Journal of Algebra and Its Applications

Let $R$ be a unital ∗-ring. As is well known, idempotents and projections can be constructed by the Moore–Penrose inverse and the core inverse of an element in $R$. In this paper, we mainly investigate characterizations and properties of these types of idempotents and projections. Also, several results in [Hartwig and Spindelböck, Matrices for which $A^{\ast }$ and $A^{†}$ commute, Linear Multilinear Algebra 14 (1984) 241–256] are extended to a general ∗-ring without ∗-cancellable conditions. As applications, the characterization of EP elements is given.

This paper studies Lie superalgebras graded by an arbitrary set ${𝒮}$ (set grading). We show that the set-graded Lie superalgebra ${ℒ}$ decomposes as the sum of well-described set-graded ideals plus a certain linear subspace. Under certain conditions, the simplicity of ${ℒ}$ is characterized and it is shown that the above decomposition is exactly the direct sum of the family of its minimal set-graded ideals, each one being a simple set-graded Lie superalgebra.

In this paper, we construct some families of rotated unimodular lattices and rotated direct sum of Barnes–Wall lattices ${\mathrm{\text{BW}}}_n$ for $n=4,8$ and $16$ via ideals of the ring of the integers $\mathbb{Z}[{\zeta }_{2^{r}q}+{\zeta }_{2^{r}q}^{-1}]$ for $q=3,5$ and $15$. We also construct rotated ${\mathrm{\text{BW}}}_{16}$ and ${\mathrm{\text{BW}}}_{32}$-lattices via $\mathbb{Z}$-submodules of $\mathbb{Z}[{\zeta }_{2^{r}15}+{\zeta }_{2^{r}15}^{-1}]$. Our focus is on totally real number fields since the associated lattices have full diversity and then may be suitable for signal transmission over both Gaussian and Rayleigh fading channels. The minimum product distances of such constructions are also presented here.

In this paper, we determine all almost simple groups each of whose character degrees has at most two distinct prime divisors. More generally, we show that a finite non-solvable group $G$ with this property is an extension of an almost simple group $S$ by a solvable group and $|\rho (G)|\le |\rho (S)|+2$, where $\rho (G)$ is the set of all primes dividing some character degree of $G$.

A module $M$ is called coseparable ($𝔪$-coseparable) if for every submodule $U$ of $M$ such that $M/U$ is finitely generated ($M/U$ is simple), there exists a direct summand $V$ of $M$ such that $V\subseteq U$ and $M/V$ is finitely generated. In this paper, we show that free modules are coseparable. We also investigate whether or not the ($𝔪$-)coseparability is stable under taking submodules, factor modules, direct summands, direct sums and direct products. We show that a finite direct sum of coseparable modules is not, in general, coseparable. But the class of $𝔪$-coseparable modules is closed under finite direct sums. Moreover, it is shown that the class of coseparable modules over noetherian rings is closed under finite direct sums. A characterization of coseparable modules over noetherian rings is provided. It is also shown that every lifting (H-supplemented) module is coseparable ($𝔪$-coseparable).

We provide an algorithm that computes a set of generators for any complete ideal in a smooth complex surface. More interestingly, these generators admit a presentation as monomials in a set of maximal contact elements associated to the minimal log-resolution of the ideal. Furthermore, the monomial expression given by our method is an equisingularity invariant of the ideal. As an outcome, we provide a geometric method to compute the integral closure of a planar ideal and we apply our algorithm to some families of complete ideals.

Let $G$ be a finite group. The power graph of $G$ is the undirected graph whose vertex set is $G$, and two distinct vertices are adjacent if one is a power of the other. The reduced power graph of $G$ is the subgraph of the power graph of $G$ obtained by deleting all edges $\{x,y\}$ with $〈x〉=〈y〉$, where $x$ and $y$ are two distinct elements of $G$. In this paper, we determine the proper connection number of the reduced power graph of $G$. As an application, we also determine the proper connection number of the power graph of $G$.

Let $X$ be a finite partially ordered set, $R$ a commutative domain and $\sigma$ an endomorphism of $R$. We define the skew incidence ring $I(X,R,\sigma )$ and we prove that each Jordan isomorphism of $I(X,R,\sigma )$ onto an associative ring $S$ is a near-sum of a homomorphism and an anti-homomorphism.

Let $A$ be a ring and $\phi$ its automorphism. It is proved that the skew Laurent series ring $A((x,\phi ))$ is a right serial ring if and only if $A$ is a right serial right Artinian ring.

We study the twisted Hochschild homology of quantum full flag manifolds, with the twist being the modular automorphism of the Haar state. We show that nontrivial $2$-cycles can be constructed from appropriate invariant projections. Moreover, we show that ${\mathrm{\text{HH}}}_2^{𝜃}({ℂ}_q[G/T])$ has dimension at least rank $(𝔤)$. We also discuss the case of generalized flag manifolds and present the example of the quantum Grassmannians.

For a commutative ring $S$ and self-orthogonal subcategory $C$ of $\mathrm{\text{Mod}}\left(S\right)$, we consider matrix factorizations whose modules belong to $C$. Let $f\in S$ be a regular element. If $f$ is $M$-regular for every $M\in C$, we show there is a natural embedding of the homotopy category of $C$-factorizations of $f$ into a corresponding homotopy category of totally acyclic complexes. Moreover, we prove this is an equivalence if $C$ is the category of projective or flat-cotorsion $S$-modules. Dually, using divisibility in place of regularity, we observe there is a parallel equivalence when $C$ is the category of injective $S$-modules.

Let $D$ be an integral domain, $D[X]$ be the polynomial ring over $D$, $t$ be the so-called $t$-operation on $D$, and $t$-Spec$(D)$ be the set of prime $t$-ideals of $D$. A nonzero nonunit of $D$ is said to be homogeneous if it is contained in a unique maximal $t$-ideal of $D$. We say that $D$ is a homogeneous factorization domain (HoFD) if each nonzero nonunit of $D$ can be written as a finite product of pairwise $t$-comaximal homogeneous elements. In this paper, among other things, we show that (1) a Prüfer $v$-multiplication domain (P$v$MD) $D$ is an HoFD if and only if $D[X]$ is an HoFD (2) if $D$ is integrally closed, then $D$ is a P$v$MD if and only if $t$-Spec$(D[X])$ is treed, and (3) $D$ is a weakly Matlis GCD-domain if and only if $D[X]$ is an HoFD with $t$-Spec$(D[X])$ treed. We also study the HoFD property of $A+XB[X]$ constructions, pullbacks, and semigroup rings.

Let $F$ be a field and ${\mathrm{\text{GL}}}_n(F)$ the general linear group of degree $n>1$ over $F$. The intersection graph $\mathrm{\Gamma }({\mathrm{\text{GL}}}_n(F))$ of ${\mathrm{\text{GL}}}_n(F)$ is a simple undirected graph whose vertex set includes all nontrivial proper subgroups of ${\mathrm{\text{GL}}}_n(F)$. Two vertices $A$ and $B$ of $\mathrm{\Gamma }({\mathrm{\text{GL}}}_n(F))$ are adjacent if $A\ne B$ and $A\cap B\ne \{I_n\}$. In this paper, we show that if $F$ is a finite field containing at least three elements, then the diameter $\delta (\mathrm{\Gamma }({\mathrm{\text{GL}}}_n(F)))$ is $2$ or $3$. We also classify ${\mathrm{\text{GL}}}_n(F)$ according to $\delta (\mathrm{\Gamma }({\mathrm{\text{GL}}}_n(F)))$. In case $F$ is infinite, we prove that $\mathrm{\Gamma }({\mathrm{\text{GL}}}_n(F))$ is one-ended of diameter $2$ and its unique end is thick.

In this paper, we prove that if every non-abelian self-centralizing subgroup of a finite group $G$ is a TI-subgroup or a subnormal subgroup of $G$, then every non-abelian subgroup of $G$ must be subnormal in $G$.

Let ${ℛ}$ be a prime ring with $\mathrm{\text{char}}({ℛ})\ne 2$, ${ℒ}$ a non-central Lie ideal of ${ℛ}$, ${𝒬}$ its Martindale quotient ring and ${𝒞}$ its extended centroid. Let $F:{ℛ}\to {ℛ}$ and $G:{ℛ}\to {ℛ}$ be nonzero generalized derivations on ${ℛ}$ such that

The main focus of this paper is on the problem of relating an ideal $I$ in the polynomial ring $\mathbb{Q}[x_1,\dots ,x_n]$ to a corresponding ideal in ${𝔽}_p[x_1,\dots ,x_n]$ where $p$ is a prime number; in other words, the reduction modulo$p$ of $I$. We first define a new notion of $\sigma$-good prime for $I$ which does depends on the term ordering $\sigma$, but not on the given generators of $I$. We relate our notion of $\sigma$-good primes to some other similar notions already in the literature. Then we introduce and describe a new invariant called the universal denominator which frees our definition of reduction modulo $p$ from the term ordering, thus letting us show that all but finitely many primes are good for $I$. One characteristic of our approach is that it enables us to easily detect some bad primes, a distinct advantage when using modular methods.

Let $𝔛$ be a class of simple groups with a completeness property $\pi (𝔛)=\mathrm{\text{char}}𝔛$. Förster introduced the concept of $𝔛$-local formation in order to obtain a common extension of well-known theorems of Gaschütz–Lubeseder–Schmid and Baer [Publ. Mat. UAB 29(2–3) (1985) 39–76]. In this paper, it is proved that the lattice of all $𝔛$-local formations of finite groups is modular.

The power graph ${𝒫}(G)$ of a finite group $G$ is the undirected simple graph whose vertex set is $G$, in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer $n\ge 2$, let $C_n$ denote the cyclic group of order $n$ and let $r$ be the number of distinct prime divisors of $n$. The minimum degree $\delta ({𝒫}(C_n))$ of ${𝒫}(C_n)$ is known for $r\in \{1,2\}$, see [R. P. Panda and K. V. Krishna, On the minimum degree, edge-connectivity and connectivity of power graphs of finite groups, Comm. Algebra 46(7) (2018) 3182–3197]. For $r\ge 3$, under certain conditions involving the prime divisors of $n$, we identify at most $r-1$ vertices such that $\delta ({𝒫}(C_n))$ is equal to the degree of at least one of these vertices. If $r=3$, or that $n$ is a product of distinct primes, we are able to identify two such vertices without any condition on the prime divisors of $n$.

A quandle is an algebra whose axioms are motivated from knot theory. A linear extension of a quandle can be described by using a pair of maps called an Alexander pair. In this paper, we show that a linear extension of a multiple conjugation quandle can be described by using a pair of maps called an MCQ Alexander pair, where a MCQ is an algebra whose axioms are motivated from handlebody-knot theory.

Let $A$ be an artin algebra, $M$ be a ${𝒲}$-Gorenstein $A$-module and $B={\mathrm{\text{End}}}_{A}M$, then $M$ is a $A$-$B$-bimodule. We use the restricted flat dimension of $M_B$ and the finitistic ${𝒲}$-dimension of $A$ to characterize the finitistic dimension of $B$, and obtain the following main result: if $A$ is ${{𝒢}}_{{𝒲}}$-finite with ${{𝒢}}_{{𝒲}}(A)={\mathrm{\text{add}}}_{A}E$, then we have: (1) If ${\mathrm{\text{findim}}}_{{𝒲}}A=0$ or $1$, then $\mathrm{\text{findim}}B\le 2+\mathrm{\text{rfd}}(M_B)+{\mathrm{\text{pd}}}_B{\mathrm{\text{Hom}}}_A(M,E);$ (2) If ${\mathrm{\text{findim}}}_{{𝒲}}A\ge 2$, then $\mathrm{\text{findim}}B\le {\mathrm{\text{findim}}}_{{𝒲}}(A)+\mathrm{\text{rfd}}(M_B)+{\mathrm{\text{pd}}}_B{\mathrm{\text{Hom}}}_A(M,E).$

We describe the left regular module of a quantum complete intersection $A(q,n_1,\dots ,n_t)$ by the property that it is the unique finite-dimensional indecomposable left $A(q,n_1,\dots ,n_t)$-module of Loewy length ${\sum }_{1\le i\le t}{n}_i-t+1.$ Using a reduction to $A(q,4,4)$-modules, we classify the $4$-dimensional indecomposable left modules over quantum complete intersection $A(q,m,n)$ in two variables, by explicitly giving their diagram presentations. Together with the existed work on indecomposable $A(q,m,n)$-modules of dimension $\le 3$, we then know all the indecomposable $A(q,m,n)$-modules of dimension $\le 4$.