# Questions tagged [ra.rings-and-algebras]

Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.

2,444
questions

**-2**

votes

**0**answers

54 views

### An special ideal in an integral domain [closed]

I am looking for an integral domain $D $ and an ideal $I $ of $D $ such that $I $ has infinitely many minimal prime ideal( overy $I $).

**0**

votes

**0**answers

48 views

### Symmetric functions for multidimensional variables

I have $N$ variables (let's call them $X$) of dimensionality $D$, that I want to symmetrize.
For $D=1$ I know I can use e.g. the elementary symmetric polynomials to accomplish this. What if $D>1$...

**3**

votes

**0**answers

45 views

### On grades of torsion modules in noetherian rings

Let $A$ be a (not necessarily commutative) two-sided noetherian ring with minimal injective coresolution $(I_i)$ of the regular module $A$ as a right module.
Say that $A$ has dominant dimension $n$ ...

**14**

votes

**4**answers

617 views

### What is known about ordinary character values at involutions?

Let $G$ be a finite group and let $\chi$ be the character of an irreducible complex representation $\rho$ of $G$ on $V$.
Let $x$ be an involution in $G$.
I'd like to ask the following
Question 1:
...

**3**

votes

**2**answers

133 views

### Polynomial identities of supercommutative-gradable algebras

All algebras below are associative, and not assumed unital, and, to fix ideas, over the complex numbers.
An algebra $A$ is supercommutative-gradable if it admits a grading $A=A_0\oplus A_1$ in $\...

**-3**

votes

**0**answers

75 views

### Prove that a ring homomorphism $f: \mathbb{Z}_m \rightarrow \mathbb{Z}_n $ is injective iff $n|m$ and $\gcd(n/m,m) = 1$ [closed]

Prove that a ring homomorphism $f: \mathbb{Z}_m \rightarrow \mathbb{Z}_n $ is injective iff $n|m$ and $\gcd(n/m,m) = 1$
I already did the proof of f being injective under the hypothesis, but I have ...

**-1**

votes

**0**answers

31 views

### Direct product of rings and its automorphisms [closed]

What are all the possible automorphisms over a direct product of rings? Are there any way to prove that they are the only ones?

**0**

votes

**0**answers

30 views

### Similar polynomial forms over different rings [closed]

The aim of this post is to know about results related to polynomial expressions evaluated over different rings (with respect to different operations of the ring).
Suppose we have a polynomial $P_d$ ...

**4**

votes

**2**answers

163 views

### The correct homotopically relevant notion of ideals of dg-algebras (or $\mathbb E_1$-rings)

I'm trying to figure out what an ideal of a, say, dg-algebra (or, if you prefer, $\mathbb E_1$-ring) $R$ is in a homotopically relevant fashion, but I can't actually figure it out. I can assume that $...

**4**

votes

**0**answers

114 views

### The Jacobson radical as a bimodule

Let $A$ be a finite dimensional algebra with Jacobson radical $J$.
Question 1: In case $A$ is a Nakayama algebra with a linear quiver corresponding to a Dyck path $D$ (via its Auslander-Reiten ...

**1**

vote

**0**answers

121 views

### When localization is indecomposable

We know that if $R $ is a domain then any localization of $R $ at any multiplicative subset of $R $ is indecomposable, that is, has no non trivial idempotents. Now let $R $ be a commutative ring with ...

**3**

votes

**1**answer

86 views

### Cohn localization examples

I'm working on my master's thesis, part of which involves an exposition on Cohn localization. (nlab discussion)
In Free ideal rings and localization in general rings, Sec 7.4, Cohn gives a ...

**1**

vote

**0**answers

102 views

### Smallest ring whose field of fractions includes all the reals (subring of omnific integers?)

The surreal numbers have a subring, the ring of "omnific integers" or $\mathbf{Oz}$, which have the property that every surreal number is a quotient of two omnific integers. That is, the field of ...

**4**

votes

**0**answers

160 views

### Cohomology and higher structures

Classically, cohomologies of Lie groups/algebras parametrize extensions. To be precise, given an linear $G$-action on $M$, there is an bijection between $H^2(G;M)$ and the set of extension $E$ of $G$ ...

**1**

vote

**0**answers

56 views

### On submodules of vector fields

I don't know much about modules aside from their basic definition and that they are more complicated than vector spaces.
I am asking this question because I wish to have a more "algebraic" ...

**3**

votes

**1**answer

126 views

### Coinvariants of tensor products of Hopf algebras

Let $G$ be a Hopf algebra, considered as a right $G$-comodule in the obvious way.
The axioms of Hopf algebras imply that
$$
G^{coinv(G)} == \{g \in G : \Delta(g) = g \otimes 1\} = \mathbb{C}1.
$$
...

**1**

vote

**0**answers

44 views

### Existence of nontrivial transfinite divisibility in $R$-modules

Let $R$ be a unital, possibly noncommutative ring and $s \in R$. For a right $R$-module $M$, define $Ms = \{ms \mid m \in M\}$; this is an additive subgroup of $M$, which is a module over the ...

**1**

vote

**1**answer

50 views

### Principal ideal of a non-associative magma

The definitions of an ideal of an algebraic structure $A$ (as a substructure $I$ such that the product of $A$ and $I$ is a subset of $I$) do not involve associativity.
However, the definitions of a ...

**0**

votes

**1**answer

51 views

### Can we extract an injective envelope from a monomorphism?

Let $A$ be an artinian ring and $f : X \rightarrow \bigoplus_{j=1}^{n}I_{j}$ be a morphism of $A$-modules, where each $I_{j}$ is injective and indecomposable. If $f$ is a monomorphism, then can we ...

**2**

votes

**0**answers

82 views

### Rings whose finitely-generated modules are co-hopfian

Let $A$ be a unital, possibly noncommutative ring. Dischinger showed [1] that the following are equivalent:
For every $a \in A$, there exists $n \in \mathbb N$ such that $a^n A = a^{n+1} A$;
For ...

**3**

votes

**1**answer

68 views

### Effect of extending scalars on maps of modules

Let $k$ be a field and $R$ be a $k$-algebra. Let $M$ and $N$ be left $R$-modules. Finally, let $\ell$ be a field extension of $k$. We thus have an $\ell$-algebra $\ell \otimes R$, and both $\ell \...

**4**

votes

**1**answer

104 views

### Global splitting field for algebras

Let $A$ be a finite dimensional algebra.
A field $K$ is a splitting field for an indecomposable $A$-module $M$ in case the local algebra $End_A(M)/(rad(End_A(M))$ is 1-dimensional.
$K$ is called a ...

**1**

vote

**1**answer

38 views

### Decreasing sequences in a finitely generated closure algebra

I am interested in finitely generated closure algebras (as a special case of Heyting algebras), and in decreasing sequences of elements within such an algebra that have no lower bound.
Call two ...

**1**

vote

**0**answers

48 views

### Compatibility with multiplication of a cyclic order on a ring

I am copying my question from here: https://math.stackexchange.com/q/3233462/427611.
Is it correct that $\mathbb Z/3\mathbb Z$ and $\mathbb Z/4\mathbb Z$ are the only rings with three or more ...

**-2**

votes

**0**answers

137 views

### How to prove that $\mathrm{Aut}(\mathcal{M})\cong\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$?

I want to study the structure of the rig of L-functions $\mathcal{M}$, which is defined as the maximal set of automorphic L-functions of $\mathrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ for some $n$ that be ...

**2**

votes

**0**answers

55 views

### Question on a subcategory being extension-closed

In the article "Homological theory of noetherian rings" by Idun Reiten from 1996, it was stated that it seems to be not known whether the subcategory $\operatorname{Tr}(\Omega^i(\mathrm{mod}\text{-}A))...

**0**

votes

**0**answers

29 views

### Do there exist characterizations of divisors of a particular element for all algebra structures on a vector space

Let $k$ be a field and $V$ a $k$-vector space. Let $M$ be the subset of $\operatorname{Hom}_k(V \otimes_{k} V,V)$ formed by all elements giving $V$ the structure of a commutative (associative) $k$-...

**3**

votes

**0**answers

32 views

### A non-singularity property for sets of real matrices

Let $M_N(\mathbb{R})$ be the ring of $N\times N$ real matrices. We say that a couple $(\mathcal{U},\mathcal{V})$, with $\mathcal{U},\mathcal{V}\subseteq M_N(\mathbb{R})$ is admissible if, for every $A\...

**5**

votes

**0**answers

75 views

### Kac-Moody Lie algebra as derivations of associative algebras

The set of derivations of an algebra $\Bbb A$ forms a Lie algebra. This is one aspect of why Lie algebras are interesting. When $\Bbb A$ is polynomial algebra in $n$ variable then $\text{Der } \Bbb A$ ...

**6**

votes

**0**answers

165 views

### On properties of an algebra as a bimodule

Let $A$ be a two-sided artinian ring.
Recall that a module $M$ is said to have dominant dimension at least $n$ in case the terms $I_i$ in the minimal injective coresolution of $M$ are projective for $...

**5**

votes

**1**answer

200 views

### Examples of noetherian local rings $R$ such that $K'_0(R)$ is not isomorphic to $\mathbb Z$

Does there exist a simple example of a commutative noetherian local ring $R$ such that $K'_0(R) = K_0(\mbox{Mod-}R)$ (by $\mbox{Mod-}R$ I mean the abelian category of finitely generated $R$-modules) ...

**0**

votes

**0**answers

52 views

### A semifield of characteristic zero may have a finite number of elements

A commutative semiring $(S, +, \cdot, 0, 1)$ with unity is said to be a semifield if for all $a, b\in S$, $a+b=0$ implies that $a=0$ and $b=0$, and $a.b=0$ implies that either $a=0$ or, $b=0$.
I ...

**2**

votes

**0**answers

74 views

### Dimension of center of $k[G]/\mathrm{rad}k[G]$ when characteristic of $k$ divides the order of $G$

Let $G$ be a finite group and consider $k[G]$ where $k$ is a field. In the scenario where $\mathrm{char}(k)$ divides $|G|$, how can one show that the dimension of $Z(k[G]/\operatorname{rad}k[G])$ is ...

**7**

votes

**1**answer

225 views

### Skew differential graded algebra

A sigma, or skew, derivation is a natural generalisation of the
notion of derivation depending on an algebra automorphism $\sigma$ which
when equal to $id = \sigma$ reduces to the usual notion of a
...

**1**

vote

**0**answers

74 views

### On reflexive bialgebras

Let $A$ be a bialgebra. We can consider $A$ as a relfexive algebra (i.e. $A\cong A^{o*}$) or relfexive coalgebra (i.e. $A\cong A^{*o}$ where in each case $o$ denotes what is sometimes called ...

**2**

votes

**0**answers

19 views

### Is there a unique way to define an Euclidean Jordan Algebra as the product of two simple Euclidean Jordan Algebras?

Let $A$ and $B$ be two simple Euclidean Jordan Algebras. It is well known that $A\times B$ can be made into a Jordan Algebra in a unique way by defining the operation component-wise. Is it true that ...

**3**

votes

**0**answers

56 views

### Frobenius algebras of small dimensions

In Classification of commutative Frobenius algebras , Jeremy Rickard showed that there are infinitely many commutative (local without loss of generality) Frobenius algebras of vector space dimension ...

**3**

votes

**1**answer

87 views

### Irreducibility of product bicomodules

Let $H$ be a Hopf algebra, and $V$ and $W$ a left, and a right,
$H$-comodule respectively. The tensor product
$$
V \otimes W
$$
has an obvious $H$-$H$-bicomodule structure.
If $V$ and $W$ are ...

**1**

vote

**0**answers

44 views

### Universal bimodule for homotopy biderivations

Recall the commutative story: for a commutative algebra $A$, its module of differentials $\Omega (A)$ is characterized by the universal property that any derivation $\delta \colon A \to M$ is in a ...

**2**

votes

**0**answers

89 views

### Reference request: Differential graded structures in mixed characteristic

I am looking for references/papers on differential graded structures and their applications in mixed characteristic. The following I have discuss differential graded algebras in the general, not in ...

**1**

vote

**0**answers

202 views

### Algebraic relation given by a 3x3 determinant

I just encountered a very curious relation in an algebra. A bit simplified, I am working in a (particular) non-commutative algebra, with some relations.
One particular relation is the following:
For (...

**13**

votes

**2**answers

619 views

### Does any derivation of commutative algebra preserve its nil-radical?

Given a commutative associative unital algebra over a field of characteristic zero.
Is it true that any derivation of it preseves its nil-radical?
More explicitly, let $D$ be a derivation of an ...

**3**

votes

**1**answer

196 views

### Elementary classification of division rings

Are there examples (other than the two mentioned below) of fields $K$ such that the classification of all finite dimensional division $K$-algebras is possible using only elementary theory (lets say a ...

**1**

vote

**0**answers

50 views

### Simplicial differential graded algebra and a filtration

Let $A$ be a simplicial differential algebra, i.e. for each $n \in \mathbb{N}$ a differential graded algebra $(A_n,d_n)$ and for each weakly increasing map $f \colon [n] \to [m]$ a morphism $f_* \...

**2**

votes

**0**answers

94 views

### Generalising injective modules

Free modules over a ring generalise to projective modules over a ring, which generalise to flat modules, which generalise to torsion free modules:
$$
\textrm{free} \to
\textrm{projective}
\to
\textrm{...

**2**

votes

**0**answers

94 views

### Flat augmentation ideal of a group-ring

If $G$ is a group and $I$ the augmentation ideal $I=Ker(\mathbb{Z}G\rightarrow \mathbb{Z})$ suppose that:
$I$ is a flat (right) $\mathbb{Z}G$-module.
$I$ is a finitely generated (right) $\mathbb{Z}G$...

**2**

votes

**0**answers

209 views

### Module structure for $\mathbb{Z}$

I am interested to know which module structures we can define in the additive group of integers $\mathbb{Z}$.
It is easy to prove that $\mathbb{Z}$ does not admit a vector space structure. (For more ...

**3**

votes

**0**answers

46 views

### Weakly symmetric rings and derived equivalences

A ring $R$ with Jacobson radical $J$ is called Frobenius in case $R/J \cong soc(R)$ as left and right $R$-modules and weakly symmetric in case we even have $R/J \cong soc(R)$ as $R$-bimodules.
...

**3**

votes

**0**answers

32 views

### Positive roots of the Tits unit form and dimension vectors

Let $A$ be a finite dimensional quiver algebra such that two indecomposable modules are isomorphic iff their dimension vectors are the same. Let $T_A$ be the tits unit form of $A$ and $r_A$ the set of ...

**3**

votes

**1**answer

204 views

### augmentation ideal is always finitely generated?

$G$ is a finitely presented group (but not a finite group), and $\mathbb{Z}G$ is the corresponding group ring.
$I$ is the kernel of the augmentation morphism $\mathbb{Z}G\rightarrow \mathbb{Z}$.
Is $...