Algebra Colloquium

We study the structure of rings which satisfy the von Neumann regularity of commutators, and call a ring $R$ C-regular if $ab-ba\in (ab-ba)R(ab-ba)$ for all $a$, $b$ in $R$. For a C-regular ring $R$, we prove $J(R[X])=N^{\ast }(R[X])=N^{\ast }(R)[X]=W(R)[X]\subseteq Z(R[X])$, where $J(A)$, $N^{\ast }(A)$, $W(A)$, $Z(A)$ are the Jacobson radical, upper nilradical, Wedderburn radical, and center of a given ring $A$, respectively, and $A[X]$ denotes the polynomial ring with a set $X$ of commuting indeterminates over $A$; we also prove that $R$ is semiprime if and only if the right (left) singular ideal of $R$ is zero. We provide methods to construct C-regular rings which are neither commutative nor von Neumann regular, from any given ring. Moreover, for a C-regular ring $R$, the following are proved to be equivalent: (i) $R$ is Abelian; (ii) every prime factor ring of $R$ is a duo domain; (iii) $R$ is quasi-duo; and (iv) $R/W(R)$ is reduced.

A simple undirected regular graph is said to be semisymmetric if it is edge-transitive but not vertex-transitive. For a semisymmetric graph $\mathrm{\Gamma }$ of order $2p^3$, $p$ a prime, it is well known that $\mathrm{\Gamma }$ is bipartite with two biparts having equal size. The complete classification of such graphs has been given for the full automorphism group $\mathrm{Aut}(\mathrm{\Gamma })$ acting unfaithfully on at least one bipart of $\mathrm{\Gamma }$, which shows that there is only one infinite family of such graphs with valency $p^2$. The graphs of this kind have been determined when $\mathrm{Aut}(\mathrm{\Gamma })$ acts faithfully and primitively on at least one bipart of $\mathrm{\Gamma }$, and thus there is only one remaining case for classifying such graphs of valency $p^2$, $\mathrm{Aut}(\mathrm{\Gamma })$ acting faithfully and imprimitively on both biparts of $\mathrm{\Gamma }$, which is dealt with in this paper. As a result, there is only one infinite family of semisymmetric graphs of order $2p^3$ with valency $p^2$.

We develop and study the generalization of rational Schur algebras to the super setting. Similar to the classical case, this provides a new method for studying rational supermodules of the general linear supergroup $\mathrm{GL}(m|n)$. Furthermore, we establish a Schur-Weyl duality result for rational Schur superalgebras and conclude that under certain conditions these objects will be semisimple.

The metric dimension problem is called navigation problem due to its application to robot navigation in space. Further this concept has wide applications in motion planning, sonar and loran station, and so on. In this paper, we study certain results on the metric dimension, upper dimension and resolving number of extended annihilating-ideal graph $\mathbb{E}\mathbb{A}\mathbb{G}(R)$ associated to a commutative ring $R$, denoted by ${\dim }_M(\mathbb{E}\mathbb{A}\mathbb{G}(R))$, ${\dim }^{+}(\mathbb{E}\mathbb{A}\mathbb{G}(R))$ and $\mathrm{res}(\mathbb{E}\mathbb{A}\mathbb{G}(R))$, respectively. Here we prove the finiteness conditions of ${\dim }_M(\mathbb{E}\mathbb{A}\mathbb{G}(R))$ and ${\dim }^{+}(\mathbb{E}\mathbb{A}\mathbb{G}(R))$. In addition, we characterize ${\dim }_M(\mathbb{E}\mathbb{A}\mathbb{G}(R))$, ${\dim }^{+}(\mathbb{E}\mathbb{A}\mathbb{G}(R))$ and $\mathrm{res}(\mathbb{E}\mathbb{A}\mathbb{G}(R))$ for artinian rings and the direct product of rings.

We give some sufficient conditions for non-congruent numbers in terms of the Monsky matrices. Many known criteria for non-congruent numbers can be viewed as special cases of our results.

Let $R$ be a ring and $e$, $g$ in $E(R)$, the set of idempotents of $R$. Then $R$ is called $(g,e)$-symmetric if $abc=0$ implies $gacbe=0$ for any $a$, $b$, $c\in R$. Clearly, $R$ is an $e$-symmetric ring if and only if $R$ is a $(1,e)$-symmetric ring; in particular, $R$ is a symmetric ring if and only if $R$ is a $(1,1)$-symmetric ring. We show that $e\in E(R)$ is left semicentral if and only if $R$ is a $(1-e,e)$-symmetric ring; in particular, $R$ is an Abel ring if and only if $R$ is a $(1-e,e)$-symmetric ring for each $e\in E(R)$. We also show that $R$ is $(g,e)$-symmetric if and only if $ge\in E(R)$, $geRge$ is symmetric, and $gxye=gxeye=gxgye$ for any $x$, $y\in R$. Using $e$-symmetric rings, we construct some new classes of rings.

In this paper, we prove that the electrical Lie algebra ${\mathfrak{e}}_{D_5}$ is isomorphic to the semidirect product of ${\mathfrak{s}\mathfrak{p}}_4$ and a 2-step nilpotent Lie algebra. Furthermore, we classify the irreducible highest weight modules for ${\mathfrak{e}}_{D_5}$.

Let $\mathcal{T}$ be a triangulated category with a proper class $\xi$ of triangles. We introduce the notions of left Frobenius pairs, left ($n$-)cotorsion pairs and left (weak) Auslander-Buchweitz contexts with respect to $\xi$ in $\mathcal{T}$. We show how to construct left cotorsion pais from left $n$-cotorsion pairs, and establish a one-to-one correspondence between left Frobenius pairs and left (weak) Auslander-Buchweitz contexts. Some applications are given in the Gorenstein homological theory of triangulated categories.

In this paper we continue the study of various ring theoretic properties of Morita contexts. Necessary and sufficient conditions are obtained for a general Morita context or a trivial Morita context or a formal triangular matrix ring to satisfy a certain ring property which is among being Kasch, completely primary, quasi-duo, 2-primal, NI, semiprimitive, projective-free, etc. We also characterize when a general Morita context is weakly principally quasi-Baer or strongly right mininjective.

In this article, we extend the Zassenhaus lemma (also called the butterfly lemma) and the Schreier refinement theorem to the case of gyrogroups. In addition, we use these results to prove the Jordan–Hölder theorem for gyrogroups as well as some theorems regarding subgyrogroup lattices.

Let $A$ be a finite dimensional $k$-algebra and $T$ be a support $\tau$-tilting right $A$-module. In this note, we give lower and upper bounds for the global dimension of the endomorphism algebra ${\mathsf{End}}_A(T)$ under some mild conditions. Finally, we give some examples to illustrate that both the upper and lower bounds can be reached.

Let $G$ be a finite group and $\mathrm{Irr}(G)$ the set of all irreducible complex characters of $G$. Let $\mathrm{cd}(G)$ be the set of all irreducible complex character degrees of $G$ and denote by $\rho (G)$ the set of all primes which divide some character degree of $G$. The character-prime graph $\mathrm{\Gamma }(G)$ associated to $G$ is a simple undirected graph whose vertex set is $\rho (G)$ and there is an edge between two distinct primes $p$ and $q$ if and only if the product $pq$ divides some character degree of $G$. We show that the finite nonabelian simple groups $A_7$, $J_1$, $J_3$, $J_4$, $L_3(3)$ and $U_3(4)$ are uniquely determined by their degree-patterns and orders.

The natural filtrations of infinite-dimensional Lie superalgebras of $\mathcal{O}$-type over a field of positive characteristic are proved to be invariant under automorphisms by studying ad-nilpotent elements. We observe that two Lie superalgebras of $\mathcal{O}$-type are isomorphic if and only if the integers in the definition of these Lie superalgebras are intrinsic. Thereby, all the Lie superalgebras of $\mathcal{O}$-type are classified in the sense of isomorphism.