# 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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48 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 $I$).
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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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### 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: ...
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### 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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### 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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### 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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### 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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### 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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### 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" ...
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### 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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### 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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### 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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### 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 ...
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### Rings whose finitely-generated modules are co-hopfian

Let $A$ be a unital, possibly noncommutative ring. Dischinger showed  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 ...
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### 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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### 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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### 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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### 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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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 (...
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### 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 ...
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 ...