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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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 $).
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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 $D>1$...
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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 $n$ ...
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3answers
502 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
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2answers
123 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 $\...
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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 ...
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29 views

Direct product of rings and its automorphisms

What are all the possible automorphisms over a direct product of rings? Are there any way to prove that they are the only ones?
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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
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2answers
156 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
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0answers
112 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 ...
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120 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 ...
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1answer
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 ...
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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 ...
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159 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 $G$ ...
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55 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
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1answer
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 $$ G^{coinv(G)} == \{g \in G : \Delta(g) = g \otimes 1\} = \mathbb{C}1. $$ ...
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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 ...
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1answer
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 ...
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1answer
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
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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
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1answer
67 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
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1answer
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 ...
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1answer
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 ...
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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 ...
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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 ...
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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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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) $k$-...
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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\...
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0answers
74 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 $\text{Der } \Bbb A$ ...
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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
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1answer
198 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) ...
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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 ...
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0answers
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 ...
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1answer
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 ...
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0answers
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 ...
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0answers
18 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 ...
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0answers
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 ...
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1answer
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 ...
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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 ...
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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 ...
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0answers
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
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2answers
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
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1answer
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 ...
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0answers
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
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0answers
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{...
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0answers
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
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0answers
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
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0answers
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
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0answers
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
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1answer
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 $...

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