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

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8
votes
1answer
211 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
230 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
462 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
481 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
261 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
322 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
173 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
160 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
313 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
497 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
221 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
477 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
299 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
187 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 ...
5
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
588 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
303 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
383 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
534 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
291 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
259 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
199 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.
8
votes
2answers
357 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 ?
0
votes
1answer
189 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 ?
2
votes
1answer
136 views

How to compute that Socle of the full matrix rings

Ler $R$ be a ring with identity. I believed that $M_n(\mbox{Soc}(R_R)) = \mbox{Soc}(M_n(R)_{M_n(R)})$ but I can not prove it. Any help would be helpful.
4
votes
1answer
181 views

When are infinite dimensional path algebras hereditary?

I allready asked this on MO, but did not get any answer. Given a finite quiver with relations. When is the path algebra modulo relations hereditary? If the path algebra is finite dimensional or ...
1
vote
1answer
351 views

The space $\psi$

Is the space $\psi$ (described in problem 5I of L. Gillman and M. Jerison, Rings of continuous functions, Springer Verlag, 1976) a F-Z-space (i.e, space with $cl(X-Z(f))$ is a zero set for every $f$ ...
3
votes
2answers
290 views

How to understand a solenoid?

Consider the invertible extension of the circle-doubling map T(x)=2x (mod 1), the new system can be represented as X={(x_k)|x_{k+1}=T(x_k)}(see GTM 259 ...
0
votes
1answer
117 views

Number of minimal left ideals

Is there any way to compute the number of minimal left ideals of $M_n(K)$, the full $n\times n$ matrix ring with entries in the field $K$ ?
6
votes
4answers
368 views

How universal is operadic approach to studying algebras?

I have just started to read about operads, so this question might be silly. So it seems to me that any "reasonable" class of algebras can actually be defined as a class of all algebras over a certain ...
1
vote
1answer
168 views

elementary question on a completion of a ring

Let $k$ a field, and $k[\epsilon]=k[X]/(X^{2})$ , what is the completion of the ring $k[\epsilon][t]$ with respect to the ideal $(t^{2}+\epsilon)$?
0
votes
1answer
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
0answers
107 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
1answer
127 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
0answers
148 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
2answers
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
0answers
139 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
2answers
545 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
2answers
220 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 ...
4
votes
1answer
132 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
0answers
200 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
1answer
215 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
1answer
176 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
1answer
345 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
1answer
103 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
1answer
534 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
2answers
573 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
1answer
133 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. ...