answer
64 views
question related to the krull schmidt theorem
Let $M$ be a finitely generated projective $R$-module, where $R$ is an Artinian ring. Then I must show that $M$ is isomorphic to a direct sum of principal indecomposable $R$-modules.
We have a ...
2
votes
1answer
67 views
zero-dimensional ideals and finite-dimensional algebras
I encountered in the literature the term "zero-dimensional ideal", however i can not find a relevant definition anywhere in Atiyah-MacDonald or Matsumura. In fact, i encounted the statement:
$I$ ...
1
vote
2answers
42 views
A complete set of orthogonal idempotents in a commutative ring
I'm reading David Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry. At page 13, Chapter $0$, he says: "... if $e_1,\ldots,e_n$ is a complete set of orthogonal idempotents in a ...
8
votes
1answer
80 views
Commutativity characterization?
Let $R$ be a ring (not necessarily unital) and for any $x\in R$ there is an integer $n \geq 2$ s.t. $x=x^2+\cdots+x^n.$
Does it imply that $R$ is commutative?
2
votes
1answer
56 views
Constructing an example s.t. $\operatorname{Hom}_R(M,N)$ is not finitely generated [duplicate]
Let $R$ be a commutative ring and $M$ and $N$ be two finitely generated $R$-modules. I wanna construct an example s.t. $\operatorname{Hom}_R(M,N)$ is not finitely generated.
It's well-known that if ...
4
votes
1answer
74 views
simple application of Bezout's Theorem
Let $f(x),g(x) \in \mathbb{C}[x_1,\cdots,x_n]$ be two irreducible homogeneous polynomials of degree $n,m$ respectively. Does Bezout's Theorem say that the system of equations $f(x)=0, g(x)=0$ has ...
2
votes
2answers
34 views
square-root of a square-free element (Matsumura)
Example 4, page 65 in Matsumura's Commutative Ring Theory reads as follows: "Let $A$ be a UFD in which $2$ is a unit. Let $f \in A$ be square-free (that is, not divisible by the square of any prime of ...
2
votes
1answer
48 views
Poles of formal power series (Hilbert-Poincaré series)
How are poles and orders of poles of formal power series defined?
The particular case, I am interested in, is the following definition from [Atiyah-Macdonald, Introduction to commutative algebra, ...
1
vote
1answer
28 views
Finitely generated integral domain and finitely generated $k$-algebra.
Let $k$ be a field and $R$ a finitely generated domain over $k$. Then there are $y_1, \ldots, y_n$ such that $$R=k[y_1, \ldots, y_n]/P,$$ where $P$ is some prime ideal of $k[y_1, \ldots, y_n]$. My ...
2
votes
0answers
61 views
A commutative ring with alternating and commutativity properties with infinite distinct elements
Is there any nontrivial commutative ring without multiplicative identity that satisfies alternating property ($x \cdot x = 0$ for all $x$ where $\cdot$ is multiplication operator and $x \cdot y \neq ...
1
vote
2answers
63 views
Two principal ideals coincide if and only if their generators are associated
Suppose we have a ring $R$ and $(a),(b)$ are both ideals of $R$. Is it always true that $(a)=(b)$ if and only if there exists a unit $c$ such that $a=bc$ (i.e., $a$ and $b$ are associate)?
I ...
3
votes
1answer
96 views
Question about isomorphism of modules.
I have been reading the book of DeMeyer and Ingraham "Separable Algebras of Commutative Rings," where in page 129 they prove the following.
Let
$\bullet$ Let $S$ be a commutative ring and $G$ be a ...
13
votes
2answers
146 views
Is a linear combination of minors irreducible?
Let $X=(X_{ij})_{1\le i,j\le n}$ be a matrix of indeterminates over $\mathbb C$. For choices $I,J\subseteq\{1,\ldots,n\}$ with $|I|=|J|=k$ denote by $X_{I\times J}$ the matrix $(X_{ij})_{i\in I,j\in ...
0
votes
1answer
29 views
Extension of the radical of an ideal
Let $I$ be an ideal of a ring $R$ and let $f: R\rightarrow S$ be a homomorphism. Let $I^e = \{\sum_{j=1}^n s_j f(r_j) : s_j \in S, r_j \in I\}$ be the extension of $I$. Is it true that the radical ...
0
votes
2answers
52 views
Nilradical and Jacobson's radical. [duplicate]
Let A be a commutative ring with 1.
1) Prove that a sum of a nilpotent element and an invertible element is invertible.
2) Prove that if $f=a_0+a_1x+\dots+a_nx^n \in A[x]$
a) $\exists f^{-1}\in ...
1
vote
2answers
87 views
Determine the total ring of fractions
Determine the total ring of fractions of $\mathbb{Z}\times \mathbb{Z}$ and $\mathbb{Z}_{12}$.
