Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups.

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4
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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
0answers
105 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
1answer
249 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
1answer
194 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
1answer
155 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
1answer
207 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 ...
2
votes
2answers
322 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
0answers
178 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 ...
6
votes
0answers
148 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
1answer
429 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
1answer
116 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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0answers
73 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
1answer
107 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
1answer
253 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
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1answer
128 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
1answer
239 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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vote
0answers
56 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
1answer
285 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
0answers
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
2answers
119 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
2answers
189 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
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0answers
65 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
0answers
81 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
1answer
451 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 = ...
8
votes
1answer
213 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
2answers
234 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
3answers
479 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 ?
9
votes
2answers
487 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
0answers
282 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
1answer
326 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
2answers
175 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
1answer
127 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
0answers
164 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
3answers
315 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 ...
4
votes
3answers
577 views

Units in a group algebra

Let k be a field and let G be a finite group. I would like to know if there is any nice description of the group of units in the group algebra kG. (If there is no nice answer in this generality, ...
0
votes
1answer
222 views

Which algebra of functions can be represented as $C(X)$

I don't know if this problem is known or not, so any information would be appreciated: Let $\cal A$ be an $\Bbb{R}$-algebra of (not necessary continuous) real valued functions defined on a ...
8
votes
2answers
485 views

rings in which every element is a sum of two commuting idempotents

Is there a known characterization of the rings $R$ (containing $1$) with this property: every element of $R$ is a sum of two commuting idempotents.
4
votes
1answer
301 views

Does $\text{Mat}_n(k)$ have some universal properties similar to its universal enveloping algebra?

Let $k$ be a field and $\text{Mat}_n(k)$ be $n \times n$ matrices over $k$. Let's consider $\text{Mat}_n(k)$ as an associative algebra and denote $gl_n(k)$ be the same $k$-linear space as ...
3
votes
3answers
190 views

Modules for an idempotented algebra

Recall that an associative algebra $A$ is called idempotented provided that is the filtered union of subalgebras $eAe$ for $e \in A$ idempotent. I think sometimes people say that $A$ has approximate ...
4
votes
1answer
421 views

Uncountable Reduced ring $R$ with $R[x]$ has only a countable number of maximal left ideals

The question is following: Is there an uncountable reduced ring (i.e., a ring with no non-zero nilpotent element) $R$ (with identity) such that $R[x]$ has only a countable number of maximal ...
14
votes
3answers
598 views

Can one show the equivalence of the abstract and classical Jordan decompositions for simple Lie algebras without complete reducibility?

The following fact is basic in the theory of complex Lie algebras: Theorem. Let ${\mathfrak g} \subset {\mathfrak gl}_n({\bf C})$ be a simple Lie algebra, and let $x \in {\mathfrak g}$. Let $x = ...
4
votes
2answers
308 views

Is there a Tychonoff space $X$ of cardinality not of the form $2^\alpha$ such that $|C(X)| = |X|$

Let $X$ be the real line with the usual topology. Then clearly $|C(X)| = c = |X|$ and on the other hand $|X| = 2^{\aleph_0}$. Now my question is as in the title: Is there a Tychonoff space $X$ of ...
10
votes
2answers
387 views

Are these rings of functions isomorphic?

Let $R$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are continuous outside $(-1,1)$ and let $S$ be the ring of all functions $f : \Bbb{R}\longrightarrow \Bbb{R}$ which are ...
12
votes
1answer
535 views

Counting representations of $k[x,y]$ when $k$ is finite

$\newcommand{\GFq}{\mathbf F_q}$ Let $r_n(q)$ denote the number of isomorphism classes of $n$-dimensional modules of the $\GFq$-algebra $\GFq[x,y]$. Is it known whether there exists a polynomial ...
5
votes
1answer
292 views

Dualizing Complexes

Let $R$ be a dualizing complex of a Noetherian graded algebra $A$ (not necessary commutative). For any $M\in D_c^b(A)$, there is a natural morphism $$ \theta: R\Gamma_m(M) \to ...
9
votes
2answers
267 views

Non-commutative reduced rings of order $p^2$

Let $p$ be a prime number. Is there a non-commutative reduced ring of order $p^2$? (Note that any ring of order $p^2$ with identity is commutative).
3
votes
1answer
201 views

Characterization of non-commutative local rings of orders 64 and 128

I need the characterization (up to isomorphism) of non-commutative local rings (with identity) of orders 64 and 128. If you know the characterization or a reference, please let me know.
9
votes
2answers
377 views

Is it true that if $M$ is injective then $S^{-1}M$ is also injective

Let $R$ be a commutative ring with identity and let $S$ be a multiplicative subset of $R$. Is it true that for any injective $R$-module like $M$, $S^{-1}M$ (as the $S^{-1}R$-module) is also injective ...
11
votes
2answers
1k views

Writing a matrix as a sum of two invertible matrices

Let $n\geq 2$. Is it true that any $n\times n$ matrix with entries from a given ring (with identity) can be written as a sum of two invertible matrices with entries from the same ring ?
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1answer
204 views

Number of Minimal left ideals in the full matrix ring over a finite commutative local ring

Inspired with another QUESTION I would like to know the number of minimal left ideals of $M_n(R)$ in terms of $n$ and $R$ where $R$ is a finite local commutative ring with identity ?