# Tagged Questions

**0**

votes

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64 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 ...

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63 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 ...

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vote

**1**answer

98 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 $M$)

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158 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 ...

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votes

**1**answer

216 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 ...

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votes

**2**answers

249 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 ...

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99 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 ...

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242 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 ...

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101 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?

**7**

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163 views

### Square of primary ideals

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

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170 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 ...

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74 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 ...

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334 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 ...

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votes

**1**answer

368 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 ...

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**1**answer

190 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 ...

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118 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?

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145 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
...

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381 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 ...

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196 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, ...

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votes

**1**answer

195 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.
...

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votes

**1**answer

111 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 ?

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235 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)?

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237 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 ...

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94 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
...

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339 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 ...

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94 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 ...

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226 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 ...

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votes

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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 ...

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votes

**1**answer

244 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 ...

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182 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 ...

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votes

**1**answer

152 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 ...

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**1**answer

200 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 ...

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votes

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320 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 ...

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votes

**0**answers

177 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 ...

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146 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 ...

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votes

**1**answer

406 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 ...

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**1**answer

113 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!?

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72 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.

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vote

**1**answer

103 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

241 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 ...

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votes

**1**answer

124 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!?

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**1**answer

238 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 ...

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50 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.

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votes

**1**answer

265 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 ...

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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 $t$?
...

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vote

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116 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 ?

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185 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 ...

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62 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 ...

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79 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 ...

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**1**answer

447 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 = ...