**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 = ...

**8**

votes

**1**answer

205 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

439 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

455 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

217 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

314 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

172 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

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

**0**answers

155 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

310 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

**3**answers

448 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

**1**answer

216 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

**2**answers

461 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

**1**answer

290 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 ...

**1**

vote

**1**answer

127 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

**1**answer

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

**3**answers

568 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

**2**answers

291 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

**2**answers

381 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 ...

**11**

votes

**1**answer

507 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

**1**answer

288 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

**2**answers

246 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).

**2**

votes

**1**answer

186 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

**2**answers

351 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

**2**answers

997 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

**1**answer

167 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

**1**answer

135 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

**1**answer

176 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

**1**answer

339 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

**2**answers

275 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

**1**answer

113 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

**4**answers

364 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

**1**answer

165 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

**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

101 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

120 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

142 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

406 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

540 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

206 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

192 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

207 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

163 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

334 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

98 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

452 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 ...