Journal of Algebra and Its Applications

In the present note, we continue the study of skew inverse Laurent series ring $R((x^{-1};\alpha ,\delta ))$ and skew inverse power series ring $R[[x^{-1};\alpha ,\delta ]]$, where $R$ is a ring equipped with an automorphism $\alpha$ and an $\alpha$-derivation $\delta$. Necessary and sufficient conditions are obtained for $R[[x^{-1};\alpha ,\delta ]]$ to satisfy a certain ring property which is among being local, semilocal, semiperfect, semiregular, left quasi-duo, (uniquely) clean, exchange, projective-free and $I$-ring, respectively. It is shown here that $R((x^{-1};\alpha ,\delta ))$ (respectively $R[[x^{-1};\alpha ,\delta ]]$) is a domain satisfying the ascending chain condition (Acc) on principal left (respectively right) ideals if and only if so does $R$. Also, we investigate the problem when a skew inverse Laurent series ring $R((x^{-1};\alpha ,\delta ))$ has the same Goldie rank as the ring $R$ and is proved that, if $R$ is a semiprime right Goldie ring, then $R((x^{-1};\alpha ,\delta ))$ is semiprimitive. Furthermore, we study on the relationship between the simplicity, semiprimeness, quasi-Baerness and Baerness property of a ring $R$ and these of the skew inverse Laurent series ring. Finally, we consider the problem of determining when $f(x)\in R((x^{-1};\alpha ,\delta ))$ is nilpotent.

Let $S$ be a semigroup and $b,c\in S$. The concept of $(b,c)$-inverses was introduced by Drazin in 2012. It is well known that the Moore–Penrose inverse, the Drazin inverse, the Bott–Duffin inverse, the inverse along an element, the core inverse and dual core inverse are all special cases of the $(b,c)$-inverse. In this paper, a new relationship between the $(b,c)$-inverse and the Bott–Duffin $(e,f)$-inverse is established. The relations between the $(b,c)$-inverse of $paq$ and certain classes of generalized inverses of $pa$ and $aq$, and the $(b^{\prime },c^{\prime })$-inverse of $a$ are characterized for some $b^{\prime },c^{\prime }\in S$, where $p,a,q\in S$. Necessary and sufficient conditions for the existence of the $(B,C)$-inverse of a lower triangular matrix over an associative ring $R$ are also given, and its expression is derived, where $B,C$ are regular triangular matrices.

For $n\in \mathbb{N}$, let $X_n=\{a_1,a_2,\dots ,a_n\}$ be an $n$-element set and let $F=(X_n;{<}_f)$ be a fence, also called a zigzag poset. As usual, we denote by $I_n$ the symmetric inverse semigroup on $X_n$. We say that a transformation $\alpha \in I_n$ is fence-preserving if $x{<}_{f}y$ implies that $x\alpha {<}_{f}y\alpha$, for all $x,y$ in the domain of $\alpha$. In this paper, we study the semigroup $PF{I}_n$ of all partial fence-preserving injections of $X_n$ and its subsemigroup $I{F}_n=\{\alpha \in PF{I}_n:{\alpha }^{-1}\in PF{I}_n\}$. Clearly, $I{F}_n$ is an inverse semigroup and contains all regular elements of $PF{I}_n.$ We characterize the Green’s relations for the semigroup $I{F}_n$. Further, we prove that the semigroup $I{F}_n$ is generated by its elements with rank $\ge n-2$. Moreover, for $n\in 2\mathbb{N}$, we find the least generating set and calculate the rank of $I{F}_n$.

A subgroup $H$ of a finite group $G$ is said to be $s$-conditionally permutable in $G$ if for every Sylow subgroup $X$ of $G$, there exists an element $g\in G$ such that $H{X}^g=X^{g}H$. In this paper, the structure of solvable group $G$ in which every $p$-subgroup of $G$ or every subnormal subgroup of $G$ is $s$-conditionally permutable in $G$ is described. Let $G$ be a solvable group and $q$ the largest prime dividing $\left|G\right|$. Suppose further that $Q$ is the Sylow $q$-subgroup of $G$ and $G^{𝔑}\ne Q$. We are going to show that $G$ is a PST-group if and only if every subnormal subgroup of $G$ is $s$-conditionally permutable in $G$.

It is known that a ring $R$ is semiprime right Goldie if and only if $R_R$ is a strongly compressible module if and only if every right ideal of $R$ is strongly compressible, and that a ring $R$ is semiprime Goldie ring if and only if every right ideal of $R$ is weakly injective if and only if every essential right ideal of $R$ is weakly injective. In this paper, we characterize the rings whose proper (principal) right ideals are strongly compressible, and the rings whose proper (essential) right ideals are weakly injective.

Let $F_q$ be a finite field with $q$ elements, $R_0=M_n(F_q)$ be the ring of all $n\times n$ matrices over $F_q$, $L(R_0)$ be the set of all nontrivial left ideals of $R_0$. The co-maximal ideal graph of $R_0$, denoted by $C(R_0)$, is a graph with $L(R_0)$ as vertex set and two nontrivial left ideals $I,J$ of $R_0$ are adjacent if and only if $I+J=R_0$. If $n=2$, it is easy to see that $C(R_0)$ is a complete graph, thus any permutation of vertices of $C(R_0)$ is an automorphism of $C(R_0)$. A natural problem is: How about the automorphisms of $C(R_0)$ when $n\ge 3$. In this paper, we aim to solve this problem. When $n\ge 3$, a mapping $\sigma$ on $L(R_0)$ is proved to be an automorphism of $C(R_0)$ if and only if there is an invertible matrix $x\in R_0$ and a field automorphism $f$ of $F_q$ such that $\sigma (I)=f(I)x$ for any $I\in L(R_0)$, where $f(I)x=\{f(z)x|z\in I\}$ and $f(z)={[f(z_{ij})]}_{n\times n}$ for $z={[z_{ij}]}_{n\times n}\in R_0$.

Let ${\mathbb{Z}}_{p^s}$ be the ring of integers modulo $p^s$ where $p$ is a prime and $s(\ge 1)$ is a positive integer, $R=M_{2\times 2}({\mathbb{Z}}_{p^s})$ the $2\times 2$ matrix ring over ${\mathbb{Z}}_{p^s}$. The zero-divisor graph of $R$, written as $\mathrm{\Gamma }(R)$, is a directed graph whose vertices are nonzero zero-divisors of $R$, and there is a directed edge from a vertex $A$ to a vertex $B$ if and only if $AB=0$. In this paper, we completely determine the automorphisms of $\mathrm{\Gamma }(R)$.

Let $G$ be a finite group and let $\mathrm{\text{cd}}(G)$ be the set of all irreducible character degrees of $G$. We consider finite groups $G$ with the property that $\mathrm{\text{cd}}(G)$ has at most two composite members. We derive a bound 6 for the size of character degree sets of such groups. There are examples in both solvable and nonsolvable groups where this bound is met. In the case of nonsolvable groups, we are able to determine the structure of such groups with $|\mathrm{\text{cd}}(G)|\le 6$.

In this paper, we introduce a semiprime closure operation on submodules of a module. Several properties of this closure operation are studied and some applications are provided.

Using a conic $(=\mathrm{\text{degree}}\text{-}2)$ algebra $B$ over an arbitrary commutative ring, a scalar $\mu$ and a linear form $s$ on $B$ as input, the non-orthogonal Cayley–Dickson construction produces a conic algebra $C:=\mathrm{\text{Cay}}(B;\mu ,s)$ and collapses to the standard (orthogonal) Cayley–Dickson construction for $s=0$. Conditions on $B,\mu ,s$ that are necessary and sufficient for $C$ to satisfy various algebraic properties (like associativity or alternativity) are derived. Sufficient conditions guaranteeing non-singularity of $C$ even if $B$ is singular are also given. As an application, we show how the algebras of Hurwitz quaternions and of Dickson or Coxeter octonions over the rational integers can be obtained from the non-orthogonal Cayley–Dickson construction.

Let $G$ be a finite solvable group, and let $p$ be a prime. We obtain some conditions for $G$ to be $p$-nilpotent or $p$-closed in terms of irreducible monomial characters.

We investigate the similarities between adic finiteness and homological finiteness for chain complexes over a commutative Noetherian ring. In particular, we extend the isomorphism properties of certain natural morphisms from homologically finite complexes to adically finite complexes. We do the same for characterizations of certain homological dimensions. In addition, we study transfer of adic finiteness along ring homomorphisms, all with a view toward subsequent applications.

A ring $R$ is strongly weakly nil-clean if every element in $R$ is the sum or difference of a nilpotent and an idempotent that commutes. We prove, in this paper, that a ring $R$ is strongly weakly nil-clean if and only if for any $a\in R$, there exists an idempotent $e\in \mathbb{Z}[a]$ such that $a\pm e$ is nilpotent if and only if $N(R)$ forms an ideal and $R/N(R)$ is weakly Boolean if and only if $R\cong R_1,R_2$ or $R_1\oplus R_2$, where $R_1$ is strongly nil-clean and $R_2$ is an IU ring if and only if $R$ has no homomorphic image ${\mathbb{Z}}_3\oplus {\mathbb{Z}}_3$ and for any $a\in R$, there exists $n\in \mathbb{N}$ (depending on $a$) such that $a^n-a^{n+2}\in N(R)$. These also extend known theorems of Danchev, Kosan, Zhou, etc.

Let $H$ be a commutative semigroup with unit element such that every non-unit can be written as a finite product of irreducible elements (atoms). For every $k\in \mathbb{N}$, let ${\mathcal{𝒰}}_k(H)$ denote the set of all $\ell \in \mathbb{N}$ with the property that there are atoms $u_1,\dots ,u_k,v_1,\dots ,v_{\ell }$ such that $u_1\cdot \dots \cdot u_k=v_1\cdot \dots \cdot v_{\ell }$ (thus, ${\mathcal{𝒰}}_k(H)$ is the union of all sets of lengths containing $k$). The Structure Theorem for Unions states that, for all sufficiently large $k$, the sets ${\mathcal{𝒰}}_k(H)$ are almost arithmetical progressions with the same difference and global bound. We present a new approach to this result in the framework of arithmetic combinatorics, by deriving, for suitably defined families of subsets of the non-negative integers, a characterization of when the Structure Theorem holds. This abstract approach allows us to verify, for the first time, the Structure Theorem for a variety of possibly non-cancellative semigroups, including semigroups of (not necessarily invertible) ideals and semigroups of modules. Furthermore, we provide the very first example of a semigroup (actually, a locally tame Krull monoid) that does not satisfy the Structure Theorem.

We consider the quantum algebra $U_q({𝔰𝔩}_2)$ in the equitable presentation. From this point of view, we describe the Lusztig automorphism and the corresponding Lusztig operator.

We study weight modules of the Lie algebra $W_2$ of vector fields on ${ℂ}^2$. A classification of all simple weight modules of $W_2$ with a uniformly bounded set of weight multiplicities is provided. To achieve this classification, we introduce a new family of generalized tensor $W_n$-modules. Our classification result is an important step in the classification of all simple weight $W_n$-modules with finite weight multiplicities.

A polycyclic group $G$ is called an $NAF{Q}_n$-group if every normal abelian subgroup of any finite quotient of $G$ is generated by $n$, or fewer, elements and $n$ is the least integer with this property. In this paper, the structure of $NAF{Q}_1$-groups and $NAF{Q}_2$-groups is determined.

In this paper, the left (right) $h$-nucleus of an invertible algebra is defined and its connection with regular permutations of the invertible algebra is investigated. Using the notions of the $h$-nucleus, we have obtained the characterizations of linear invertible algebras and those for left (right) linear invertible algebras by the second-order formulas.

We study the structure of genetic coalgebras and prove the existence of a decomposition as the direct sum of indecomposable subcoalgebras with genetic realization. To obtain such a decomposition, we first define a new multiplication for their related cubic stochastic matrices of type (1,2) and then, considering these matrices as cubic non-negative, we show how this new multiplication induces an index classification which can be used to study the genetic coalgebra structure. Genetically, the given coalgebra decomposition can be understood as a genetic clustering of the different types for the genetic trait defining the genetic coalgebra, which allows us to identify isolated ancestral populations.

Let $𝔞$ be an ideal of a commutative noetherian ring $R$ with unity and $M$ an $R$-module supported at $\mathrm{\text{V}}(𝔞)$. Let $n$ be the supermum of the integers $i$ for which $H_i^{𝔞}(M)\ne 0$. We show that $M$ is $𝔞$-cofinite if and only if the $R$-module ${Tor}_i^R(R/𝔞,M)$ is finitely generated for every $0\le i\le n$. This provides a hands-on and computable finitely-many-steps criterion to examine $𝔞$-confiniteness. Our approach relies heavily on the theory of local homology which demonstrates the effectiveness and indispensability of this tool.

In this paper, we discuss the concept and examples of algebras of uniform exponential growth. We prove that Golod–Shafarevich algebras and group algebras of Golod–Shafarevich groups are of uniform exponential growth. We prove that uniform exponential growth of the universal enveloping algebra of a Lie algebra $L$ implies uniform exponential growth of $L$, and conversely should $L$ be graded by the natural numbers. We prove that a restricted Lie algebra is of uniform exponential growth if and only if its universal enveloping algebra is. We proceed to give several conditions equivalent to the uniform exponential growth of the graded algebra associated to a group algebra filtered by powers of its fundamental ideal.