# 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.

2,469
questions

**6**

votes

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97 views

### Is $\mathbb{Q}_p \otimes_{\mathbb{Q}}\mathbb{Q}_p $ coherent?

Let $\mathbb{Q}_p$ denote the field of fractions of $\mathbb{Z}_p$. By the answers to this quesition the tensor product $\mathbb{Q}_p \otimes_{\mathbb{Q}} \mathbb{Q}_p$ cannot be a Noetherian ring (...

**2**

votes

**1**answer

90 views

### Behavior of invariants under reduction mod p

Let $R$ be a finitely generated $\mathbb{Z}$-algebra with an [edit: linear algebraic] action of $G(\mathbb{Z})$ where $G$ is a split simply-connected semisimple group.
Then for any prime $p$ we have a ...

**6**

votes

**1**answer

209 views

### Category of modules over an Azumaya algebra and the Brauer group

Let $k$ be a field, and let $\alpha \in \mathrm{Br}(k)$. Let $A$ be an Azumaya algebra representing $\alpha$. Then the category $A$–$\mathrm{mod}$ depends only on $\alpha$.
I would like to know ...

**0**

votes

**0**answers

56 views

### When the sum of two ideals is indecomposable

I am looking for a commutative ring $R$ and two ideals $I$ and $J$ of $R$ and two different maximal ideals $m_1$ and $m_2$ of $R$ such that $ann(I)=m_1$ and $ann(J)=m_2$ and $I+J$ is an ...

**0**

votes

**0**answers

73 views

### Field theory, Abel-Ruffini theorem, technical question

Let me put the question first.
Let $F,K$ be subfields of $\mathbb{C}$. Suppose that $t,\rho\in \mathbb{C}$ are algebraic over $F$ and $\rho \in K$. If $F(t)\cap K\subset F$, is it true that $F(t,\rho)\...

**0**

votes

**1**answer

60 views

### Rings or algebras with many nilpotent elements and efficient computation

Crossposted from quantum.SE
where comment appears to suggest that solving modulo 2 might
be possible.
Searching the web for '"quantum computer" nilpotent'
returns many results, so maybe the ...

**4**

votes

**1**answer

101 views

### Exterior algebra of normed spaces

This question is related to my prior question, but this one is aimed, even though it's more general. If $V$ is a vector space, we define the exterior algebra of $V$ do be:
$$\bigwedge V := \bigoplus_{...

**1**

vote

**0**answers

78 views

### if $I$ is finitely presented nilpotent and $M/IM$ is finitely presented, then $M$ is finitely presented

Let $R$ be a commutative ring, and let $I \subseteq R$ be a nilpotent ideal. Let moreover $M$ be an $R$-module, and let $IM$ be the submodule generated by the products $xm$ with $x \in I$ and $m \in M$...

**2**

votes

**3**answers

132 views

### Efficient algorithm for matrix equation $AXB + BXA = F$

For $n\in\mathbb{N}$, let $A\in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{n\times n}$ be two symmetric positive definite matrices and let $F\in\mathbb{R}^{n\times n}$ be arbitrary.
Is there any ...

**3**

votes

**1**answer

56 views

### Question concerning Brauer's second main theorem, Brauer correspondent blocks and blocks covered by nilpotent blocks

A version of Brauer's second main theorem is as follows:
Let $G$ be a finite group, $x$ be a $p$-element of $G$, $B\in\mathcal{Bl}(G)$, and $\chi\in$ Irr$(B)$.
If $d_{\chi\mu}^x\neq 0$ and $\mu$ ...

**1**

vote

**1**answer

84 views

### Second summand to make projective module free

Suppose there's a projective $R$-module $P$ (non-free). We know that there is another $R$-module $M$ such that $P\oplus M$ is free over $R$. Is there a way to write down such an $M$ in terms of $P$?
...

**5**

votes

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44 views

### Reference request: associative subalgebras of Cayley algebras are at most 4-dimensional

By a Cayley algebra I mean an 8-dimensional algebra (over an arbitrary field) formed in the Cayley-Dickson process. (They are also called octonion algebras, but I prefer to reserve the term octonion ...

**4**

votes

**1**answer

152 views

### Example of a projective module with non-superfluous radical

Let $R$ be a ring with unit. A submodule $N$ of an $R$-module $M$ is called superfluous if the only sumbodule $T$ of $M$ for which $N+T = M$ is $M$ itself.
It is shown, for example, in
[1] F. W....

**3**

votes

**0**answers

46 views

### Gelfand-Kirillov dimension of the first Weyl algebra

How can we compute the Gelfand-Kirillov dimension (GK for short) of the first Weyl algebra?
As we know we can look at the Weyl algebra as a generalized Weyl algebra in the following way:
Let $A=\...

**1**

vote

**1**answer

36 views

### Contraction elements in unital *-rings

Let $A$ be a unital *-ring. Let us put, $A^+=\{\sum_1^N x^*_ix_i: x_i\in A , N\in \mathbb{N} \}$. We write $x\geq0$ if $x\in A^+$.
Suppose that for every $n\in \mathbb{N}$ and $a\in A$, there exsits $...

**0**

votes

**0**answers

38 views

### Classify all 4D real associative unital algebras with quadratic minimal polynomials?

Is there a classification of all 4D real associative unital algebras where each element is the root of some quadratic polynomial with real coefficients?
Such a classification should contain some ...

**24**

votes

**1**answer

542 views

### Is this formal noncommutative power series identity known?

I recently discovered the following cute formal noncommutative power series identity: if $(x_i)_{i \in I}$ is some finite collection of noncommuting variables, then the formal power series
$$ 1 + \...

**4**

votes

**0**answers

89 views

### Is the average associator over a finite subloop of octonions necessarily zero?

For any three octonions $a,b,c$, their associator is defined as \begin{equation*} [a,b,c]=a(bc)-(ab)c \end{equation*} and measures their non-associativity so to speak.
Now suppose that $L$ is a finite ...

**6**

votes

**0**answers

217 views

### How much does Ext tell me about isomorphisms?

So this was a question I sort of stumbled on and realised I was quite stumped. Suppose we have two finitely generated $R$-modules $M, N$ (I have the group ring $R=\mathbb{Z}[G]$ in mind) which appear ...

**2**

votes

**0**answers

76 views

### Projective group of Plucker quadric over the reals

A somewhat elementary question but seemingly difficult to find a suitable reference:
Consider the six-dimensional real space $\wedge^2(\mathbb R^4)$ with basis $e_i \wedge e_j \ (i < j)$ where $...

**1**

vote

**1**answer

70 views

### References about transfinite socle series

I would like to know if there is any literature that discusses "transfinite socle series" of a ring module. Below is my attempt at defining the series.
Let $R$ be an associative unital ring and $...

**0**

votes

**0**answers

92 views

### Stabilizers in the action of $\mathrm{GL}(n, \mathbb Z)$ on $\mathbb Z^n$

How can we calculate effectively the subgroups of $G: = \mathrm{GL}(n, \mathbb Z)$ which fix pointwise a given submodule $S$ of $\mathbb Z^n$ in the action of $G$ on $\mathbb Z^n$ by left ...

**5**

votes

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135 views

### Is there a list of all real unital subalgebras of M(2,C)?

Is there a complete classification of all real unital subalgebras of $M(2,\mathbb C)$ up to isomorphism? The list should include $M(2,\mathbb C)$, the quaternions, complex numbers, split-complex ...

**5**

votes

**0**answers

50 views

### Formally real non-Jordan algebras

Jordan, von Neumann and Wigner [1] showed that for any finite-dimensional real vector space $A$ with a bilinear commutative power-associative operation $\circ : A \times A \to A$, the formal reality ...

**1**

vote

**0**answers

78 views

### Derivations of differential operators

For a smooth affine variety $\operatorname{Spec} A$ over a ring $R$ we have an algebra of differential operators $\mathcal{D}_A$ (here I mean not the Grothendieck differential operators but PD-ones). ...

**3**

votes

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55 views

### Family of Lie algebras parametrized by a discrete valuation ring

I have a family of Lie algebras parametrized by a discrete valuation ring, whose generic fiber is reductive and whose special fiber is nilpotent. I'd like to learn about the relationship between the ...

**5**

votes

**1**answer

286 views

### Jordan algebra identities

A Jordan algebra is a vector space with a commutative bilinear operation $\circ$ obeying an identity that's often written as
$$ (x \circ y) \circ (x \circ x) = x \circ (y \circ (x \circ x)) . $$
...

**3**

votes

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66 views

### Realizing a *-algebra as an endomorphism algebra

A $*$-ring is a (unitial, associative, not necessarily commutative) ring $A$ with a map $* : A \rightarrow A$ (we write $*(a) = a^*$), such that
$1^* = 1$.
$\forall a, b \in A, (a + b)^* = a^* + b^*$....

**3**

votes

**1**answer

115 views

### How to identify the two copies of $D_{24}$ in the homomorphisms of the 2 musical actions? [closed]

Let $S$ be the set of minor and major triads. Two sets of actions are defined on the set:
1) Musical transposition and inversion;
2) P, L, R actions
$$P(C-major) = c-minor,$$
$$L(C-major) = e-minor,$...

**1**

vote

**0**answers

95 views

### Symmetric functions for multidimensional variables

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

**3**

votes

**0**answers

50 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 $n$ in ...

**15**

votes

**4**answers

666 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

143 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 $\...

**5**

votes

**2**answers

178 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

118 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

130 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 ...

**4**

votes

**1**answer

109 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 ...

**2**

votes

**1**answer

159 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 ...

**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

58 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

**3**answers

169 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^{\operatorname{coinv}(G)} == \{g \in G : \Delta(g) = g \otimes 1\} = \...

**1**

vote

**0**answers

45 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 ...

**2**

votes

**1**answer

52 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

53 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

83 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

71 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

105 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

44 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

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))...