**0**

votes

**1**answer

85 views

### $\mathrm{rk}_R M$ vs $\mathrm{rk}_S M$ - how nice need $R,S$ be?

Let $R\hookrightarrow S$ be Noetherian (noncommutative) rings without zero divisors with $\mathrm{rk}_{R} S < \infty$ (e.g. $S=R*G$ the crossed product of $R$ with a finite group $G$). Let $M$ be a ...

**2**

votes

**0**answers

104 views

### Artinian rings where all injective hulls of simple modules have finite length

Let $A$ be a unitary artinian ring. Suppose additionally that the injective hulls of all simple modules have finite length (or, equivalently, are finitely generated).
Does the ring $A$ have to ...

**2**

votes

**1**answer

123 views

### A class of rings related to rings with IBN property [closed]

A ring $R$ (with 1) has IBN property if free $R$-modules have unique rank (e.g., commutative rings). In the same fashion, lets call $R$ a good ring if in every free $R$-module any independent set can ...

**6**

votes

**0**answers

144 views

### Is it possible to define Hopf algebra as an object in the category of (associative) algebras?

I think this must be well-known, but I can't find references, so my apologies from the very beginning.
Consider first the notion of bialgebra. It is usually defined as an object $B$ in the category ...

**5**

votes

**2**answers

408 views

### Isomorphic rings of functions

Let $X$ and $Y$ be two topological spaces with $C(X) \cong C(Y)$ (where $C(X)$ is the ring of all continuous real valued functions on $X$). I know that we can not conclude that $X$ and $Y$ are ...

**4**

votes

**0**answers

138 views

### Injectivity criterion for surjective coalgebra maps: does it hold in full generality?

Let $k$ be a commutative ring.
Let $C$ be a filtered $k$-coalgebra. This means a $k$-coalgebra equipped with an increasing $k$-module filtration $C^0 \subseteq C^1 \subseteq C^2 \subseteq ...$ ...

**9**

votes

**2**answers

542 views

### Countable Maximal Ideals

This may be simple but I can not see a way. I am looking for an uncountable ring (with 1) containing a countable maximal left ideal which is not a direct summand (as a left ideal).

**5**

votes

**2**answers

211 views

### How big can a commutative subalgebra of Weyl algebra be?

Consider the smallest Weyl algebra $A_1=\{q,p; qp-pq=1\}$. It is known that there exist pairs of commuting elements, say $L$ and $M$, that obey various polynomial relations, e.g. elliptic curves. I ...

**3**

votes

**1**answer

128 views

### Are the heredity ideals in an heredity chain always finitely generated?

An ideal $J$ of a ring $A$ is a heredity ideal of $A$ if:
$J^2=J$;
$J$ is a projective $A$-module;
$J (\operatorname{Rad}A)J=0$.
A (unitary) semiprimary ring $A$ is said to be quasihereditary if ...

**2**

votes

**0**answers

195 views

### Using the Affine Maxima Package

The Maxima computer algebra system has a package called Affine for doing the calculations implicit in Bergman's diamond lemma for rings. It can be viewed as a kind of noncommutative analogue of ...

**2**

votes

**1**answer

212 views

### Symmetric algebra of an ideal and syzygies

Let $(R,\mathfrak{m})$ be a Cohen Macaulay local ring and $I=(a_1,\ldots,a_g,a_{g+1})$ an almost complete intersection ideal of codimension $g.$ Let $R^k\longrightarrow R^{g+1}\longrightarrow ...

**3**

votes

**1**answer

171 views

### Is the definition of Gerstenhaber bracket related to operads?

I was reading these notes by Keller. On the page 19 he defines the Gerstenhaber bracket on the Hochschild cochain complex. But first of all he defines the operation $\bullet$ on the cochains by
$$
...

**8**

votes

**1**answer

338 views

### Generalizing detropicalization

Given an identity in max,plus arithmetic, are there ways to turn it into an ordinary algebraic identity it other than by replacing addition by multiplication and replacing max by series-plus or by ...

**1**

vote

**1**answer

100 views

### Invariance of reduced trace of Azumaya algebras

Let $A$ be an Azumaya algebra over the commutative ring $R$ and let
$\newcommand{\tr}{\operatorname{tr}} \tr\colon A \to R$ be the reduced trace as defined (for example) in Section IV.2 of M.-A. Knus, ...

**2**

votes

**1**answer

476 views

### Commutative associative rational binary operations

What are all the nondegenerate rational binary operations that are commutative and associative? (Examples: $(x,y) \mapsto x+y$, $xy+x+y$, $xy/(x+y)$.)
Feel free to re-tag if you can think of ...

**4**

votes

**2**answers

552 views

### Rings with a finite maximal ideal

A field has a finite maximal ideal. Of course the converse is not true. So is there a characterization for those rings having a finite maximal ideal ??
EDIT: The characterization below is very nice. ...

**1**

vote

**1**answer

132 views

### Idempotents in Green J classes

I recently read this article Syntactic semigroups. In page $8$, he speaks about a J class having an idempotent is called regular:
A $\mathcal J$-class containing an idempotent is called regular. ...

**4**

votes

**1**answer

280 views

### Rings with group of units cyclic of prime order

For what prime numbers $p$ there exists a ring with identity and exactly $p$ invertible elements ?
REMARK It can be shown that for $p=5$ there is no such ring, so I am wondering for what values of ...

**3**

votes

**2**answers

262 views

### Do morphisms of finitely-decomposable Quiver representations map indecomposables nicely?

Consider two quivers $Q$ and $Q'$ of type $A_n$, laid out horizontally like so:
Given representations of $Q$ and $Q'$, Gabriel's theorem guarantees the existence of finitely many indecomposables ...

**2**

votes

**1**answer

114 views

### A non-orthomodular orthocomplemented lattice identity?

Assume you have an orthocomplemented (but possibly not orthomodular) lattice $L$. For $q,r\in L$ say "$q$ and $r$ are in position $p'$" to mean that $q\wedge r^\perp=0$ and $r\wedge q^\perp=0$. Is ...

**4**

votes

**1**answer

400 views

### “as close to being semisimple as it can possibly be.”

I had originally asked this question on math stack exchange but I think maybe it's more appropriate to ask it here.
In the paper of Beilinson, Ginzburg and Soergel entitled "Koszul Duality ...

**2**

votes

**0**answers

79 views

### Usefulness of polynomial ideals for graphs

Given a graph $G=(V,E)$ on $n$ vertices, one can associate to it the polynomial $f \in k[x_1,\dots,x_n]$ given by $f = \prod_{\{i,j\}\in E}(x_i - x_j)$. Then one can map various graph properties into ...

**1**

vote

**0**answers

133 views

### simple tensor product of modules over algebras

Let $M$, $N$ be simple modules over associative algebras $A$ and $B$ (over $\mathbb{C}$), respectively. When is $M\otimes N$ simple as a $A\otimes B$-module?
It is right if $A$ or $B$ has a ...

**3**

votes

**2**answers

489 views

### Shape of axioms in abstract algebra

When defining abstract algebraic structures (like monoids, groups, etc...), are there some constraints on the shape of the axioms, for the structure to have good properties that we implicitly use in ...

**7**

votes

**2**answers

205 views

### Two infinite dimensional algebras such that the center of their tensor product is bigger than the tensor product of their centers

I am searching for two infinite dimensional algebras such that the center of their tensor product is bigger than the tensor product of their centers. Who knows of such examples? Thanks a lot.

**2**

votes

**1**answer

101 views

### Degree principles for non-symmetric polynomials

A theorem of Timofte says that a symmetric polynomial inequality of degree $d$ holds on $\mathbb{R}^{n}_{+}$ if and only if it holds for all vectors in $\mathbb{R}^{n}_{+}$ with at most $\max\{\lfloor ...

**11**

votes

**1**answer

323 views

### $\mathbb{Z}G$ (left) Noetherian$\Rightarrow$ $l^1(G)$ is a flat $\mathbb{Z}G$-(right) module?

Let $G$ be a countable discrete group (not necessarily abelian), and suppose the group ring $\mathbb{Z}G$ is a left-Noetherian ring, for example, when $G$ is a polycyclic-by finite group.
Denote the ...

**2**

votes

**2**answers

124 views

### Are there atomistic ortholattices which are not modular?

Let $L$ be an atomic ortholattice. We say that two elements $a$ and $b$ of $L$ are orthogonal if $a\leq b^\perp$. If $L$ is orthomodular then every element of $L$ can be written as a join of pairwise ...

**7**

votes

**1**answer

170 views

### $K_0$ group of graph underlying an approximately finite (AF) C* algebra

Say we have an AF C* algebra $A$ described by some Bratteli diagram $E$. If $M_\infty (A)=\displaystyle{\lim_\rightarrow M_n(A)}$ and $P(A)$ are the projections in this algebra, we know that ...

**0**

votes

**1**answer

272 views

### GCD computation for multiple polynomials and degree of Bezout coefficients

Assuming two polynomials $P_1,P_2 \in \mathbb{Z}_p[r]$ of degree $n$, with no common factors, we know that there exist polynomials $Q_1,Q_2$ s.t.: $Q_1P_1 + Q_2P_2 =1$. From Bezout's identity we also ...

**0**

votes

**1**answer

225 views

### Finite extension of a field [closed]

Is it true, that if $A$ is finitely generated commutatative algebra over a field $k$, not necessary algebraically closed, then prime ideal $p \subset A$ is maximal if and only if $k \subset Quot(A/p)$ ...

**4**

votes

**2**answers

535 views

### Example of a reduced ring whose completion is not reduced

Let $(R,\mathfrak{m})$ be a local Noetherian ring. Give an example such that R is reduced but $\mathfrak{m}$-adic completion of R is not reduced.

**7**

votes

**2**answers

280 views

### Where should I search for resolutions?

In my research, to test some conjectures or just to illustrate some facts, I often need to compute some explicit examples of derived functors (in the sence of Quillen's model categories). Mainly I ...

**1**

vote

**0**answers

124 views

### Ring of Witt Vectors and Tensor product of Fields

Let $p > 2$ be a prime, and let $\textbf{F}_{p} =
\textbf{Z}/p\textbf{Z}$. Let $k_{1}$ be a finite field over
$\textbf{F}_{p}$, and let $k$ be a perfect field of characteristic
$p$. Then we have ...

**3**

votes

**1**answer

195 views

### Polynomial identities for mod p matrices

Can there be a polynomial over the field $F_p$ of $p$ elements ($p$ prime) in non-commuting variables $X_1,..., X_r$ such that:
1) $f(A_1,...,A_r)=0$ for every $n \times n$ matrices $A_1,...,A_r$ ...

**3**

votes

**1**answer

120 views

### something like a kleene algebra of rational functions?

There's a procedure that control engineers (used to) do to calculate the transfer function of a linearized system, gradually reducing a block diagram to a rational function of s. It's justified by ...

**1**

vote

**1**answer

110 views

### cobar construction of a direct sum of coalgebras

I have the following question. Let's denote by $\Omega$ the cobar functor, by $C$ some DG-coalgebra (not co-commutative) over a field $k$. Also suppose $V=k\times\dots\times k$ is just the product of ...

**15**

votes

**1**answer

385 views

### Geometry of numbers for three by three matrices?

While trying to use Minkowski's theorem to calculate the (left) class number of a noncommutative ring, I ran into the following problem:
What is the volume of the largest symmetric convex subset ...

**2**

votes

**0**answers

83 views

### Nontrivial examples of rings of relative stable rank 1

Given a (commutative) ring $R$, one says that the stable rank $sr(R)\leq n$ if any unimodular row $(a_1,\ldots,a_{n+1})$ of length $n+1$ is stable, i.e. there exist $x_1,\ldots,x_n$ such that ...

**2**

votes

**2**answers

108 views

### Specialization of PBW-algebras over rational function field

I have been thinking about the concept of specialization of algebras defined over the field of rational functions $k = \mathbb{C}(t)$ (I'm using $\mathbb{C}$ for my work, but the question can be asked ...

**3**

votes

**1**answer

150 views

### Morphism of algebras

Let $Q(i)$ be the extension of the rational numbers $Q$ obtained by adjoining a root i of the polynomial $X^2 + 1$.
Consider the algebra B defined by the Hilbert symbol $(-2, -5)$ over $Q(i)$. So, by ...

**11**

votes

**2**answers

424 views

### What is a Kelley ring?

I've heard that in some book by someone named Kelley, perhaps an early edition of John L. Kelley's General Topology, the author gave a definition of a ring which turned out to be weaker than the usual ...

**4**

votes

**0**answers

105 views

### What is the composition in SesquiAlg?

To motivate my question, I will describe a famous 2-category. First, there is the 1-category $\text{Vect}$ of vector spaces (over some fixed field). This category has a tensor product $\otimes$, and ...

**4**

votes

**1**answer

189 views

### Fermat’s Two Squares for polynomials

Is there an analog of Fermat’s Two Squares theorem over polynomial rings $\Bbb Z[x]$ by considering a version for non-negative polynomials?
Fermat two squares says primes $p$ is sum of two squares ...

**1**

vote

**1**answer

122 views

### Finding reducible polynomials with restricted factors

Given $f(x),g(x) \in \mathbb{Z}[x]$, two irreducible polynomials, is there a polynomial $h(x) \in \mathbb{Z}[x]$ coprime to $f(x)$ such that $f(x) + g(x)h(x)$ is reducible over $\mathbb{Z}[x]$ with ...

**3**

votes

**1**answer

193 views

### Existence of a generalized matrix inverse over an arbitrary field?

Let $A\in M_n(K)$ be a square matrix over a field $K$. The notion of inverse matrix was generalized by Moore and Penrose for real and complex matrices
(also called pseudo-inverse $A^{\dagger}$ of $A$, ...

**4**

votes

**1**answer

201 views

### A question about Weil algebra

Let $G$ ba a compact Lie group with Lie algebra $\mathfrak{g}$, $\mathfrak{g}^{*}$ be the dual of $\mathfrak{g}$. We known the Weil algebra is
$$W(\mathfrak{g})=\wedge(\mathfrak{g}^{*})\otimes ...

**7**

votes

**2**answers

571 views

### Quintic polynomial solution by Jacobi Theta function.

Does someone have a good and rigorous reference for the solution of quintic ploynomial equation with Jacobi Theta function, in English?
Mathworld and Wikipedia don't give a good English reference, at ...

**3**

votes

**1**answer

136 views

### Thinking about the quadratic dual graphically

Say you have a quadratic algebra $A$ , that is, an algebra defined by a finite list of generators over a ground ring $k$ (either a field or a direct product of fields, which will be assumed ...

**5**

votes

**0**answers

199 views

### On the linear transformation between matrix space

Suppose we have two matrix subspaces, $n\times n$ matrix subspace $S_1$ and $m\times m$ matrix subspace $S_2$. Every element of $S_1$ and $S_2$ is complex symmetric matrix.
Suppose there exists ...