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        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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        56 views

        $\Omega^2(S) \cong \tau(S)$ for simple modules

        Let $A$ be an Artin algebra with $\Omega^2(S) \cong \tau(S)$ for each simple module $S$. Is $A$ a quasi-Frobenius (=selfinjective) algebra? Here $\tau$ denotes the Auslander-Reiten translate, which is ...
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        vote
        0answers
        135 views

        spectral property of irreducible matrices

        We have an irreducible non-negative matrix $B$ in which all the diagonal elements are zero and the other entries can be 0 or 1. Moreover, $B$ is diagonalizable. We partition the rows of this matrix to ...
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        votes
        1answer
        223 views

        A peculiar operation on $M_2(\mathbb Z)$ which along with the usual matrix addition, makes $M_2(\mathbb Z)$ into a commutative ring with unity [closed]

        For $A=\begin{pmatrix} a_1 & b_1 \\ c_1&d_1 \end{pmatrix}, B=\begin{pmatrix} a_2 & b_2 \\ c_2&d_2 \end{pmatrix}\in M_2(\mathbb Z)$, define $A*B:=a_1L_1BR_1+b_1L_1BR_2+c_1L_2BR_1+...
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        votes
        0answers
        102 views

        Cellular and primary binomial ideals

        Let $I \subseteq \mathbb{K}[x_1, \dots, x_n]$ be an ideal of a polynomial ring over a field $\mathbb{K}$. $I$ is called cellular if every variable $x_i$, with $i=1, \dots, n$, is either a ...
        8
        votes
        1answer
        83 views

        Injective indecomposable modules over Laurent polynomial rings

        What does the injective envelope of $\mathbb C[x,x^{-1}]/(p(x,x^{-1}))$ as a $\mathbb C[x,x^{-1}]$-module look like where $p(x,x^{-1})$ is an irreducible element? I’m sure this is well known, but ...
        6
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        1answer
        170 views

        Zero divisors in complex group algebras of residually finite groups

        Conjecture. There exists a function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that if $\alpha$ and $\beta$ are non-zero elements of the complex group algebra $\mathbb{C}[G]$ of a finite group $G$ ...
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        57 views

        Characterisation of projective modules over tensor products of fields

        Let $k$ be a field and $L_1$ and $L_2$ finite field extensions of $k$. Let $A:=L_1 \otimes_k L_2$ as an algebra. Question: Given a finitely generated $A$-module $M$, do we have that $M$ is ...
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        89 views

        Unitary element of the group algebra

        Let $G$ be a torsion free group. Are unitary elements of $\mathbb CG$ studied? By unitary element I mean an element $\alpha$ in $\mathbb CG$, such that $\alpha^*\alpha=1$? Do the triviality of unitary ...
        3
        votes
        0answers
        84 views

        Does this element belong to $\mathbb CG$?

        Let $G$ be a torsion-free group. Let $\alpha$ be a symmetric element of $\mathbb CG$, i.e. $\alpha^*=\alpha$, with $\|\alpha\|_1=\sum|\alpha(g)|<1$, so $\beta:=\sum_{n\ge 0}(-1)^n\alpha^n$ is an ...
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        votes
        2answers
        289 views

        When is $\Omega^1$ an equivalence?

        Let $C$ be an abelian category with enough projectives and $\underline{C}$ the stable category of $C$ that is obtained by factoring out projective modules. When is the functor $\Omega^1 : \underline{...
        3
        votes
        0answers
        73 views

        Sparsest similar matrix

        Given a square matrix A (say with complex entries), which is the sparsest matrix which is similar to A? I guess it has to be its Jordan normal form but I am not sure. Remarks: A matrix is sparser ...
        4
        votes
        2answers
        169 views

        Naturality of PD model of a CDGA

        In the paper "Poincaré duality and commutative differential graded algebras", Lambrechts and Stanley constructed PD model for cdga with simply connected cohomology. My question is: if $A$ and $B$ are ...
        3
        votes
        0answers
        58 views

        conditions for algebra cohomology finiteness

        Let $A=\bigoplus\limits_{n\geq 0}A_n$ be a graded (unital associative) ring, say an algebra over a field with finite dimensional graded components and $A_0$ semisimple. Are there reasonable conditions ...
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        0answers
        79 views

        Duality between coalgebras and (pseudocompact) algebras - uniqueness

        The following result is well-known. It can for example be found in [Iovanov: The representation theory of profinite algebras, Theorem 1.0.2]. For definitions, see below. Let $k$ be a field. The ...
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        votes
        2answers
        402 views

        Is there a good name for the operation that turns $A\operatorname{-mod}$ and $B\operatorname{-mod}$ into $A\otimes B\operatorname{-mod}$?

        The name pretty much says it all: there's a well-defined operation on categories equivalent to modules over some ring: if $\mathcal{C}_1=A\operatorname{-mod}$ and $\mathcal{C}_2=B\operatorname{-mod}$, ...

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