Journal of Algebra and Its Applications

The paper shows an extension of the Lichtenbaum–Hartshorne Vanishing theorem for local cohomology modules with respect to a pair of ideals. We also study the attached primes of top local cohomology module $H_{I,J}^c(M),$ where $c=sup\{i|H_{I,J}^i(M)\ne 0\}.$ In the case, where $c=dimM-1,$ we show that ${\mathrm{\text{Att}}}_R(H_{I,J}^{\mathrm{\text{dim}}M-1}(M))\cap V(J)=\{𝔭\in {\mathrm{\text{Supp}}}_{R}M\cap V(J)|\mathrm{\text{cd}}(I,R/𝔭)=\mathrm{\text{dim}}M-1\}.$

It is proved that the algebra of multiplications of the free commutative alternative algebra of finite rank $r\ge 3$ is strongly Lie nilpotent of class $2(r-1)$. It is found the class of nilpotency of the ideal, generated by commutators in the free $r$-generated associative algebra with identity of Lie nilpotency of degree $n$ under the condition that $r\le 5$ or $n\le 5$.

Given a certain factorization property of a ring $R$, we can ask if this property extends to the polynomial ring over $R$ or vice versa. For example, it is well known that $R$ is a unique factorization domain if and only if $R[X]$ is a unique factorization domain. If $R$ is not a domain, this is no longer true. In this paper, we survey unique factorization in commutative rings with zero divisors, and characterize when a polynomial ring over an arbitrary commutative ring has unique factorization.

For a commutative integral domain $D$ with field of fractions $K$, the ring of integer-valued polynomials on $D$ is $\mathrm{\text{Int}}(D)=\{f\in K[x]|f(a)\in D\mathrm{\text{for all }}a\in D\}$. In this paper, we extend this construction to skew polynomial rings. Given an automorphism $\sigma$ of $K$, the skew polynomial ring $K[x;\sigma ]$ consists of polynomials with coefficients from $K$, and with multiplication given by $xa=\sigma (a)x$ for all $a\in K$. We define $\mathrm{\text{Int}}(D;\sigma )=\{f\in K[x;\sigma ]|f(a)\in D\mathrm{\text{for all}}a\in D\}$, which is the set of integer-valued skew polynomials on $D$. When $\sigma$ is not the identity, $K[x;\sigma ]$ is noncommutative and evaluation behaves differently than it does for ordinary polynomials. Nevertheless, we are able to prove that $\mathrm{\text{Int}}(D;\sigma )$ has a ring structure in many cases. We show how to produce elements of $\mathrm{\text{Int}}(D;\sigma )$ and investigate its properties regarding localization and Noetherian conditions. Particular attention is paid to the case where $D$ is a discrete valuation ring with finite residue field.

Denote by ${\nu }_p(G)$ the number of Sylow $p$-subgroups of $G$. For every subgroup $H$ of $G$, it is easy to see that ${\nu }_p(H)\le {\nu }_p(G)$, but ${\nu }_p(H)$ does not divide ${\nu }_p(G)$ in general. Following [W. Guo and E. P. Vdovin, Number of Sylow subgroups in finite groups, J. Group Theory 21(4) (2018) 695–712], we say that a group $G$ satisfies DivSyl(p) if ${\nu }_p(H)$ divides ${\nu }_p(G)$ for every subgroup $H$ of $G$. In this paper, we show that “almost for every” finite simple group $S$, there exists a prime $p$ such that $S$ does not satisfy DivSyl(p).

We investigate constructions of higher arity self-distributive operations, and give relations between cohomology groups corresponding to operations of different arities. For this purpose we introduce the notion of mutually distributive $n$-ary operations generalizing those for the binary case, and define a cohomology theory labeled by these operations. A geometric interpretation in terms of framed links is described, with the scope of providing algebraic background of constructing $2$-cocycles for framed link invariants. This theory is also studied in the context of symmetric monoidal categories. Examples from Lie algebras, coalgebras and Hopf algebras are given.

A subgroup $H$ of a finite group $G$ is said to be $S$-semipermutable in $G$, if $H$ permutes with all Sylow $q$-subgroups of $G$ for the primes $q$ not dividing $\left|H\right|$. In this paper, we consider the $S$-semipermutability of some $p$-subgroups to investigate the $p$-supersolubility of a finite group. Some interesting results are obtained which extend some known results.

Injectivity is one of the useful notions in algebra, as well as in many other branches of mathematics, and the study of injectivity with respect to different classes of monomorphisms is crucial in many categories. Also, essentiality is an important notion closely related to injectivity. Down closed monomorphisms and injectivity with respect to these monomorphisms, so-called dc-injectivity, were first introduced and studied by the authors for $S$-posets, posets with an action of a pomonoid $S$ on them. They gave a criterion for dc-injectivity and studied such injectivity for $S$ itself, and for its poideals. In this paper, we give results about dc-injectivity of $S$-posets, also we find some homological characterization of pomonoids and pogroups by dc-injectivity. In particular, we give a characterization of pomonoids over which dc-injectivity is equivalent to having a zero top element. Also, introducing the notion of $T$-injectivity for $S$-posets, where $S=T\dot{\cup }\{1\}$ and $1$ is externally adjoined to the posemigroup $T$, we find some classes of pomonoids such that for $S$-posets over them the Baer Criterion holds. Further, several kinds of essentiality of down closed monomorphisms of $S$-posets, and their relations with each other and with dc-injectivity is studied. It is proved that although these essential extensions are not necessarily equivalent, they behave almost equivalently with respect to dc-injectivity. Finally, we give an explicit description of dc-injective hulls of $S$-posets for some classes of pomonoids $S$.

The aim of this paper is to reveal the relationship between the proper class generated projectively by g-semiartinian modules and the subprojectivity domains of g-semiartinian modules. A module $M$ is called g-semiartinian if every nonzero homomorphic image of $M$ has a singular simple submodule. It is proven that every g-semiartinian right $R$-module has an epic projective envelope if and only if $R$ is a right PS ring if and only if every subprojectivity domain of any g-semiartinian right $R$-module is closed under submodules. A g-semiartinian module whose domain of subprojectivity as small as possible is called gsap-indigent. We investigated the structure of rings whose (simple, coatomic) g-semiartinian right modules are gsap-indigent or projective. Furthermore, over right PS rings, necessary and sufficient condition to be gsap-indigent module was determined.

In this paper, we introduce a class of 2-cocycles on monoidal Hom–Hopf algebras, study their properties, and extend neat lazy 2-cocycles to a Radford $(m,n)$-biproduct monoidal Hom–Hopf algebra.

Let $q$ be a $p$-power where $p$ is a fixed prime. In this paper, we look at the $p$-power maps on unitriangular group $U(n,q)$ and triangular group $T(n,q)$. In the spirit of Borel dominance theorem for algebraic groups, we show that the image of this map contains large size conjugacy classes. For the triangular group we give a recursive formula to count the image size.

For the bounded derived category of an abelian category, bounds of the dimension with respect to a complete hereditary cotorsion pair are given. We also characterize levels of DG-modules and study how levels involved in a recollement of derived categories over DG-rings are related.

The set of standard Capelli bitableaux and the set of standard Young–Capelli bitableaux are bases of $U(\mathrm{\text{gl}}(n))$, whose action on the Gordan–Capelli basis of polynomial algebra $ℂ[M_{n,n}]$ have remarkable properties (see, e.g. [A. Brini, A. Palareti and A. Teolis, Gordan–Capelli series in superalgebras, Proc. Natl. Acad. Sci. USA 85 (1988) 1330–1333; A. Brini and A. Teolis, Young–Capelli symmetrizers in superalgebras, Proc. Natl. Acad. Sci. USA 86 (1989) 775–778; A. Brini and A. Teolis, Capelli bitableaux and $\mathbb{Z}$-forms of general linear Lie superalgebras, Proc. Natl. Acad. Sci. USA 87 (1990) 56–60; A. Brini and A. Teolis, Capelli’s theory, Koszul maps, and superalgebras, Proc. Natl. Acad. Sci. USA 90 (1993) 10245–10249.]). We introduce a new class of elements of $U(\mathrm{\text{gl}}(n))$, called the Capelli immanants, that can be efficiently computed and provide a system of linear generators of $U(\mathrm{\text{gl}}(n))$. The Okounkov quantum immanants [A. Okounkov, Quantum immanants and higher Capelli identities, Transform Groups 1 (1996) 99–126; A. Okounkov, Young basis, Wick formula, and higher Capelli identities, Int. Math. Res. Not. 1996(17) (1996) 817–839.] — quantum immanants, for short — are proved to be simple linear combinations of diagonal Capelli immanants, with explicit coefficients. Quantum immanants can also be expressed as sums of double Young–Capelli bitableaux. Since double Young–Capelli bitableaux uniquely expands into linear combinations of standard Young–Capelli bitableaux, this leads to canonical presentations of quantum immanants, and, furthermore, it does not involve the computation of the irreducible characters of symmetric groups.

We prove that a quasi-bialgebra admits a preantipode if and only if the associated free quasi-Hopf bimodule functor is Frobenius, if and only if the related (opmonoidal) monad is a Hopf monad. The same results hold in particular for a bialgebra, tightening the connection between Hopf and Frobenius properties.

A class of integer-valued functions defined on the set of ideals of an integral domain $R$ is investigated. We show that this class of functions, which we call ideal valuations, are in one-to-one correspondence with countable descending chains of finite type, stable semistar operations with largest element equal to the $e$-operation. We use this class of functions to recover familiar semistar operations such as the $w$-operation and to give a solution to a conjecture by Chapman and Glaz when the ring is a valuation domain.

Let $R$ be an associative ring with identity. A right $R$-module $M_R$ is said to have Property ($A$), if each finitely generated ideal $I\subseteq Z(M_R)$ has a nonzero annihilator in $M_R$. Evans [Zero divisors in Noetherian-like rings, Trans. Amer. Math. Soc. 155(2) (1971) 505–512.] proved that, over a commutative ring, zero-divisor modules have Property ($A$). We study and construct various classes of modules with Property ($A$). Following Anderson and Chun [McCoy modules and related modules over commutative rings, Comm. Algebra 45(6) (2017) 2593–2601.], we introduce $G$-dual McCoy modules and show that, for every strictly totally ordered monoid $G$, faithful symmetric modules are $G$-dual McCoy. We then use this notion to give a characterization for modules with Property ($A$). For a faithful symmetric right $R$-module $M_R$ and a strictly totally ordered monoid $G$, it is proved that the right $R[G]$-module $M{[G]}_{R[G]}$ is primal if and only if $M_R$ is primal with Property ($A$).

Let $R$ be a polynomial ring in $n$ indeterminates with coefficients in the field $K$ of characteristic $p>0$ and ${𝒟}$ be the ring of differential operators over $R$. In this paper, we introduce the notion of generalized Eulerian ${𝒟}$-modules for characteristic $p>0$ and establish their properties. We show that if ${𝒯}$ is any graded Lyubeznik functor on the category of modules over $R,$ then ${𝒯}(R)$ is a generalized Eulerian ${𝒟}$-module. As a consequence, we prove that all socle elements of module $H_{𝔪}^i({𝒯}(R))$ are concentrated in degree $-n,$ where $𝔪$ is an irrelevant maximal ideal of $R$.

Let $n\ge 1$ be a squarefree integer, and let $M$, $A$ be two groups of order $n$. Using our previous results on the enumeration of Hopf–Galois structures on Galois extensions of fields of squarefree degree, we determine the number of skew braces (up to isomorphism) with multiplicative group $M$ and additive group $A$. As an application, we enumerate skew braces whose order is the product of three distinct primes, in particular proving a conjecture of Bardakov, Neshchadim and Yadav on the number of skew braces of order $2pq$ for primes $q>p\ge 3$.

In this paper, we study the category of finitely generated modules over a class of right $4$-Nakayama artin algebras. This class of algebras appear naturally in the study of representation-finite artin algebras. First, we give a characterization of right $4$-Nakayama artin algebras. Then, we classify finitely generated indecomposable right modules over right $4$-Nakayama artin algebras. We also compute almost split sequences for the class of right $4$-Nakayama artin algebras.

It was shown in [Y. Katsov, Tensor products and injective envelopes of semimodules over additively regular semirings, Algebra Colloq. 4 (1997) 121–131.] that each semimodule over an additively regular semiring has an injective envelope. We show the converse statement is true as well; moreover, the result holds even if each finitely generated semimodule has an injective envelope.

Let $A=A(\alpha ,\beta )$ be a graded down-up algebra with $(degx,degy)=(1,n)$ and $\beta \ne 0$, and let $\nabla A$ be the Beilinson algebra of $A$. If $n=1$, then a description of the Hochschild cohomology group of $\nabla A$ is known. In this paper, we calculate the Hochschild cohomology group of $\nabla A$ for the case $n\ge 2$. As an application, we see that the structure of the bounded derived category of the noncommutative projective scheme of $A$ is different depending on whether (10) $\left(\right.\begin{array}{cc}\alpha \hfill & 1\hfill \\ \beta \hfill & 0\hfill \end{array}{\left)\right.}^n$ $\left(\right.\begin{array}{c}1\hfill \\ 0\hfill \end{array}\left)\right.$ is zero or not. Moreover, it turns out that there is a difference between the cases $n=2$ and $n\ge 3$ in the context of Grothendieck groups.