Journal of Algebra and Its Applications

Given a finite ring $A$ which is a free left module over a subring $R$ of $A$, two types of $R$-bases, pseudo-self-dual bases (similar to trace orthogonal bases) and symmetric bases, are defined which in turn are used to define duality preserving maps from codes over $A$ to codes over $R$. Both types of bases are generalizations of similar concepts for fields. Many illustrative examples are given to shed light on the advantages to such mappings as well as their abundance.

Let $0\to A\to L\to B\to 0$ be a short exact sequence of Lie algebras over a field $F$, where $A$ is abelian. We show that the obstruction for a pair of automorphisms in $Aut(A)\times Aut(B)$ to be induced by an automorphism in $Aut(L)$ lies in the Lie algebra cohomology $H^2(B;A)$. As a consequence, we obtain a four term exact sequence relating automorphisms, derivations and cohomology of Lie algebras. We also obtain a more explicit necessary and sufficient condition for a pair of automorphisms in $Aut\left(\right.L_{n,2}^{(1)}\left)\right.\times Aut\left(\right.L_{n,2}^{ab}\left)\right.$ to be induced by an automorphism in $Aut\left(\right.L_{n,2}\left)\right.$, where $L_{n,2}$ is a free nilpotent Lie algebra of rank $n$ and step $2$.

Let $(R,𝔪,k)$ be a Noetherian local ring with maximal ideal $𝔪$ and residue field $k$. The linearity defect of a finitely generated $R$-module $M$, which is denoted ${ld}_R(M)$, is a numerical measure of how far $M$ is from having linear resolution. We study the linearity defect of the residue field. We give a positive answer to the question raised by Herzog and Iyengar of whether ${ld}_R(k)<\infty$ implies ${ld}_R(k)=0$, in the case when ${𝔪}^4=0$.

The study of rings with right Property ($A$), has done an important role in noncommutative ring theory. Following literature, a ring $R$ has right Property ($A$) if every finitely generated two-sided ideal consisting entirely of left zero-divisors has a nonzero right annihilator. Our results in this paper concerns the right Property ($A$) of Ore extensions as well as skew power series rings. We will show that if $R$ is a right duo ring, then the skew power series ring $R[[x;\alpha ]]$ has right Property ($A$), when $R$ is right Noetherian and $\alpha$-compatible. Moreover, for a right duo ring $R$ which is $(\alpha ,\delta )$-compatible, it is shown that (i) the Ore extension ring $R[x;\alpha ,\delta ]$ has right Property ($A$) and (ii) $R[x;\alpha ,\delta ]$ is right zip if and only if $R$ is right zip. As a corollary of our results, we provide answers to some open questions related to Property $(A)$, raised in [C. Y. Hong, N. K. Kim, Y. Lee and S. J. Ryu, Rings with Property ($A$) and their extensions, J. Algebra 315 (2007) 612–628].

We introduce and study the notion of w-Rickart modules (i.e. modules $M$ such that for every nonzero endomorphism $\phi$ of $M$, the kernel of $\phi$ is contained in a proper direct summand of $M$). We show that the class of right w-Rickart modules lies properly between the class of right ${𝒦}$-nonsingular modules and the class of right Rickart modules. Many properties of w-Rickart modules are established. Some relevant counterexamples are indicated.

A module $M$ is called a $D4$-module if, whenever $A$ and $B$ are submodules of $M$ with $M=A\oplus B$ and $f:A\to B$ is a homomorphism with $\mathrm{\text{Im}}f{\subseteq }^{\oplus }B$, we have $kerf{\subseteq }^{\oplus }A$. The class of $D4$-modules contains the $D3$-modules as well as the dual-square-free (DSF) modules. Furthermore, a $D4$-module $M$ is called pseudo-discrete if $M$ is also a lifting module. In this paper, we study the $D4$-, the DSF, and the pseudo-discrete modules, and show that a pseudo-discrete module is clean iff it has the finite exchange property iff it has the full exchange property.

We study the algebra of bilinear multiplications of an $n$-dimensional vector space. In particular, we study the Kantor product of some well-known (associative, Lie, alternative, Novikov and some other) multiplications.

We compute the continuous Hochschild cohomology of four reduced incidence algebras: the algebra of formal power series, the algebra of exponential power series, the algebra of Eulerian power series, and the algebra of formal Dirichlet series. We achieve the result by carrying out the computation for the coalgebra Cotor-groups of their pre-dual coalgebras.

In this paper, prime as well as primitive Kumjian–Pask algebras ${\mathrm{\text{KP}}}_R(\mathrm{\Lambda })$ of a row-finite $k$-graph $\mathrm{\Lambda }$ over a unital commutative ring $R$ are completely characterized in graph-theoretic and algebraic terms. By applying quotient $k$-graphs, these results describe prime and primitive graded basic ideals of Kumjian–Pask algebras. In particular, when $\mathrm{\Lambda }$ is strongly aperiodic and $R$ is a field, all prime and primitive ideals of a Kumjian–Pask algebra ${\mathrm{\text{KP}}}_R(\mathrm{\Lambda })$ are determined.

Let $G$ be a group, $F$ a field of characteristic $p\ge 0$, and ${𝒰}(FG)$ the unit group of the group algebra $FG$. In this paper, among other results, we show that if either (1) $FG$ satisfies a non-matrix polynomial identity, or (2) $G$ is locally finite, $F$ is infinite and ${𝒰}(FG)$ is an Engel-by-finite group, then the $p$-elements of $G$ form a (normal) subgroup $P$ and $G/P$ is abelian (here, of course, $P=1$ if $p=0$).

In this paper, we introduce the notions of a $3$-$Li{e}_{\infty }$-algebra and a 3-Lie 2-algebra. The former is a model for a 3-Lie algebra that satisfy the fundamental identity up to all higher homotopies, and the latter is the categorification of a 3-Lie algebra. We prove that the 2-category of 2-term $3$-$Li{e}_{\infty }$-algebras is equivalent to the 2-category of 3-Lie 2-algebras. Skeletal and strict 3-Lie 2-algebras are studied in detail. A construction of a 3-Lie 2-algebra from a symplectic 3-Lie algebra is given.

We investigate the topological structure of a cellular decomposition of the fixed locus of a unipotent operator of regular Jordan type acting on the wonderful compactification of the variety of complete quadrics and the variety of complete skew forms. The Poincaré polynomial is computed in each case and the poset of cell closures under inclusion is described in the complete quadrics case.

A graph is called weakly perfect if its vertex chromatic number equals its clique number. Let $𝕍$ be a vector space over a field $𝔽$ with $\{{\alpha }_1,\dots ,{\alpha }_n\}$ as a basis and $𝜃$ as the null vector. The non-zero component graph of $𝕍$ with respect to $\{{\alpha }_1,\dots ,{\alpha }_n\}$, denoted by $\mathrm{\Gamma }(𝕍)$, is a graph with the vertex set $𝕍\backslash 𝜃$ and two distinct vertices $a$ and $b$ are adjacent if and only if there exists at least one ${\alpha }_i$ along which both $a$ and $b$ have non-zero components. In this paper, it is shown that $\mathrm{\Gamma }(𝕍)$ is a weakly perfect graph. Also, we give an explicit formula for the vertex chromatic number of $\mathrm{\Gamma }(𝕍)$. Furthermore, it is proved that the edge chromatic number of $\mathrm{\Gamma }(𝕍)$ is equal to the maximum degree of $\mathrm{\Gamma }(𝕍)$.

We answer the question of splitting of the normalizer of a maximal torus in orthogonal groups of type $B_n$ and $D_n$ over finite fields and algebraically closed fields of positive characteristic.

Let $A={\mathbb{Z}}_p{S}_n$ with $p$ a prime and $S_n$ a symmetric group. We prove in this paper that if $n<2p$, then $\mathrm{\text{rad}}A=I$, where $I$ is the nilpotent ideal constructed in [Radicals of symmetric cellular algebras, Collog. Math. 133 (2013) 67–83]. Finally we give two remarks on algebras $A={\mathbb{Z}}_p{S}_n$ with $n<2p$.

In this paper, we construct three exact sequences on automorphism groups of abelian Lie ring extensions which relate automorphisms with group cohomology. The sequences resemble well-known Wells exact sequence for group extensions. An application of our construction is given.

Let $R=S/I$ be a graded algebra with $t_i$ and $T_i$ being the minimal and maximal shifts in the minimal graded free resolution of $R$ at degree $i$. We prove that $t_n\le t_1+T_{n-1}$ for all $n$. As a consequence, we show that for Gorenstein algebras of codimension $h$, the subadditivity of maximal shifts $T_n$ in the minimal graded free resolution holds for $n\ge h-1$, i.e. we show that $T_n\le T_a+T_{n-a}$ for $n\ge h-1$.

A ring $R$ is strongly 2-nil-clean if every element in $R$ is the sum of two idempotents and a nilpotent that commute. Fundamental properties of such rings are obtained. We prove that a ring $R$ is strongly 2-nil-clean if and only if for all $a\in R$, $a-a^3\in R$ is nilpotent, if and only if for all $a\in R$, $a^2\in R$ is strongly nil-clean, if and only if every element in $R$ is the sum of a tripotent and a nilpotent that commute. Furthermore, we prove that a ring $R$ is strongly 2-nil-clean if and only if $R/J(R)$ is tripotent and $J(R)$ is nil, if and only if $R\cong R_1,R_2$ or $R_1\times R_2$, where $R_1/J(R_1)$ is a Boolean ring and $J(R_1)$ is nil; $R_2/J(R_2)$ is a Yaqub ring and $J(R_2)$ is nil. Strongly 2-nil-clean group algebras are investigated as well.

We attach a diagraph with generalized Quaternion group of order $4n$ by utilizing the power map $f:Q_{4n}\to Q_{4n}$ defined by $f(x)=x^k$ for all $x\in Q_{4n}$, where $k$ is a fixed natural number. We examine the structure of these power digraphs and establish numerous results encapsulating the existence of cycle vertices, derivation of different formulae concerning the number of cycles, length of cycles and most importantly in-degree of vertices. Moreover, we categorize the regular and semi-regular power digraphs.

Four geometric conditions on a geodesic metric space, which are stronger variants of classical conditions characterizing hyperbolicity (featuring $\delta$-thin polygons, the Gromov product or the mesh of triangles), are proved to be equivalent. They define the class of polygon hyperbolic geodesic metric spaces. In the particular case of the Cayley graph of a finitely generated group, it is shown that they characterize virtually free groups.