**1**

vote

**0**answers

57 views

### Possibilities for dimensions of $\mathfrak{m}^i/\mathfrak{m}^{i+1}$ for a local ring

Let $R$ be a local commutative ring with maximal ideal $\mathfrak{m}$, and denote by $k$ the residue field $R/\mathfrak{m}$. Then we can look at the sequence of $k$-vectorspaces
$$R/\mathfrak{m}, ...

**0**

votes

**0**answers

27 views

### Grassmann algebra morphism with universal property

I'm pretty sure that the following doesn't work, but nevertheless i wanted to ask, maybe this is a kind of well-known construction i've never heard of:
Let $\Lambda(\mathbb{R}^n)$ be a finite ...

**-2**

votes

**1**answer

41 views

### How to find a matrix by its characteristic value and characteristic vectors? [on hold]

Now I am studying linear algebra course, In that for a given matrix we are finding the characteristic values (eigen vlaues) and characteristic vectors (eigen vectors). But my qustion is why cant we ...

**1**

vote

**1**answer

77 views

### Weyl algebras $A_n(k)$ as tensor product of the first Weyl algebra

In afew threads I've read that the Weyl algebra $A_{n+1}(k)$ is isomorphic to the $k$-tensor product of $A_n(k)$ with $A_1(k)$, why is this true?

**1**

vote

**2**answers

128 views

### Is there an intuitionistic generalized boolean algebra (of Stone)?

A "boolean algebra without the greatest element" was called by Stone "generalized boolean algebra" and he axiomatized it. Is there any publication about "preudo-boolean algebras without the greatest ...

**1**

vote

**1**answer

57 views

### Does a BCL algebra define a partial order?

A BCL algebra is a universal algebra with a binary operation denoted as "$*$" and a $0$-ary operation (constant) denoted as "$0$", satisying the following axioms:
(1) $x * x = 0$;
(2) if $x * y = 0$ ...

**1**

vote

**0**answers

40 views

### Alternative Generating Sets of the Quantum Special Linear Group

The Hopf algebra ${\cal O}_q(SL(N))$ has as generators the elements $u^i_j$, subject to certain $q$-relations. Moreover, the antipode is bijective, with square equal to a multiple of the identity on ...

**6**

votes

**1**answer

68 views

### Infinite dimensional simple algebras of finite degree

Let $F$ be a field and let $B$ be an $F$-algebra. The degree of $B$ over $F$ is the smallest positive integer $\deg_F B = d \geq 1$ such that every element of $B$ satisfies a (monic) polynomial of ...

**1**

vote

**0**answers

41 views

### On generating Euler Square of index q, q-1 (where q is any prime power)

Can somebody help me in generating Euler Square of index q, q-1 (where q is any prime power). Also, kidly tell me if there is any code availeble for generating the stated Euler Square. The details of ...

**1**

vote

**0**answers

102 views

### Cycles in Quivers and Path Algebras

I cannot find anything giving the algebra of a quiver with a single cycle on three or more vertices. In other words if your quiver consists of n vertices (n>2), and e_i is connected to e_{i+1} (taking ...

**4**

votes

**0**answers

244 views

### The logarithm over $\mathbb F_1$

In 'Cyclotomy and analytic geometry over F1', Manin proposes a version of the notion of `analytic function' over the 'field with one element $\mathbb F_1$'.
Question 1: can somebody explain or give ...

**5**

votes

**1**answer

163 views

### Equivariant Formality

Let $G$ be a finite group and $\mathcal{A}$ be a $dg$-algebra. Assume $G$ acts on $\mathcal{A}$, i.e. there exists a homomorphism $G\to {\rm Aut}_{dg}(\mathcal{A})$.
Assume further there exists a ...

**8**

votes

**6**answers

525 views

### Properties of rings that have an elegant description in terms of the associated category of modules

Suppose $A$ is a ring. Then $A$ happens to be a division ring iff every left $A$-module is free. (See here for proofs). I think this is very beautiful; what other properties of rings have an elegant ...

**3**

votes

**0**answers

93 views

### Formal DG-algebras

Sorry for this question but I really have difficulties with model categories.
Usually a $dg$-algebra $A$ is called formal, if there exists a $dg$-algebra $B$ and quasi-isomorphisms $$A\leftarrow B\to ...

**3**

votes

**0**answers

62 views

### Quasi-isomorphisms and Subalgebras

Let $A$ and $B$ $dg$-algebras over $\mathbb{C}$. If there exists an isomorphism $f:A\to B$, then every subalgebra $A'$ of $A$ is isomorphic to the subalgebra $f(A')$ of $B$.
What is if $f$ is ...

**15**

votes

**5**answers

2k views

### Origin of exact sequences

I have seen exact sequences appearing a lot in algebraic texts with different purposes. But I've never seen names of the people associated with it. Also I don't understand what's so good about showing ...

**2**

votes

**1**answer

125 views

### Jacobson radical and group rings/subalgebras

Let $G$ be a finite group and $N\le G$ be a subgroup. Consider the group algebra $kN$ as a subalgebra of $kG$ over an algebraically closed field $k$ of positive characteristic.
What can we deduce ...

**2**

votes

**1**answer

131 views

### coherent modules

Let $R$ be a nontrivial ring. A right $R$-module $M$ is called coherent if ${\rm Ker} (f)$ is finitely generated for any $R$-module homomorphism $f: L\to M$ with $L$ finitely generated.
It is ...

**1**

vote

**0**answers

65 views

### Units in residue classes

Let $K$ be a CM-field of degree $2n$. (Quadratic extension of totally real number field)
Let $\mathcal{O}_K$ be the ring of integers in $K$, and $m\geq 1$ an integer. Let $U_K$ be the group of units ...

**7**

votes

**0**answers

126 views

### What happens to simple modules under Ringel duality?

If $A$ is a quasi-hereditary algebra then its Ringel dual $A'=End_A(T)$ is the endomorphism algebra of a (minimal) full tilting module $T$ for $A$. The algebra $A'$ is again quasi-hereditary, although ...

**4**

votes

**0**answers

108 views

### Which endomorphism algebras are not Morita-trivial?

Let $\mathcal C$ be a symmetric monoidal category — for example, $\mathcal C$ might be the category of $R$-modules for a commutative ring $R$. At a minimum, I request further that:
$\mathcal C$ is ...

**0**

votes

**0**answers

69 views

### Relationships between finiteness of stable rank and IBN property of rings

Does any ring of finite stable rank have IBN property? Where can we find this result?

**2**

votes

**0**answers

57 views

### Isomorphic bound quiver algebras for different admissible Ideals

We know that for a path algebra KQ, whether or not KQ is finite dimensional (namely, Q may or may not have oriented cycles), there might be different admissible ideals I and J of KQ for which the ...

**2**

votes

**0**answers

105 views

### Re-interpreting vector spaces in a choice-less model of ZF as modules over a regular ring in ZFC

I am searching a module $M$ over a (von Neumann) regular ring $A$ ($\forall a\in A$, $\exists x\in A$: $axa=a$) with two properties:
(1) every finitely generated submodule of $M$ is projective ...

**0**

votes

**0**answers

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

**1**answer

87 views

### Existence of a left adjoint to tensor product implies projectivity

Let $S$ and $R$ be two (not necessarily commutative) $k$-algebras for $k$ a field. If I have a $S$-$R$ bimodule $_SM_R$, I can form the functor $_SM_R\otimes_R (-):R\text{Mod} \rightarrow ...

**0**

votes

**1**answer

77 views

### Norm of a number in an algebraically closed field

I consider an algebraically closed field of characteristic zero $F$ as a vector space over a real closed field $R \subset F$. I would like to define a norm on $F$ in an invariant fashion, i.e. if we ...

**5**

votes

**1**answer

218 views

### A generalized K- theory via generalized idempotents

Edit After the answer by Neil Strickland, I add the word "a ring" in this new version.
In the literature, there is a concept of generalized idempotent: an n- idempotent is an element $a$ of a Banach ...

**1**

vote

**1**answer

100 views

### Universal constructions that factor through endomorphisms

If $\cal A$ is a variety of algebras (e.g., all groups) and $\cal B$ is a subvariety defined by some set of identities $X$ (e.g., abelian groups with $X = \{xy \simeq yx\}$), then there is a functor ...

**2**

votes

**1**answer

112 views

### Trace of finitely generated projective module

Let $k$ be a field and $A$ a $k$-algebra with unit. The trace module is
$$
T(A)=A/[A,A],
$$
where $[A,A]$ is the left $A$-module generated by all elements of the form $ab-ba$ for $a,b\in A$. The ...

**1**

vote

**0**answers

64 views

### Simplest (?) example of bicrossed product Hopf algebra

Suppose we have two Hopf algebras, H and A and additionally A is (left) H-module algebra and H is (right) A comodule coalgebra. This means that A is left module over H and moreover that
...

**3**

votes

**2**answers

278 views

### Are algebraic structures uniquely identifed by their free objects?

It might be a naive question, as I am not a specialist in this field.
This is a follow-up to this question.
I want to study varieties of objects generalizing ordered monoids, in particular using an ...

**2**

votes

**2**answers

123 views

### Examples of complete distributive lattices that are not Heyting algebras

Here is a short question with a possibly simple and short answer:
I need an example of a complete distributive lattice that is not a Heyting algebra which should be an infinite complete lattice that ...

**0**

votes

**0**answers

63 views

### approximate coordinates in a one dimensional lattice

suppose I have a finite set of real numbers ${r_1, \ldots r_n \in \mathbf{R} }$ and a single real number $x \in \mathbf{R}$. Is there a fast algorithm for finding integer numbers ${i_1, \ldots i_n \in ...

**4**

votes

**1**answer

151 views

### Potentially identity elements in an Abelian group

I didn't see this problem before. I motivated by the questions
Is every commutative group structure underlying at least one (unitary, commutative) ring structure
A basic question about rings
...

**1**

vote

**0**answers

103 views

### Non commutative analogy of compact-open topology

Let $R$ be a ring, define a topology on $AUT(R)$(Or End(R)) with the following subbase:
For every two 2-sided Ideal $I$ and $J$, a subbase element is $B(I,J)=\{f\in AUT(R) \mid f(I)+J=R\}$.
We can ...

**1**

vote

**1**answer

189 views

### A question in ring theory

Is there an example of two groups $G_{1}, G_{2}$ such that there are
two non isomorphic ring $R_{1}$ and $R_{2}$ such that the additive group of both rings is isomorphic to $G_{1}$ and their unit ...

**1**

vote

**2**answers

290 views

### A basic question about rings

Perhaps this is a trivial question, but I have no idea how to justify it.
Call a pair of groups $(G_1, G_2)$ ring-compatible if $G_1$ is abelian and there exists a ring $R$ with addition and ...

**3**

votes

**2**answers

153 views

### Examples of cancellative normal semigroups

I've got a couple of things to test against normality in cancellative semigroups. A normal semigroup $S$ is one in which for any $x\in S$ we have $xS=Sx.$ This implies the Ore condition $$x,y\in ...

**1**

vote

**0**answers

106 views

### subring of the matrix algebra

Let $Mat_2(\mathbb{Z})$ be the $\mathbb{Z}$-algebra of $2\times2$ matrices with integer entries.
Let $A$ be a $\mathbb{Z}$-submodule of $Mat_2(\mathbb{Z})$ containing $\mathbb{Z}$. We want to show ...

**3**

votes

**1**answer

182 views

### On rings $R$ for which $R \cong \frac RI$ for any proper two-sided ideal $I$

This is a problem I asked in SE, but it seems the question is more suitable for MO.
Consider a ring $R$ (not necessary with identity or commutative) such that for any proper two-sided ideal $I$, ...

**2**

votes

**0**answers

47 views

### Cyclic modules over serial rings

Let $R$ be a serial ring. What can be said about the uniform dimension of cyclic $R$-modules? Specially I would like to know Is it true that every cyclic $R$-module has finite uniform dimension ?

**3**

votes

**0**answers

87 views

### Sum of projective submodules of a projective over a semihereditary ring

Sorry in advance if this is too silly. Let $R$ be a right semihereditary ring and $P$ a projective right $R$-module. It is well-known that finitely generated (thus projective) submodules of $P$ form a ...

**2**

votes

**2**answers

195 views

### How much information does the multiplicative semigroup of an algebra contain?

How much do we know about an given algebra when we only know its semigroup strucure under the product law?
How far can two algebras be distinguished by knowing only their semigroup strucure?
The ...

**0**

votes

**0**answers

130 views

### How to find $n$ such that the group of units $U(\mathbb{Z}/n\mathbb{Z})$ has a given abelian subgroup?

Given an integer $n$, we can determine the structure of the multiplicative group of integers modulo $n$ ($U(\mathbb{Z}/n\mathbb{Z})$) by the factorization of $n$. Hence we can easily find all the ...

**4**

votes

**1**answer

205 views

### Inverse limit of Gorenstein local rings is again Gorenstein?

If we have the system of surjective ring homomorphisms
$f_{i,i+1}: R_{i+1} \twoheadrightarrow R_i$
for an arbitrary $i \geq 0$ such that all $R_i$ are Gorenstein local ring. Let us put
...

**0**

votes

**0**answers

81 views

### Super-Gorenstein ideal of ${\Bbb F}_p[[X_1,\ldots,X_n]]$

Let $A \colon= {\Bbb F}_p[[X_1,\ldots,X_n]]$ be a $n$-variable power series ring over a finite field ${\Bbb F}_p$. We put ${\frak m}_A \colon= (X_1,\ldots,X_n)$.
Definition(Super-Gorenstein ideal): ...

**5**

votes

**2**answers

182 views

### Integration on Compact Semirings

I want to know if integration of functions $f:X\rightarrow G$ where $G$ is a compact semiring has been defined and if it is possible to ensure that all continuous functions are integrable. This is ...

**0**

votes

**0**answers

24 views

### Regular and Primary Polynomials

does anyone know how to prove "A regular polynomial f is primary if and only if uf(Mio f) is primary in K[x]"?
(just consider that u: R[x]-->R/m[x]
R is finite local ring and m is maximal ideal)
...

**0**

votes

**2**answers

269 views

### Dual of a module

Let $M$ be a $ \mathbb{Z}_{p}[[T]] $-module and $X=Hom(M,\mathbb{Q}_{p}/\mathbb{Z}_{p})$ be the dual of $M$. Let $X[p^n]$ denotes the $p^n$-torsion points of $X$. Is $X/X[p^n]$ the dual of $M[p^n]$ ...