**2**

votes

**1**answer

70 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

326 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

353 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

183 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

116 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?

**2**

votes

**1**answer

140 views

### $Aut(\mathbb{Z}G)=?$ for $G=\mathbb{Z}^2\rtimes_n\mathbb{Z}$

I am interested in the automorphism group of the group ring $\mathbb{Z}G$ for some noncommutative group $G$ of the form $\mathbb{Z}^2\rtimes_n\mathbb{Z}$, say
...

**11**

votes

**1**answer

359 views

### Automorphisms of $P(\Bbb N)$

I believe I've proved that the power semigroup of non-negative integers with addition has a trivial automorphism group. The proof is a bit long, completely elementary and rather unremarkable (as the ...

**4**

votes

**2**answers

170 views

### Maximal centralizer in full matrix ring

I will be so thankful if someone can help me with the following question.
Is it possible to obtain all maximal centralizers in the full matrix ring, $M_n(F)$, for an arbitrary finite field $F$? Here, ...

**3**

votes

**1**answer

184 views

### What is twisted in a twisted Poisson structure?

The usual definition of a twisted Poisson algebra involving a 3-form
does not seem to refer to a Poisson algebra that is then twisted,
i.e. twisting either the commutative product or the Lie bracket.
...

**0**

votes

**1**answer

108 views

### On separable field extensions [closed]

Let $F\subseteq K$ be a finite separable field extension with $a_1,..., a_n$ an $F$-basis for $K$. Is it true that the matrix $A := [\mbox{tr}(a_ia_j)]$ is non-singular ?

**7**

votes

**2**answers

226 views

### Is there a cohomology for magmas?

Is there a cohomology theory for magmas? Or cohomology theories for any class of non-associative algebras (other than Lie and maybe Jordan)?

**1**

vote

**1**answer

220 views

### when does one want to use the Reynolds operator in GIT?

The role of Reynolds operator in GIT has always been a little mystery to me. Actually I see that in some proofs it gets used in an efficient way, but what I cannot grasp is the general philosophy. I ...

**2**

votes

**0**answers

91 views

### integrals in Hopf algebras

Let $H$ be a Hopf algebra and $g$ be a group-like element in
$H^{*}$. Define $L_{g}=\{x\in H\mid hx=g(h)x,~\forall h\in H\}$. If $H$
is not semisimple, when can we find a group-like element $g$ in
...

**7**

votes

**2**answers

306 views

### Number of ways to write an integer as a product of irreducibles

Is there any way to tell the number of distinct ways to factor $a\in\mathcal{O}_k$ (up to units, of course) when $k$ is not a PID? A simple investigation in $\mathbb{Q}(\sqrt{-5})$ with integer ring ...

**2**

votes

**0**answers

92 views

### Separable extension

Let $A$ be an $R$-algebra, $B$ an $R$-subalgebra of $A$ and $I$ an
ideal of $A$. Kazuhiko Hirata and Kozo Sugano proved that: if $A$ is
a separable extension of $B$, then $A/I$ is a separable ...

**4**

votes

**0**answers

219 views

### Ext groups of affine scheme

Let $A$ be a commutative ring. $\textbf{Spec}(A)$ is the the
spectrum of $A$. $M$ and $N$ are $A$-modules. $\tilde{M}$ and
$\tilde{N}$ are sheaves associated on $\textbf{Spec}(A)$
respectively.
I ...

**2**

votes

**0**answers

103 views

### How do I compute the Azumaya locus?

I have an algebra $A$ that is finite over its centre $Z(A)$ and I want to compute the Azumaya locus of $Z(A)$ (or equivalently, its ramification locus).
Looking in McConnell-Robson Noncommutative ...

**4**

votes

**1**answer

238 views

### Two definitions of modules in monoidal category

The standard definition of a (left) simplicial module $V$ over some simplicial algebra $A$ is the map of simplicial vector spaces $A\otimes V\to V$ that gives the usual modules component-wise. Here ...

**2**

votes

**1**answer

165 views

### Monoids and groups of fractions

Let $G$ be a group containing a monoid $M$ that spans $G$ as a group. Is it possible to have a proper quotient $\varphi \colon G \to Q$ of $G$ such that the restriction of $\varphi$ to $M$ is ...

**0**

votes

**1**answer

146 views

### Depth of polynomial ring $S=\Bbb{R}[x_1,x_2,x_3,…,x_n,…]$

Consider the polynomial ring of countable variables with coefficients in the real numbers, i.e, $S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$. My Question is about the depth of this ring.
Question: Could we ...

**1**

vote

**1**answer

181 views

### chain of prime ideals in polynomial ring $S=\Bbb{R}[x_1,x_2,…,x_n,…]$

Consider the polynomial ring of countable variables with coefficients in the real numbers, i.e, $S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$. Is the following question true?
Question: Is there any maximal ...

**3**

votes

**2**answers

312 views

### noncommutative polynomials equality

Suppose $x$, $y$, $z$ are three variables satisfying $yz=zy$, $zx=xz$, $xy=yzx$.
Could anyone give me two (non-commutative) polynomials $f$ and $g$ in the above three variables such that the ...

**2**

votes

**0**answers

169 views

### Automorphisms of the affine quadric $z_1^2+\dots+z_n^2=1$

Denote by $V$ the vanishing locus of the affine quadric $z_1^2+\dots+z_n^2-1$ in $\mathbb{C}^n$. Clearly $V$ is a complex manifold. Let $g:V\to V$ be an automorphism of $V$ (a biholomorphism, not ...

**5**

votes

**0**answers

137 views

### Series in topological rings that only converge if almost all summands are zero

While trying to understand a certain topological ring better, I stumbled onto the following question.
Suppose $I$ is a fixed infinite index set, $R$ is a topological ring and $(x_i)_{i\in I}$ is a ...

**4**

votes

**1**answer

326 views

### Topological Problems Solved by Lattice Duality

It is well known the success of lattice dualities (as Pontryagin duality for abelian groups, Stone duality for Boolean algebras and Priestley duality for distributive lattices) to solve algebraic ...

**3**

votes

**1**answer

104 views

### left primitive but not right primitive

There is a known example due to Bergman (goes back to 1963) of a ring which is left primitive but not right primitive. I am wondering if there is another (preferably simpler) example of such a ring!?

**0**

votes

**0**answers

19 views

### SSC conditions for connected graded FBN algebras

Let $A$ is prime PI (or prime FBN) conected graded Noetherian algebras. Let
$U$ be a uniform graded right ideals of $A$, can we get a proper graded submodule of $U$ which is isomorphic to $U(-l)$ for ...

**0**

votes

**0**answers

69 views

### algebras of infinite injective dimension

Are there any connected graded Noetherian algebra of infinite injective dimension but has a balanced dualizing complex?
Thanks a lot.

**1**

vote

**1**answer

90 views

### How to calculate $(A^{-1})_{ii}$ for an invertible hyperhermitian quaternionic matrix $A$?

The article
Alesker, S. (2003). Quaternionic Monge-Ampere equations. The Journal of Geometric Analysis, 13(2), 205-238.
has the following CLAIM:
Claim. Let $A$ be an invertible hyperhermitian ...

**4**

votes

**1**answer

198 views

### Two-sided bar construction

On page 4 of this paper by H. Abbaspour, the author defines the two-sided bar construction
$$B(A,A,A):=A\otimes T(s\bar{A})\otimes A$$
of a differential graded algebra $(A,d_A)$ (over a field).
The ...

**6**

votes

**1**answer

116 views

### Abelian groups injective over their endomorphism

Let $M$ be an abelian group and let $R = \mbox{End}_\Bbb{Z}(M)$. Under what conditions (on $M$), $_RM$ is injective!?

**0**

votes

**1**answer

232 views

### Example of a ring satisfying this variant definition of “symmetric” on nilpotent elements

I want an example to show that if $a,b$ are nilpotent elements of a ring $R$ with 1 and if $c$ is any element of $R$, then $abc=0\Rightarrow acb=0$ but $cab=0$ does not imply $acb=0$.
This is unlike ...

**1**

vote

**0**answers

49 views

### injective dimension of dualizing complex

Let $R$ be a balanced dualizing complex of a Noetherian connected graded algebras $A$. Dose one always have $\text{id}_A R = \text{id}_{A^{op}} R$?
Thanks a lot.

**9**

votes

**1**answer

221 views

### Bernstein-Sato polynomial (one variable)

Let $R = \mathbb{C}[x_1,...,x_n]$, $p \in R$. There exists a monic (of lowest degree) $b_p(x) \in \mathbb{C}[s]$ and a differential operator $D(s)$ such that
$$b_p(s) p^s = D(x)p^{s+1}.$$
The ...

**1**

vote

**0**answers

44 views

### Duality of morphisms induced by multiplication of regular normal element

I was confused by the sequence of modules in [Yekiutieli and Zhang, Rings with Auslander dualizing complex, p.33, l.-10]. The question is that: why is the second morphism right multiplication of $t$?
...

**1**

vote

**2**answers

113 views

### On modules with finite uniform dimension

Is it true that if a module $M$ has finite uniform dimension then the same is true for its homomorphic images ?

**2**

votes

**2**answers

184 views

### Tychonoff spaces and ideals

Let $X$ be a tychonoff space and let $T$ be the set of all $f \in C(X)$ such that for any $g$ the equation $fg = 1$ has at most finitely many solutions. Under what conditions on $X$, the set $T$ is an ...

**2**

votes

**0**answers

56 views

### Quotient categories and essential extension

Let $A$ be a right Noetherian positively graded ring. Let $Gr(A)$ be the category of right graded $A$-modules, and $Tors(A)$ be the full subcategory of $Gr(A)$ of torsion modules. Let $QGr(A)$ be the ...

**0**

votes

**0**answers

72 views

### Krull dimension of Noetherian positively graded rings

Suppose that $A$ is a positively graded Noetherian ring and $x\in A$ is a homogeneous normal element of positive degree. Do we have $K\dim A/x < K\dim A$?
Krull dimension of rings is the one ...

**7**

votes

**1**answer

445 views

### On the full rings of matrices

Let $R$ be a ring with elements $a,b$ such that $a^2 = 0 = b^2$ and $a+b$ is a unit. How to show that $R$ is a ring of $2\times2$ matrices?
COMMENT: The converse is clearly true, just take $a = ...

**7**

votes

**1**answer

171 views

### Rings in which every module has an injective image

Consider the class of rings $R$ with identity such that any left $R$-module has a non-zero injective homomorphic image. Any such ring is clearly a left V-ring. Is it true that any such ring must be ...

**3**

votes

**2**answers

222 views

### on a characterisation of regular D-modules

Let $X$ be a smooth variety over a field of characteristic zero. Let $M$ be in the derived category of holonomic $\mathcal{D}_{X}$-modules, $D^{b}_{h}(\mathcal{D}_{X})$.
We know that if we assume ...

**2**

votes

**3**answers

433 views

### Is it true that simple projective modules are injective?

It is known that simple modules are either projective or singular. Is it true that simple projective modules over (commutative) rings are injective ?

**8**

votes

**2**answers

451 views

### Origin of the Socle of a module

Where does the notation $\mbox{Soc}(M)$ (the sum of all simple submodules of a module $M$) first appear?

**4**

votes

**0**answers

209 views

### Hermitian forms over quaternion algebra

Notations: Let $Q=(a,b)$ be a quaternion algebra over a field of characteristic $\neq 2$, i.e. $i^2=a, j^2=b, k=ij, ij=-ji$. Consider $K=k(t)(\alpha)$, where $\alpha=\sqrt{at^2+b}$. Let ...

**-2**

votes

**1**answer

304 views

### For a ring $A$, is $A$ Morita equivalent to $M_\infty(A)$? [closed]

Let $A$ be a ring, let $M_n(A)$ be the ring of $n$-by-$n$ matrices with elements in $A$, $A$ is Morita equivalent to $M_n(A)$, I was wondering if this also applied to infinite matrices? That is, if ...

**3**

votes

**2**answers

171 views

### A ring with 2 minimal left ideal and 3 minimal right ideal

Is there a (finite) ring with exactly 2 minimal left ideal and exactly 3 minimal right ideal? (rings are assumed to have identity)

**0**

votes

**1**answer

126 views

### Local rings with simple radical

Is there a (finite) non-commutative local ring $R$ (containing identity) such that $J(R)$ is simple as a left module?

**3**

votes

**0**answers

153 views

### Comparing different Euclidean algorithms on a Euclidean domain

I have posted this question at stackexchange (502413), without responses until now.
In the papers by T. Motzkin: The Euclidean Algorithm, Bull. AMS 55, 1949, pp. 1142--1146 and P. Samuel: About ...

**3**

votes

**3**answers

309 views

### A question concerning the isomorphic type of continuous functions

let $S$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are continuous outside a bounded open interval containing zero (depended on $f$). Is it possible to consider $S$ as ...