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3
votes
1answer
175 views

operations on ideals in a subring of number field

For three ideals $I, J$ and $K$ of a subring $R$ in a number field $L$, does this equality hold in general? $(I+J) \cap K = (I \cap K) + (J \cap K)$ I have no counterexample yet but I couldn't prove ...
10
votes
0answers
127 views

Does every commutative variety of algebras have a cogenerator?

By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes. Equivalently (i) every interpretation of ...
7
votes
0answers
219 views

Formally undecidable problems on finitely presented quandles

In the literature, one sometimes sees the claim that finitely presented quandles (in particular, knot quandles) are "hard to deal with". Hence, a great deal of effort has gone into studying finite ...
3
votes
0answers
139 views

What can we say about a semigroup that's acted on by an ideal of polynomials?

This got no response on MSE, so posting here. Let $S = \{ (a,b) \in \Bbb{Z}^2 : \gcd(a,b) \neq 1 \} \cup (1,1)$. Then $S$ forms a semigroup. The operation being componentwise multiplication. Let ...
3
votes
0answers
79 views

How much do idempotent ultrafilters generate in terms of semigroups?

It is known that the set of ultrafilters on, say, the natural numbers $\mathbb{N}$, can naturally be endowed with the structure of a compact topological left semigroup (which fails to be anything ...
3
votes
0answers
105 views

Making an algebra the uniqe maximal one-sided ideal in another unital algebra

If $R$ is an algebra without a unit, then the standard unitisation $R^\sharp$ can have maximal one-sided ideals other than $R$. Thus, it is natural to ask about the following. Let $R$ be an algebra ...
2
votes
0answers
95 views

Hall's paper on the profinite groups and Andre Weils “voisinage” notion

I am reading through a classical paper A Topology for Free Groups and Related Groups by Marshall Hall Jr. in which profinite groups are defined for the first time. There he defines on p. 129: ...
2
votes
0answers
123 views

Abelian Subgroup in an infinite non-abelian 3-group

Does an infinite non-abelian 3-group of exponent greater than or equal to 9 has an infinite abelian subgroup? I know that 2-groups and 3-group of exponent 3 has an infinite abelian subgroup. I wonder ...
2
votes
0answers
89 views

Lazard's $\Gamma_n(f)$ as cocycle

In Michel Lazard's "Commutative Formal Groups" Springer Lecture Notes, he defines an operator on a polynomial 3-cochain $f$ denoted $\Gamma_n(f)$, which defines as the $n^{th}$ homogeneous piece of ...
2
votes
0answers
98 views

classification of rank $2$ $\mathbb{Z}/p^n\mathbb{Z}$-algebra with invertible discriminant

Let $p$ be a prime number and $n$ be an integer. Let $A$ be an $\mathbb{Z}/p^n\mathbb{Z}$-algebra of rank $2$ whose discriminant is non invertible. In Serre's book lecture on the mordell Weil theorem ...
2
votes
0answers
103 views

Quivers for algebras which are not basic or unital.

Are there any definitions of quivers for algebras which are not basic or unital? I am reading the book Elements of the representation theory of associative algebras: volume one. The ordinary quivers ...
2
votes
0answers
118 views

Clifford algebra is graded separable

Let $D$ be an algebra of odd differential operators on a free module $V$, this algebra is isomorphic to the Clifford algebra $Cl(V^* \oplus V)$. Let $m$ denote multiplication map $$m : D\otimes D \to ...
2
votes
0answers
169 views

Idea behind choosing $\small f(x)$ as $c^{s}x^{p-1} \frac{[\theta(x)]^{p}}{(p-1)!}$ in the proof that $\pi$ is transcendental.

I am going through the article at this link, where the author proves that: "$\pi$ is $\text{transcendental}$ over $\mathbb{Q}$". Although, I understand the proof, I have some doubts. At page $6$, ...
1
vote
0answers
66 views

Hochschild Homology Subalgebra

MY question is simple, suppose $R$ is a $CRing$, and $A$ is an $R$-a subaglebra of the $R$-algebra $B$. Then, how is the Hochschild homology of $A$ with values in the $A$ bimoduile $M$ related to ...
1
vote
0answers
78 views

Positivstellensatz for non-polynomial term

Can we use Positivstellensatz (P-satz) below for a non-polynomial term? P-satz: Let $R$ be real closed field. Let $f,g,h$ be finite families of polynomials in $R[X_{1} ,...,X_{n}]$. Denote by P the ...
1
vote
0answers
75 views

lattice coset counting

Let $m\geq 1$ be an integer and denote $B_m = \{-m, -m+1, \dots, m\}^n$. Suppose that $M\subseteq \mathbb{Z}^n$ is a module. By the structure theorem we can write $$ \mathbb{Z}^n/M \simeq ...
1
vote
0answers
127 views

Solving a system of rational functions

Given pairwise distinct numbers $c_1, c_2, \dots c_n \in \mathbb{C} \setminus \{0\}$, does the system of equations $$\frac{6}{c_k} + \sum_{i \ne k} \frac{2}{c_k - c_i} = \sum_{i = 1}^n \frac{1}{c_k - ...
1
vote
0answers
110 views

Is a certain group related to a primitive L function isomorphic to $Gal(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})$ for some $\ell$?

I define the notion of "Galois class of L functions" in the following way: $A$ is a Galois class of L functions if and only if the follwing three conditions hold simultaneously: 1) every element ...
1
vote
0answers
280 views

about union of conjugate proper subgroups in a math paper

My question is about the shaded area in this image. Does the symbol $L=\bigcup_{g \in G} T^{g}$ means that $L$ is a union of sets or $L=\langle T^{g}, g\in G \rangle$? If it means the first one, then ...
1
vote
0answers
158 views

Invariant Ideals in Split Hopf Algebroids

Given a split Hopf algebroid $(S,\Sigma)=(S,S\otimes B)$ over $K$, Ravenel leaves as an exercise the proof of the following: An ideal $J\subset S$ is invariant under the action of the group ...
0
votes
0answers
36 views

Supersets of P-finite sequences and rings

P-finite sequences are a superset of C-finite sequences. While doing programming work, the question came up what generalizations or supersets of P-finite sequences have people described. In other ...
0
votes
0answers
59 views

Problem with Smoothness and quasi-freeness

Let A be a unital associative algebra over a field k. Then A is smooth if and only if X:=Spec(A) is smooth. That is $\Omega_{X|Spec(k)}$ is locally-free. The later module is isomorphic to ...
0
votes
0answers
36 views

Decomposition results for locally commutative semigroups

Every finite abelian group is the direct product of its cyclic groups of prime order, and every commutative monoid divides a product of its cyclic submonoids. Could these results generalized to ...
0
votes
0answers
55 views

Graded Betti Numbers of a Graded Ideal with Linear Quotients

Exercise 8.8 in Monomial Ideals by Herzog and Hibi: Let $I\subset S=K[x_{1},...,x_{n}]$ be a graded ideal which has linear quotients with respect to a homogeneous system of generators ...
0
votes
0answers
33 views

Graded Betti Numbers of a Stable Monomial Ideal

Exercise 8.8 in Monomial Ideals by Herzog and Hibi: Let $I\subset S=K[x_{1},...,x_{n}]$ be a stable monomial ideal with $G(I)=\{u_{1},...,u_{m}\}$ and such that for $i<j$, either ...
0
votes
0answers
38 views

About Linear Quotients of Square of an Ideal with Linear Quotients

Let $I$ be a monomial ideal generated by quadratic monomials $u_{1},...,u_{s}$ and suppose that $I$ has linear quotients with respect to this given ordering. Is it true or false that $I^{2}$ has ...
0
votes
0answers
163 views

Monomial ideals: isomorphism problem for commutative algebras?

Theorem 5.27 in Polytopes, Rings, and K-Theory (Bruns, Gubeladze - 2009 - Springer SMM) claims: Let $K$ be a field and $I\!\unlhd\!K[x]= K[x_1,\ldots,x_n]$ and ...
0
votes
0answers
67 views

Example of a ring whose minimals are annihilators of idempotents?

I'm looking for examples† of rings with the property that for each $P={\rm Ann}_R(a)\in{\rm Min}(R)$ then $a\in R$ is idempotent (ie $a^2=a$) † other than domains!
0
votes
0answers
99 views

Operating on a set of sequences - such as adding sequences and so on possible even when sequence is coded as number?

From http://math.stackexchange.com/questions/346680/operating-on-a-set-of-sequences-such-as-adding-sequences-and-so-on-possible-ev Suppose that there is a way to code some set of sequences into ...
0
votes
0answers
113 views

Structure groups and a special class of L-functions

Hello, Let $X$ and $Y$ be two mathematical objects such that there exists a canonical embedding $f:X\hookrightarrow Y$. I define the structure group of $Y$ relatively to $X$, denoted $Str(Y/X)$, as ...
0
votes
0answers
116 views

On X-s-permutable subgroups of a finite group

I want to prove Lemma 2.1(1) in the paper On X-s-Permutable Subgroups of a Finite Group by Min Bang SU, Yang Ming LI. It is on the web. This is my proof. . Since $H$ is $X−s−$permutable in $G$, then ...