**3**

votes

**1**answer

304 views

### Primary structures in $\mathbb Q$

I'll formulate a topic restricted here to the positive rational
numbers $\ \mathbb Q_{_{>0}},\ $, then will pose a question (Q2) plus some related, to which I don't know the answers nor reference. ...

**5**

votes

**0**answers

137 views

### Flat + locally of finite presentation + monomorphism = open immersion

It is known that the following are equivalent for an epimorphism $A \to B$ in $\mathbf{CRing}$:
Let $S$ be the set of elements $a \in A$ such that $A [a^{-1}] \to B [a^{-1}]$ is an isomorphism. Then ...

**-1**

votes

**0**answers

18 views

### Uniqueness of Smith normal form in Z (ring of integers) [migrated]

It is a very well known fact that Smith Normal Form has proven useful when dealing with the development of the structure theorem of finitely generated abelian groups. In this context, there is an ...

**-2**

votes

**0**answers

66 views

### Degree of a rational Function [closed]

This might sound a very trivial question but I found different answers on the web.
Assume on has a rational function f(x)/g(x) where f(x) and g(x) are polynomials. What is the degree of the rational ...

**4**

votes

**1**answer

83 views

### Completeness of Localizations of Completions of Commutative Rings

Let $R$ be an integral domain. Let $x,y\in R\setminus\{0\}$ be distinct. Let $\hat R$ be the $x$-adic completion of $R$ (the ring of all sequences $(r_n+Rx^n)_{n\ge0}$ where for $n\ge0$, $r_n\in R$ ...

**1**

vote

**0**answers

81 views

### Invertible elements in a group algebra

Let $H$ be a torsion-free abelian group and let $\mathbb{K}$ be a field with two elements.
I would like to ask the following question:
Is the group of units of the group algebra $\mathbb{K}[H]$ ...

**3**

votes

**1**answer

191 views

### A technical question about affine grassmanian

For a commutative ring $R$, consider $R[[t]]$-modules $$t^k R[[t]]^n \subset M \subset t^{-k} R[[t]]^n \subset R((t))^n.$$ It is known that if $t^{-k} R[[t]]^n / M$ is finitely generated projective ...

**6**

votes

**1**answer

91 views

### Partial Orders realized by Prime Ideals on commutative rings

Is there a general criterion for which partial orders can be realized by the prime ideals of commutative rings (like we have for topological spaces - https://en.wikipedia.org/wiki/Spectral_space)?
...

**5**

votes

**1**answer

114 views

### Pythagorean number in Artin's theorem on nonnegative rational fractions

Emil Artin's theorem on nonnegative rational fractions says that a rational fraction $Q$ with $n$ variables with real coefficients which is non-negative on $\mathbb R^n$ is a sum of squares of ...

**0**

votes

**0**answers

110 views

### on the ``generic" modules of finite length (skyscrapers)

Let $R$ be a local or graded ring. (If it helps, can assume the ring is "good", e.g. $R=k[[x_1,..,x_p]]$, where $k$ is a field of zero characteristic.)
Let $M$ be a finitely generated $R$-module ...

**-2**

votes

**0**answers

305 views

### Complete Local Ring and Fermat's Last Theorem

Let us consider the infinitely many variables formal power series ring over a finite field, viz.
$R:= F_p[[S_1,...,S_∞]].$
Question: Choose an arbitrary finitely
generated ideal $I$ of $R$.
Is ...

**0**

votes

**0**answers

33 views

### finite orbits of transformations of the rational function field

Let $K : = k(x_1, x_2, \cdots, x_t)$ be the rational function field and consider the transformation $\tau$ of $K$ defined by $u \mapsto \frac{\alpha(u) + b}{a}$, where $a \ne 0 , b \in k$ and $\alpha$ ...

**0**

votes

**1**answer

95 views

### fixed points of an affine polynomial automorphism

Let $K:- k[x_1, x_2, \cdots, x_n]$ be the polynomial ring over a field $k$. Let $a_i, b_i \in K$ where $a_i \ne 0$. Consider the automorphism $\alpha$ of $K$ defined by
$x_i \mapsto a_ix_i + b_i$. ...

**2**

votes

**1**answer

148 views

### programming to compute kernel quotient image of a $\mathbb{Z}$-module endomorphism

Let the integers $n\geq 2$, $k\geq 1$, $v=0$ or $1$ and $n_1,\cdots,n_k\geq 1$ such that
$$
\sum_{i=1}^k n_i+v=n.
$$
Define $P_a^b=0$ if both $a,b$ are odd and $P_a^b={{[a/2]}\choose {[(a+b)/2]}}$ ...

**3**

votes

**0**answers

98 views

### Are there necessary and sufficient conditions for a chain complex $0 \to C_0 \to C_1 \to C_2 \to 0$ to be Poincare?

I am looking for necessary and/or sufficient conditions for the chain complex $0 \to C_0 \to C_1 \to C_2 \to 0$ over a principal ideal domain to be Poincare in the sense that $H_0 \cong H^2$, $H_1 ...

**14**

votes

**2**answers

552 views

### Witt-vector vectors

I've never really made my way in any detail through the Witt-vector construction. I did read all the articles that a quick Google and MSN search turned up, and none seemed to address it, but I could ...

**4**

votes

**0**answers

149 views

### Is a wild automorphism of $k[x_1,\ldots,x_n]$, $n \geq 3$, necessarily of infinite order?

Let $k[x_1,\ldots,x_n]$ be a polynomial ring over a field $k$ of characteristic zero.
When $n=2$, it is known that every automorphism of $k[x_1,x_2]$ is tame, namely, a finite product of elementary ...

**0**

votes

**1**answer

113 views

### Tensor products of simple modules over algebras [closed]

Let $A$ and $B$ be $\mathbb{C}$-algebras. Suppose that $M$ and $N$ are respectively simple $A$ and $B$ modules.
We can regard $M\otimes_{\mathbb C}N$ as $A\otimes_{\mathbb C} B$-modules in natural ...

**1**

vote

**1**answer

91 views

### the pfaffian-adjugate and its counterparts for matrices odd size

Let $R$ be a commutative ring (zero characteristic). Take a skew-symmetric matrix $A\in Mat^{skew-sym}(n,R)$.
If $n$ is even, then $det(A)=Pf^2(A)$ and there exists the "Pfaffian adjugate/adjoint" ...

**2**

votes

**0**answers

137 views

### Description of the equalizer of $\prod _j R/I_j \rightrightarrows \prod _{i,j}R/(I_i+I_j)$

This is a crosspost of this MSE question.
I have asked several questions in an attmept to get a general version of the Chinese remainder theorem without conditions on the ideals which will ...

**1**

vote

**0**answers

127 views

### ideals linked to an almost complete intersection

Is a grade $3$ type $3$ perfect ideal in a Noetherian ring linked to a grade $3$ almost complete intersection? I know that grade $3$ type $2$ perfect ideals are (by a work of Anne Brown).

**2**

votes

**0**answers

62 views

### non-Noetherian r-Noetherian ring with Noetherian total quotient ring

A commutative ring is said to be r-Noetherian if every regular ideal is finitely generated, where an ideal is said to be regular if it contains a non-zerodivisor. Does there exist a non-Noetherian ...

**30**

votes

**2**answers

400 views

### If $A$ is the ring of continuous functions on a genus $g$ surface, can the genus of $X$ be seen by simple algebra in $A$?

I was describing to a friend the result that a compact Hausdorff space is determined up to homeomorphism up to by its ring of continuous functions, and he asked how one could see the genus of a ...

**0**

votes

**2**answers

338 views

### Solving a system of equations using Gröbner basis

In Sage (or any other package) when using Gröbner basis to solve a system of equations (some of which are non-linear equations) does computing the Gröbner basis for the ideal ID generated by the ...

**0**

votes

**2**answers

152 views

### Ring with Cohen-Macaulay canonical module

Let $(R,m)$ be Noetherian local ring which is an imagine of a Gorenstein ring $(S,n)$. Set
$$ K_R:= Ext_S^{s-d}(R,S), $$
where $d=\dim R$, $s=\dim S$.
If $K_R$ is Cohen-Macaulay (i.e. $R$ is a ...

**1**

vote

**0**answers

50 views

### How to associate the following two kinds of real polynomials?

Suppose the following real polynomial of $n$ variables
$$f(X_1,X_2,\cdots,X_n)=\sum_{I=(i_1,i_2,\cdots,i_n)}a_IX_1^{i_1}X_2^{i_2}\cdots X_n^{i_n}$$
is easy or familiar to us, but I need to deal with ...

**3**

votes

**0**answers

73 views

### Shifts in the decomposition of Bott-Samelson bimodules

Let $k$ be an algebraically closed field of characteristic $0$, let $V=k^n$ be a $k$ vector space of dimension $n$, and let $R=k[V]$ be the ring of polynomial functions on $V$. Suppose that ...

**1**

vote

**1**answer

117 views

### What can be said about $A$ and $B$ given the exact sequence $0 \to R^p \to A \to R^r \to R^q \to B \to 0$?

Let $A,B$ be two $R$-modules over a commutative ring $R$ (restrict to $R = \mathbb{Z}$ or $R= \mathbb{K}$ a field where appropriate). Suppose $A$ and $B$ fit into an exact sequence
$0 \to R^p \to ...

**0**

votes

**0**answers

58 views

### grade of ideals in non-noetherian rings

Let $R$ be a commutative ring with unity, and $M$ an $R$-module. Assume that $I$ and $J$ are finitely generated ideals and $K$ another ideal of $R$. Let $\textbf{x}$ be a sequence of generators of ...

**7**

votes

**1**answer

239 views

### Ring of invariants for the regular representation

The symmetric group $S_n$ acts on $\mathbb C^n$ by permuting the coordinates. In this case the ring of invariants is generated by elementary symmetric polynomials in n-variables. Now consider the ...

**1**

vote

**1**answer

175 views

### Solving Non-Linear Equations over a Finite Field of a Large Prime Order

I want to know is there is an efficient way to figure out whether or not a ( underdetermined) system of non-linear equations have a solution over a finite field of large prime order. The equations ...

**9**

votes

**2**answers

711 views

### Algebraic independance of exponentials

First of all, a happy new year. Be it better than 2015,
healthy, wealthy, fruitful and cross-fertilizing
for you, familly and friends.
In order to cope with families of solutions of evolution ...

**1**

vote

**1**answer

147 views

### Base change for non-flat coherent sheaves and affine maps

Let $A$ be a finitely generated $k$-algebra, where $k$ is a field, let $I$ be an ideal in $A$, let $M$ be a finitely generated $A/I$-module, and let $M^{\prime}$ denote $M$ considered as an ...

**0**

votes

**0**answers

119 views

### Deeply ramified implies non discrete valuation - Almost ring theory

In their book "Almost Ring Theory" (http://arxiv.org/abs/math/0201175), Ofer Gabber and Lorenzo Ramero define a valued field $K$ to be "deeply ramified" if the module of Kähler differentials ...

**1**

vote

**0**answers

24 views

### Elimination theory for variables packaged in a matrix

I am wondering if the elimination theory in computational algebraic geometry can be more efficiently carried out if all variables lies within some given matrices.
For instance, consider the following:
...

**0**

votes

**1**answer

77 views

### Properties of Betti number of ideal

Notations:
$R$- Noetherian graded ring and $I,J$ homogeneous ideals in $R$
Definition:
The projective dimension of $R/I$, denoted $pd(R/I)$, is the length of a minimal free graded resolution of ...

**2**

votes

**1**answer

143 views

### Division and multiplication that preserve Euclidean norms

I am looking for ways to define
$$\frac{1}{x}\in \mathbb{R}^n\quad \quad and\quad \quad x\cdot y\in \mathbb{R}^n ,$$
where $x,y\in \mathbb{R}^n$ such that
...

**1**

vote

**1**answer

162 views

### Generalizing Dedekind's Factorization Theorem

A classical theorem due to Dedekind states the following:
Let $O_{K}$ be the ring of integers of a number field $K$, and assume
$K$ is generated by adjoining the algebraic integer $\alpha$ to
...

**-1**

votes

**1**answer

159 views

### Which finite cyclic groups can be characterized by Lattice isomorphism and isomorphism between their automorphism groups?

Given a finite cyclic group $G$, we denote by $L(G)$ the lattice of its subgroups, and by $\mathop{\rm Aut}(G)$ the automorphism group of $G$. Let $H$ be any group. Assume that $L(G)\cong L(H)$ and ...

**6**

votes

**0**answers

189 views

### Is $k(\!(x,y)\!)$ a topological field?

More generally, let $(R,m)$ be a Noetherian local domain with fraction field $K$. The $m$-adic topology turns $R$ into a topological ring. When $R$ is a discrete valuation ring, this topology extends ...

**0**

votes

**0**answers

52 views

### Non-local differentially smooth algebra

Let $A$ be a noetherian commutative algebra over a perfect field $k$.
The algebra $A$ is said to be differentially smooth over $k$ if
(1) $\Omega^1_{A/k}$ is a projective $A$-module, and
(2) the ...

**3**

votes

**1**answer

161 views

### Maximal ideals of polynomial ring containing a fixed element

We know that for a field $k $ and $f\in k [x]$, the only maximal ideals of $k [x]$ containing $f $ are the ideals generated by prime factors of $f $. Now, I want to know that if $R $ is an arbitrary ...

**0**

votes

**0**answers

44 views

### references for semi-linear algebra over fields

What is a good reference for semi-linear algebra? In particular semi-linear transformations and their matrix representations.

**3**

votes

**0**answers

271 views

### A relation between a ring with its polynomial ring

Let $\{f_i(x)\}_{i\in I}$ be a subset of $R[x]$ where $R[x]$ is the polynomial ring of $R$(a commutative ring with identity). If the ring $R/\langle f_i^2(n)-f_i(n)\rangle_{i\in I, n\in A}$, for every ...

**2**

votes

**1**answer

256 views

### Dimension of a commutative ring

For a commutative ring $R $ with identity, if $Nil (R)\not= J (R)$, can we deduce than $dim (R/J (R))<dim (R)$? ($R $ has finite $Krull$ dimension (=$dim$), $Nil (R)$=the set of all nilpotent ...

**53**

votes

**1**answer

1k views

### $R$ is isomorphic to $R[X,Y]$, but not to $R[X]$

Is there a commutative ring $R$ with $R \cong R[X,Y]$ and $R \not\cong R[X]$?
This is a ring-theoretic analog of my previous question about abelian groups: In fact, in any algebraic category we may ...

**2**

votes

**0**answers

73 views

### If $A$ is an integer ring such that each $P \in A_L[X]$ has a finite number of zeros in $A$, is $A$ commutative?

Let $A$ be a ring in which the product of any two nonzero elements is nonzero (we shall say that $A$ is an integral domain, even if $A$ is non commutative). It is well-known that if $A$ is ...

**1**

vote

**1**answer

254 views

### maximal ideals of polynomial ring

For a maximal ideal $ n $ of polynomial ring $ R [x] $ over a commutative ring $R$ with identity, are there conditions under which $m [x]\subset n$, for some maximal ideal $m$ of $R$?($m [x] $ is ...

**0**

votes

**0**answers

65 views

### Grobner basis for a general algebra

Let $R$ be a quotient of the polynomial ring $\mathbb{C}[x_1,\dots , x_n]$. We fix a $\mathbb{C}^*$ action on $R$ which preserve homogenous components and the multiplication. (The geometric analogue ...

**1**

vote

**0**answers

66 views

### A problem related to parametrizing $\operatorname{rank}\le r$ matrices and Segre embedding

Given a field $k$. We denote $A_{mn}=k[\{X_{ij}\}_{1\le i\le m,1\le j\le n}]$ a polynomial ring of $mn$ variables. Given $m,n,r>0$, we have a natural homomorphism $\phi\colon A_{mn}\to ...