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.

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6
votes
0answers
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
1answer
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
1answer
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
0answers
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
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0answers
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
1answer
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
1answer
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
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0answers
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
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3answers
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
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1answer
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
1answer
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
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0answers
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
1answer
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
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0answers
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
1answer
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
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0answers
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
1answer
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
0answers
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
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0answers
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
0answers
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
1answer
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
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0answers
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
0answers
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
0answers
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
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0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
4answers
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
2answers
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
2answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
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0answers
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
3answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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))...

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