**2**

votes

**1**answer

232 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 ...

**4**

votes

**1**answer

199 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

83 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

248 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 ...

**6**

votes

**0**answers

155 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

88 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?

**3**

votes

**0**answers

119 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 ...

**4**

votes

**1**answer

377 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 ...

**1**

vote

**1**answer

222 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

82 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

255 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

185 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

286 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

81 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
...

**7**

votes

**1**answer

152 views

### riemann mapping theorem for skew-fields of quaternions and beyond

Is there any known generalization of the riemann mapping theorem over skew-fields of quaternions and beyond or at least a conjectured formulation of it?
In a less focused way, how far does the main ...

**3**

votes

**2**answers

322 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 ...

**3**

votes

**2**answers

507 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

68 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

161 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

115 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

253 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

314 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

184 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

116 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

210 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

59 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 ?

**4**

votes

**0**answers

103 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

203 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

137 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 ...

**5**

votes

**1**answer

235 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

90 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): ...

**7**

votes

**2**answers

351 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

**2**answers

295 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 $M[p^n]$ ...

**13**

votes

**2**answers

475 views

### Does base extension reflect the property of being isomorphic?

Let L/K be a (separable?) field extension, let A be a finite dimensional algebra over K, and let M and N be two A-modules. Let $A' = L \otimes_K A$ be the algebra given by extension of scalars, and ...

**0**

votes

**0**answers

68 views

### What is a necessary and sufficient condition that the kernel of a semi-module homomorphism is a partitioning sub-semi-module?

I would like to identify a representation of the subcategory of a comma category of semi-rings, whose objects are abelian group objects.
When attempting to identify the representation, the following ...

**1**

vote

**0**answers

65 views

### An invariant number of modules over Auslander Gorenstein modules

Given an Auslander Gorenstein $R$ of injective dimension $\mu$, one can associate with each finitely generated module $M$ with a number $\varepsilon_{\mu}(M)$, which is the number of the ...

**1**

vote

**1**answer

99 views

### When for every module $M$, $|E(M)| = |M|$

Is there a non-semisimple ring $R$ such that for any left $R$-module $M$, $|E(M)| = |M|$ ? (where $E(M)$ is the injective hull of $M$ and $|M|$ is the cardinality of $M$)

**2**

votes

**1**answer

161 views

### “order two sequence” in a paper of Waldhausen

In F. Waldhausen's paper "Algebraic K-theory of generalized free products, Part I", page 142,line 19, there is a term "order two sequence". Can anyone explain its meaning to me?
According to the ...

**5**

votes

**2**answers

273 views

### Proving that a semigroup is regular

In a number of diverse situations of interest to me (mostly associated with something called the abelian sandpile model), one can define a nonabelian semigroup generated by commuting elements ...

**0**

votes

**2**answers

265 views

### Rank of a $ \mathbb{Z}_{p}[[T]] $ module

Let $p$ be a prime and $M$ is a finitely generated $ \mathbb{Z}_{p}[[T]] $ module. Suppose $M[p]$ denotes the $p$-torsion of $M$. Then $M[p]$ and $M/(p)$ are both $ F_{p}$ vector spaces. So we can ...

**1**

vote

**0**answers

110 views

### Universally catenary and all its formal fibers over minimal members are Cohen-Macaulay but it has a nonCohen-Macaulay formal fiber

Please help me to find a Noetherian local ring $R$ such that: $R$ is universally catenary and all its formal fibers over minimal members of $Spec(R)$ are Cohen-Macaulay but $R$ has a nonCohen-Macaulay ...

**3**

votes

**3**answers

268 views

### Weyl algebra and its nontriviality

The Weyl algebra (say, over $\mathbb{C}$) is an universal unital algebra with two generators $x,y$ subject to the relation $xy-yx=1$. This algebra can be constructed in the following way: take two ...

**1**

vote

**0**answers

112 views

### Chinese Remainder Theorem

On a non-commutative ring we have this generalization of CRT.
I wonder if there is a statement involving only left or right ideals.
do you know any?

**6**

votes

**0**answers

167 views

### Square of primary ideals

Is there any example of a $P$-primary ideal $I$ in a noetherian domain $R$ such that $I^2=PI \not=P^2$?

**4**

votes

**1**answer

178 views

### What is the formula for the commutative multiplication on CP(infinity)?

There is a classic formula for maps $\mathrm{CP}(r) \times \mathrm{CP}(s) \to \mathrm{CP}(r+s)$ or maybe $r+s+1$ using Plücker coordinates - IF memory serves. In the limit we get the abelian ...

**2**

votes

**1**answer

80 views

### Rings in which every J-matrix is non-singular

Let $R$ be a ring with identity. A matrix $A=[a_{ij}] \in M_n(R)$ is called a J-matrix if for any $i$, $a_{ii} \not \in J(R)$ but for any $i \not = j$, $a_{ij} \in J(R)$. Now suppose that every ...

**6**

votes

**1**answer

339 views

### Morita theorem for simplicial rings

My question is the following: is there an analog of Morita theorem in the simplicial setting?
I mean, we can define two simplicial rings $A,B$ to be simplicially Morita equivalent is the categories ...

**4**

votes

**1**answer

391 views

### Homotopy of quivers

The matrix ring $k^{n\times n}$ can be realized in many ways as a quotient of a path algebra: For example choose the quiver $1\leftrightarrows 2 \leftrightarrows \cdots \leftrightarrows ...

**4**

votes

**1**answer

211 views

### Algebra structure $Tor(A,A)$

This is a question i asked on math.stackexchange but i didn't get any answer.
Let $A$ be algebra over commutative ring $k$ and $P_{\bullet}=(P_i,d_i)\rightarrow A$, $k$ projective resolution. Then we ...

**4**

votes

**1**answer

130 views

### Localizations of hereditary rings

It is known that if a commutative Noetherian ring $R$ is hereditary then for any maximal ideal $M$ the localization $R_M$ is also hereditary. Is the Noetherian assumption necessary?