Journal of Algebra and Its Applications

Let R be a non-commutative prime ring of characteristic different from 2 and 3, $Q_r$ its right Martindale quotient ring and C its extended centroid. Suppose that L is a non-central Lie ideal of R, F a nonzero b-generalized skew derivation of R. If

• (a)there exists $\lambda \in C$ such that $F(x)=\lambda x$, for all $x\in R$;
• (b)$R\subseteq M_2(C)$, the ring of $2\times 2$ matrices over C, and there exist $a\in Q_r$ and $\lambda \in C$ such that $F(x)=ax+xa+\lambda x$, for all $x\in R$.

Let G be a group with identity e and R be G-graded commutative ring with $1\ne 0.$ In this paper, we introduce and study the graded versions of 1-absorbing prime ideal. We give some properties and characterizations of these ideals in graded ring, and we give a characterization of graded 1-absorbing ideal the idealization $R(+)M.$

Let ${𝒰}$ be a triangular algebra over a commutative ring ${ℛ}$. In this paper, under some mild conditions on ${𝒰}$, we prove that if $\delta :{𝒰}\to {𝒰}$ is a nonlinear map satisfying

In this paper, we consider flat epimorphisms of commutative rings $R\to U$ such that, for every ideal $I\subset R$ for which $IU=U$, the quotient ring $R/I$ is semilocal of Krull dimension zero. Under these assumptions, we show that the projective dimension of the R-module U does not exceed 1. We also describe the Geigle–Lenzing perpendicular subcategory $U^{{\perp }_{0,1}}$ in $R-Mod$. Assuming additionally that the ring U and all the rings $R/I$ are perfect, we show that all flat R-modules are U-strongly flat. Thus, we obtain a generalization of some results of the paper [], where the case of the localization $U=S^{-1}R$ of the ring R at a multiplicative subset $S\subset R$ was considered.

In this paper, we consider the 2-generator p-groups of nilpotency class 2. We find the Fibonacci length and the period of the generalized order k-Pell sequences of these groups.

It is proved that a ring A is a right or left Noetherian, right distributive, centrally essential ring if and only if $A=A_1\times \cdots \times A_n$, where each of the rings $A_i$ is either a commutative Dedekind domain or a left and right Artinian, left and right uniserial ring.

Let S be a commutative semiring with unity different than zero. In this paper, we study the concept of 2-absorbing ideals of S which can be considered as a generalization of prime ideals. Among others, it is shown that the radical of a 2-absorbing ideal is also a 2-absorbing ideal and there are at most 2 prime k-ideals of S that are minimal over a 2-absorbing ideals.

The aim of this paper is to introduce 3-Hom-ρ-Lie algebra structures generalizing the algebras of 3-Hom-Lie algebra. Also, we investigate the representations and deformations theory of this type of Hom-Lie algebras. Moreover, we introduce the definition of extensions and abelian extensions of 3-Hom-ρ-Lie algebras and show that associated to any abelian extension, there is a representation and a 2-cocycle.

We provide a table of multiplication of the Poisson algebra on the minimal set of generators of the invariants of pairs of matrices of degree three.

The object of this paper is to generalize the notion of supplement in modules to monoid acts. In contrast to the case of modules that supplements of submodules do not generally exist, here we uniquely characterize the supplement of a proper subact of an act. Supplemented acts are defined as acts whose proper subacts all have proper supplements. We discuss how the property of being supplemented relates to certain other properties of acts. In particular, we prove that being supplemented and being completely reducible coincide.

Let G and A be finite groups of relative coprime orders and A act on G via automorphisms. In this paper, we prove that when every maximal A-invariant subgroup of G that contains the normalizer of some Sylow subgroup has prime index, then G is supersolvable; if every non-nilpotent maximal A-invariant subgroup of G has prime index or is normal in G, then G is a Sylow tower group.

In this paper, we study a family of rational monomial parametrizations. We investigate a few structural properties related to the corresponding monomial ideal J generated by the parametrization. We first find the implicit equation of the closure of the image of the parametrization. Then we provide a minimal graded free resolution of the monomial ideal J, and describe the minimal graded free resolution of the symmetric algebra of J. Finally, we provide a method to compute the defining equations of the Rees algebra of J using three moving planes that follow the parametrization.

We investigate the rational powers of ideals. We find that in the case of monomial ideals, the canonical indexing leads to a characterization of the rational powers yielding that symbolic powers of squarefree monomial ideals are indeed rational powers themselves. Using the connection with symbolic powers techniques, we use splittings to show the convergence of depths and normalized Castelnuovo–Mumford regularities. We show the convergence of Stanley depths for rational powers, and as a consequence of this, we show the before-now unknown convergence of Stanley depths of integral closure powers. Additionally, we show the finiteness of asymptotic associated primes, and we find that the normalized lengths of local cohomology modules converge for rational powers, and hence for symbolic powers of squarefree monomial ideals.

A commutative ring R is said to satisfy acc on d-annihilators if for every sequence ${(a_k)}_{k\in \mathbb{N}}$ of elements of R, the sequence $\mathrm{\text{ann}}(a_1)\subseteq \mathrm{\text{ann}}(a_1{a}_2)\subseteq \cdots$ is stationary. In this paper we extend the notion of rings with acc on d-annihilators by introducing the concept of rings with S-acc on d-annihilators, where S is a multiplicative set. Let R be a commutative ring and S a multiplicative subset of R. We say that R satisfies S-acc on d-annihilators if for every sequence ${(a_k)}_{k\in \mathbb{N}}$ of elements of R, the sequence $\mathrm{\text{ann}}(a_1)\subseteq \mathrm{\text{ann}}(a_1{a}_2)\subseteq \cdots$ is S-stationary, that is, there exist a positive integer n and an $s\in S$ such that for each $k\ge n$, $s({\mathrm{\text{ann}}}_R(a_1{a}_2\cdots a_k))\subseteq {\mathrm{\text{ann}}}_R(a_1{a}_2\cdots a_n)$. We give equivalent conditions for the power series (respectively, polynomial) ring over an Armendariz ring to satisfy S-acc on d-annihilators. We also study serval properties of rings satisfying S-acc on d-annihilators. The concept of the amalgamated duplication of R along an ideal I, $R\bowtie I$ is studied.

We describe the cohomology ring of a monomial algebra in the language of dimension tree or minimal resolution graph and in this context we study the finite generation of the cohomology rings of the extension algebras, showing among others that the cohomology ring $E(\mathrm{\Lambda })$ is finitely generated $\iff E(A)$ is $\iff E(A^{\ast })$ is, where $\mathrm{\Lambda }$ is the dual extension of a monomial algebra A and $A^{\ast }$ is the opposite algebra of A.

Let m and n be positive integer numbers. In this paper, we study $𝔑(m,n)$, the class of all groups G that for all subsets m and n of G containing m and n elements, respectively, there exist $x\in M$ and $y\in N$ such that $〈x,y〉$ is nilpotent, which introduced by Zarrin in 2012. We improve some results of Zarrin and find some sharp bounds for m and n such that $G\in 𝔑(m,n)$ implies that G is nilpotent. Also we will characterize all finite $𝔫$-groups in $𝔑(m,n)$, which $m+n\le 10$.

We characterize in bicategorical terms actions of monoidal categories on the categories of representations of algebras and of relative Hopf modules. For this purpose we introduce 2-cocycles in any 2-category ${𝒦}$. We observe that under certain conditions the structures of pseudofunctors between bicategories are in one-to-one correspondence with (twisted) 2-cocycles in the image bicategory. In particular, for certain pseudofunctors to $Cat$, the 2-category of categories, one gets 2-cocycles in the free completion 2-category under Eilenberg–Moore objects, constructed by Lack and Street. We introduce (co)quasi-bimonads in ${𝒦}$ and a suitable bicategory of Tambara (co)modules over (co)quasi-bimonads in ${𝒦}$ fitting the setting of the latter pseudofuntors. We describe explicitly the involved 2-cocycles in this context and show how they are related to Sweedler’s and Hausser–Nill 2-cocycles in ${𝒦}$, which we define. This allows us to recover some results of Schauenburg, Balan, Hausser and Nill for modules over commutative rings. We fit a version of the 2-category of bimonads in ${𝒦}$, which we introduced in a previous paper, in a similar setting as above and recover a result of Laugwitz. We observe that pseudofunctors to $Cat$ in general determine what we call pseudo-actions of hom-categories, which correspond to the whole range of a 2-cocycle, so that the described actions of categories appear as restrictions of these 2-cocycles to endo-hom categories.

Let x be an element of a finite group G, and p a prime factor of the order of G. It is clear that there always exists a unique minimal subnormal subgroup containing x, say $G_x$. We call the conjugacy class of x in $G_x$ the sub-class of x in G, see [G. Qian and Y. Yang, On sub-class sizes of finite groups, J. Aust. Math. Soc. (2020) 402–411]. In this paper, assume that $G(=AB)$ is the product of the subgroups A and B, we investigate the solvability, p-nilpotence and supersolvability of the group G under the condition that the sub-class sizes of prime power order elements in $A\cup B$ are $2^2$ free, $p^2$ free and square free, respectively, so that some known results relevant to conjugacy class sizes are generalized.

Let G be a group and $\mathrm{\text{DS}}(G)=\{H^{\prime }|H\le G\}$. G is said to be a ${𝒟}C$-group if $\mathrm{\text{DS}}(G)$ is a chain under set inclusion. In this paper, we prove that a finite ${𝒟}C$-group is a semidirect product of a Sylow p-subgroup and an abelian $p^{\prime }$-subgroup. For the case of G being a finite p-group, we obtain an optimal upper bound of $d(G^{\prime })$ for a ${𝒟}C$ p-group G. We also prove that a ${𝒟}C$ p-group is metabelian when $p\le 3$ and provide an example showing that a non-abelian ${𝒟}C$ p-group is not necessarily metabelian when $p\ge 5$. In particular, ${𝒟}C$ 2-groups are characterized.

Prompted by an inquiry of Manin on whether a coacting Hopf-type structure $H$ and an algebra $A$ that is coacted upon share algebraic properties, we study the particular case of $A$ being a path algebra $𝕂Q$ of a finite quiver $Q$ and $H$ being Hayashi’s face algebra $𝔥$ attached to $Q$. This is motivated by the work of Huang, Wicks, Won and the second author, where it was established that the weak bialgebra coacting universally on $𝕂Q$ (either from the left, right, or both sides compatibly) is $𝔥$. For our study, we define the Kronecker square $\widehat{Q}$ of $Q$, and show that $𝔥\cong 𝕂\widehat{Q}$ as unital graded algebras. Then we obtain ring-theoretic and homological properties of $𝔥$ in terms of graph-theoretic properties of $Q$ by way of $\widehat{Q}$.

Exponential Puiseux semirings are additive submonoids of ${\mathbb{Q}}_{\ge 0}$ generated by almost all of the nonnegative powers of a positive rational number, and they are natural generalizations of rational cyclic semirings. In this paper, we investigate some of the factorization invariants of exponential Puiseux semirings and briefly explore the connections of these properties with semigroup-theoretical invariants. Specifically, we provide exact formulas to compute the catenary degrees of these monoids and show that minima and maxima of their sets of distances are always attained at Betti elements. Additionally, we prove that sets of lengths of atomic exponential Puiseux semirings are almost arithmetic progressions with a common bound, while unions of sets of lengths are arithmetic progressions. We conclude by providing various characterizations of the atomic exponential Puiseux semirings with finite omega functions; in particular, we completely describe them in terms of their presentations.