Algebra Colloquium

It is proved that for matrices $A$, $B$ in the $n$ by $n$ upper triangular matrix ring $T_n(R)$ over a domain $R$, if $AB$ is nonzero and central in $T_n(R)$ then $AB=BA$. The $n$ by $n$ full matrix rings over right Noetherian domains are also shown to have this property. In this article we treat a ring property that is a generalization of this result, and a ring with such a property is said to be weakly reversible-over-center. The class of weakly reversible-over-center rings contains both full matrix rings over right Noetherian domains and upper triangular matrix rings over domains. The structure of various sorts of weakly reversible-over-center rings is studied in relation to the questions raised in the process naturally. We also consider the connection between the property of being weakly reversible-over-center and the related ring properties.

Let $(A,𝔪)$ be a Noetherian local ring with maximal ideal $𝔪$, $I$ an ideal of $A$, $J$ an $𝔪$-primary ideal of $A$, $N$ a finitely generated $A$-module, $S={⨁}_{n\ge 0}{S}_n$ a finitely generated standard graded algebra over $A$ and $M={⨁}_{n\ge 0}{M}_n$ a finitely generated graded $S$-module. We characterize the multiplicity and the Cohen–Macaulayness of the fiber cone $F_J(M)={⨁}_{n\ge 0}{M}_n/J{M}_n$. As an application, we obtain some results on the multiplicity and the Cohen–Macaulayness of the fiber cone${⨁}_{n\ge 0}{I}^{n}N/J{I}^{n}N$.

This article is a contribution to the study of the automorphism groups of 3-$(v,k,3)$ designs. Let $\mathcal{S}=(\mathcal{P},\mathcal{B})$ be a non-trivial 3-$(q+1,k,3)$ design. If a two-dimensional projective linear group $\mathrm{PSL}(2,q)$ acts flag-transitively on $\mathcal{S}$, then $\mathcal{S}$ is a 3-$(q+1,4,3)$ or 3-$(q+1,5,3)$ design.

In this paper, we propose a new approach towards the classification of spherical fusion categories by their Frobenius–Schur exponents. We classify spherical fusion categories of Frobenius–Schur exponent 2 up to monoidal equivalence. We also classify modular categories of Frobenius–Schur exponent 2 up to braided monoidal equivalence. It turns out that the Gauss sum is a complete invariant for modular categories of Frobenius–Schur exponent 2. This result can be viewed as a categorical analog of Arf's theorem on the classification of non-degenerate quadratic forms over fields of characteristic 2.

Firstly, the commutativity of rings is investigated in this paper. Let $R$ be a ring with identity. Then we obtain the following commutativity conditions: (1) if for each $x\in R\backslash N(R)$ and each $y\in R$, ${(xy)}^k=x^k{y}^k$ for $k=m,m+1,n,n+1$, where $m$ and $n$ are relatively prime positive integers, then $R$ is commutative; (2) if for each $x\in R\backslash J(R)$ and each $y\in R$, ${(xy)}^k=y^k{x}^k$ for $k=m,m+1,m+2$, where $m$ is a positive integer, then $R$ is commutative. Secondly, generalized 2-CN rings, a kind of ring being commutative to some extent, are investigated. Some relations between generalized 2-CN rings and other kinds of rings, such as reduced rings, regular rings, 2-good rings, and weakly Abel rings, are presented.

A Noetherian (Artinian) Lie algebra satisfies the maximal (minimal) condition for ideals. Generalisations include quasi-Noetherian and quasi-Artinian Lie algebras. We study conditions on prime ideals relating these properties. We prove that the radical of any ideal of a quasi-Artinian Lie algebra is the intersection of finitely many prime ideals, and an ideally finite Lie algebra is quasi-Noetherian if and only if it is quasi-Artinian. Both properties are equivalent to soluble-by-finite. We also prove a structure theorem for serially finite Artinian Lie algebras.

We study the variety of binary Lie algebras defined by the identities $x^2=J(x,y,zu)=0$, where $J(a,b,c)$ denotes the Jacobian of $a$, $b$, $c$. Building on previous work by Carrillo, Rasskazova, Sabinina and Grishkov, in the present article it is shown that the Levi and Malcev theorems hold for this variety of algebras.

For a radical square zero algebra $\Lambda$ and an indecomposable right $\Lambda$-module $M$, when $\Lambda$ is Gorenstein of finite representation type or $\tau M$ is $\tau$-rigid, $M$ is $\tau$-rigid if and only if the first two projective terms of a minimal projective resolution of $M$ have no non-zero direct summands in common. In particular, we determine all $\tau$-tilting modules for Nakayama algebras with radical square zero.

Let $\widetilde{\mathfrak{s}\mathfrak{v}}$ be the extended Schrödinger–Virasoro Lie algebra and $n\ge 1$ an integer. A map $f:{\widetilde{\mathfrak{s}\mathfrak{v}}}^n=\widetilde{\mathfrak{s}\mathfrak{v}}\times \widetilde{\mathfrak{s}\mathfrak{v}}\times \cdots \times \widetilde{\mathfrak{s}\mathfrak{v}}\to \widetilde{\mathfrak{s}\mathfrak{v}}$ is called an $n$-derivation if it is a derivation in one variable while other variables fixed. We investigate $n$-derivations of the extended Schrödinger–Virasoro Lie algebra $\widetilde{\mathfrak{s}\mathfrak{v}}$. The main result when $n=2$ is then applied to characterize the linear commuting maps and the commutative post-Lie algebra structures on $\widetilde{\mathfrak{s}\mathfrak{v}}$.

We give a characterization of the cyclic subgroup separability and weak potency of the fundamental group of a graph of polycyclic-by-finite groups and free-by-finite groups amalgamating edge subgroups of the form $〈h〉\times D$, where $h$ has infinite order and $D$ is finite.

Let $\mathcal{A}$ be an abelian category, $\mathcal{C}$ an additive, full and self-orthogonal subcategory of $\mathcal{A}$ closed under direct summands, $r\mathcal{G}(\mathcal{C})$ the right Gorenstein subcategory of $\mathcal{A}$ relative to $\mathcal{C}$, and ${}^{\mathrm{\perp }}\mathcal{C}$ the left orthogonal class of $\mathcal{C}$. For an object $A$ in $\mathcal{A}$, we prove that if $A$ is in the right 1-orthogonal class of $r\mathcal{G}(\mathcal{C})$, then the $\mathcal{C}$-projective and $r\mathcal{G}(\mathcal{C})$-projective dimensions of $A$ are identical; if the $r\mathcal{G}(\mathcal{C})$-projective dimension of $A$ is finite, then the $r\mathcal{G}(\mathcal{C})$-projective and ${}^{\mathrm{\perp }}\mathcal{C}$-projective dimensions of $A$ are identical. We also prove that the supremum of the $\mathcal{C}$-projective dimensions of objects with finite $\mathcal{C}$-projective dimension and that of the $r\mathcal{G}(\mathcal{C})$-projective dimensions of objects with finite $r\mathcal{G}(\mathcal{C})$-projective dimension coincide. Then we apply these results to the category of modules.

For any positive integer $N$, we clearly describe all finite-dimensional algebras $A$ such that the upper triangular matrix algebras $T_N(A)$ are piecewise hereditary. Consequently, we describe all finite-dimensional algebras $A$ such that their derived categories of $N$-complexes are triangulated equivalent to derived categories of hereditary abelian categories, and we describe the tensor algebras $A\otimes K[X]/(X^N)$ for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.

We study the Leibniz $n$-algebra ${\mathbf{U}}_n(\mathfrak{L})$, whose multiplication is defined via the bracket of a Leibniz algebra $\mathfrak{L}$ as $[x_1,\dots ,x_{\mathrm{n}}]=[x_1,[\dots ,[x_{n-2},[x_{n-1},x_n]]\dots ]]$. We show that ${\mathbf{U}}_n(\mathfrak{L})$ is simple if and only if $\mathfrak{L}$ is a simple Lie algebra. An analog of Levi's theorem for Leibniz algebras in ${\mathbf{U}}_n(\mathbf{Lb})$ is established and it is proven that the Leibniz $n$-kernel of ${\mathbf{U}}_n(\mathfrak{L})$ for any semisimple Leibniz algebra $\mathfrak{L}$ is the $n$-algebra ${\mathbf{U}}_n(\mathfrak{L})$.

Let $\beta$ be a complex number, and a class of non-semisimple and non-solvable rank two Lie conformal algebras $\mathcal{L}(\beta )$ are introduced. In this paper, conformal derivations, conformal quasiderivations, generalized conformal derivations and conformal biderivations of $\mathcal{L}(\beta )$ are studied. Besides, central extensions and conformal modules of rank one of $\mathcal{L}(\beta )$ are determined.