Journal of Algebra and Its Applications

In this paper, we introduce the notions of non-abelian tensor and exterior products of two ideal graded crossed submodules of a given crossed module of Lie superalgebras. We also study some of their basic properties and their connection with the second homology of crossed modules of Lie superalgebras.

Let $R$ be a commutative local ring whose maximal ideal is generated by a nilpotent element, and Mat$(n,R)$ be the multiplicative monoid of the square matrices of order $n$ over $R$. In this paper, we provide the construction of Green’s ${\mathscr{H}}$-equivalence classes in the multiplicative monoid Mat$(n,R)$. Then, we enumerate these classes in the special cases $R=\mathbb{Z}/p^d\mathbb{Z}$ and $R={𝔽}_q[x]/〈x^d〉$.

Analogy between Anderson $t$-motives and abelian varieties with multiplication by an imaginary quadratic field (MIQF) is a source of 2 results:

(1) A description of abelian varieties with MIQF of dimension $r$ and signature $(n,r-n)$ in terms of ”lattices” of dimension $r$ in ${ℂ}^n$;

(2) A construction of exterior powers of abelian varieties with MIQF having $n=1$.

We study the absolute-valued algebras ${𝒜}$ satisfying the identity $(x{x}^2)x=x(x^{2}x).$ We show that, if moreover, the norm of ${𝒜}$ comes from an inner product and ${𝒜}$ contains a nonzero central element then ${𝒜}$ is finite dimensional and isomorphic to either $ℝ,ℂ$, $\stackrel{\ast }{ℂ}$, $ℍ$, $\stackrel{\ast }{ℍ}$, $𝕆$, or $\stackrel{\ast }{𝕆}$.

Let $R$ be a commutative ring and let $N$ be a submodule of $R^n$ which consists of columns of a matrix $A=(a_{ij})$ with $a_{ij}\in R$ for all $1\le i\le n$, $j\in \mathrm{\Lambda }$, where $\mathrm{\Lambda }$ is an index set. For every $\mu =\{j_1,\dots ,j_q\}\subseteq \mathrm{\Lambda }$, let I${}_{\mu }(N)$ be the ideal generated by subdeterminants of size $q$ of the matrix $(a_{ij}:1\le i\le n,j\in \mu )$. Let $M=R^n/N$. In this paper, we obtain a constructive description of T($M$) and we show that when $R$ is a local ring, $M$/T($M$) is free of rank $n-q$ if and only if I${}_{\mu }(N)$ is a principal regular ideal, for some $\mu =\{j_1,\dots ,j_q\}\subseteq \mathrm{\Lambda }$. This improves a lemma of Lipman which asserts that, if I($M$) is the $(m-q)$th Fitting ideal of $M$ then I($M$) is a regular principal ideal if and only if $N$ is finitely generated free and $M$/T($M$) is free of rank $m-q.$

We study stability conditions on the Calabi–Yau-$N$ categories associated to an affine type $A_n$ quiver which can be constructed from certain meromorphic quadratic differentials with zeroes of order $N-2$. We follow Ikeda’s work to show that this moduli space of quadratic differentials is isomorphic to the space of stability conditions quotient by the spherical subgroup of the autoequivalence group. We show that the spherical subgroup is isomorphic to the braid group of affine type $A$ based on the Khovanov–Seidel–Thomas method.

We present an elementary approach to characterizing Lie polynomials on the generators $A,B$ of an algebra with a defining relation in the form of a twisted commutation relation $AB=\sigma (BA)$. Here, the twisting map $\sigma$ is a linear polynomial with a slope parameter, which is not a root of unity. The class of algebras defined as such encompasses $q$-deformed Heisenberg algebras, rotation algebras, and some types of $q$-oscillator algebras, the deformation parameters of which, are not roots of unity. Thus, we have a general solution for the Lie polynomial characterization problem for these algebras.

Let $G$ be a group. The enhanced power graph of $G$ is a graph with vertex set $G$ and two distinct vertices $x$ and $y$ are adjacent if there exists $z\in G$ such that $x=z^m$ and $y=z^n$ for some $m,n\in \mathbb{N}$. Also, a vertex of a graph is called dominating vertex if it is adjacent to every other vertex of the vertex set. Moreover, an enhanced power graph is said to be a dominatable graph if it has a dominating vertex other than the identity element. In an article of 2018, Bera and his coauthor characterized all abelian finite groups and nonabelian finite $p$-groups such that their enhanced power graphs are dominatable (see [2]). In addition as an open problem, they suggested characterizing all finite nonabelian groups such that their enhanced power graphs are dominatable. In this paper, we try to answer their question. We prove that the enhanced power graph of finite group $G$ is dominatable if and only if there is a prime number $p$ such that $p\left|\right|Z(G)|$ and the Sylow $p$-subgroups of $G$ are isomorphic to either a cyclic group or a generalized quaternion group.

We adapt and generalize results of Loganathan on the cohomology of inverse semigroups to the cohomology of ordered groupoids. We then derive a five-term exact sequence in cohomology from an extension of ordered groupoids, and show that this sequence leads to a classification of extensions by a second cohomology group. Our methods use structural ideas in cohomology as far as possible, rather than computation with cocycles.

Wu–Liu–Ding algebras are a class of affine prime regular Hopf algebras of GK-dimension one, denoted by $D(m,d,\xi )$. In this paper, we consider their quotient algebras $D^{\prime }(m,d,\xi ),$ which are a new class of non-pointed semisimple Hopf algebras. We describe the Grothendieck rings of $D^{\prime }(m,d,\xi )$ when $d$ is odd. It turns out that the Grothendieck rings are commutative rings generated by three elements subject to some relations. Then we compute the Casimir numbers of the Grothendieck rings for $m=1$ and $m=3$.

In [E. V. Zhuravlev and A. S. Monastyreva, Compressed zero-divisor graphs of finite associative rings, Siberian Math. J.  61(1) (2020) 76–84.], we found the graphs containing at most three vertices that can be realized as the compressed zero-divisor graphs of some finite associative ring. This paper deals with associative finite rings whose compressed zero-divisor graphs have four vertices. Namely, we find all graphs containing four vertices that can be realized as the compressed zero-divisor graphs of some finite associative ring.

It is shown that every finitely generated right $R$-module is almost injective if and only if every cyclic right $R$-module is almost injective, if and only if $R/J(R)$ is a right $SV$-ring with $\mathrm{\text{Loewy}}(R_R)\le 2$ and there is a finite set of orthogonal idempotents ${\{e_i\}}_{i\in I}$ in $R$ such that $e_{i}R$ is an injective local right $R$-module of length two for every $i\in I$ and $J(R)={\oplus }_{i\in I}J(e_{i}R)$.

In 1993, Sim proved that all the faithful irreducible representations of a finite metacyclic group over any field of positive characteristic have the same degree. In this paper, we restrict our attention to non-modular representations and generalize this result for — (1) finite metabelian groups, over fields of positive characteristic coprime to the order of groups, and (2) finite groups having a cyclic quotient by an abelian normal subgroup, over number fields.

We show that the term equivalence between MV-algebras and MV-semirings lifts to involutive residuated lattices and a class of semirings called involutive semirings. The semiring perspective leads to a necessary and sufficient condition for the interval $[d,1]$ to be a subalgebra of an involutive residuated lattice, where $d$ is the dualizing element. We also import some results and techniques of semimodule theory in the study of this class of semirings, generalizing results about injective and projective MV-semimodules. Indeed, we note that the involution plays a crucial role and that the results for MV-semirings are still true for involutive semirings whenever the Mundici functor is not involved. In particular, we prove that involution is a necessary and sufficient condition in order for projective and injective semimodules to coincide.

A theorem of Lichtenbaum states, that every proper, regular curve ${𝒞}$ over a discrete valuation domain $R$ is projective. This theorem is generalized to the case of an arbitrary valuation domain $R$ using the following notion of regularity for non-noetherian rings introduced by Bertin: the local ring ${{𝒪}}_{{𝒞},x}$ of a point $x\in {𝒞}$ is called regular, if every finitely generated ideal $I$ has finite projective dimension. The generalization is a particular case of a projectivity criterion for proper, normal $R$-curves: such a curve ${𝒞}$ is projective if for every irreducible component $Y$ of its closed fiber ${{𝒞}}_m$ there exists a closed point $P$ of the generic fiber of ${𝒞}$ such that the Zariski closure $\overline{P}$ meets $Y$ and meets ${{𝒞}}_m$ in regular points only.

In the hyperalgebra ${{𝒰}}_r$ of the $r$th Frobenius kernel ${({\mathrm{\text{SL}}}_2)}_r$ of the algebraic group ${\mathrm{\text{SL}}}_2$, we construct a basis of the ${{𝒰}}_r$-module generated by a certain element which was given by the author before. As its applications, we also prove some results on the ${{𝒰}}_r$-modules and the algebra ${{𝒰}}_r$.

In this paper, we will introduce the concept of sympathetic 3-Lie algebras and show that some classical properties of semi-simple 3-Lie algebras are still valid for sympathetic 3-Lie algebras. We prove that every perfect 3-Lie algebra $L$ contains a greatest special sympathetic ideal $M$, and there exists a solvable ideal of $L$ denoted by $P(L)$ which is the greatest among the solvable ideals $I$ of $L$ for which $I\cap M=\{0\}$. And we show that there exists a sympathetic subalgebra $m$ of $L$ such that $L=m\dotplus P(L)$ and $L$ is a sympathetic 3-Lie algebra if and only if $P(L)=\{0\}$. Moreover, we also study the ideals $I$ of a 3-Lie algebra $L$ such that $L/I$ is a sympathetic 3-Lie algebra and investigate some properties about them.

We provide explicit constructions for various ingredients of right exact monoidal structures on the category of finitely presented functors. As our main tool, we prove a multilinear version of the universal property of so-called Freyd categories, which in turn is used in the proof of correctness of our constructions. Furthermore, we compare our construction with the Day convolution of arbitrary additive functors. Day convolution always yields a closed monoidal structure on the category of all additive functors. In contrast, right exact monoidal structures for finitely presented functor categories are not necessarily closed. We provide a necessary criterion for being closed that relies on the underlying category having weak kernels and a so-called finitely presented prointernal hom structure. Our results are stated in a constructive way and thus serve as a unified approach for the implementation of tensor products in various contexts.

In this paper, we have introduced copure-direct-injective modules. A right $R$-module $M$ is said to be copure-direct-injective if every copure submodule of $M$ isomorphic to a direct summand of $M$ is itself a direct summand. We have studied properties of copure-direct-injective modules. We characterized rings over which every (cofinitely generated, free, projective) module is copure-direct-injective. We have examined for which rings or under what conditions copure-direct-injective modules are direct-injective, quasi-injective, copure-injective, injective. Also we have compared copure-direct-injective modules with pure-direct-injective modules.

Given an affine algebra $R=P/I$, where $P=K[x_1,\dots ,x_n]$ is a polynomial ring over a field $K$ and $I$ is an ideal in $P$, we study re-embeddings of the affine scheme $\mathrm{\text{Spec}}(R)$, i.e. presentations $R\cong P^{\prime }/I^{\prime }$ such that $P^{\prime }$ is a polynomial ring in fewer indeterminates. To find such re-embeddings, we use polynomials $f_i$ in the ideal $I$ which are coherently separating in the sense that they are of the form $f_i=z_i-g_i$ with an indeterminate $z_i$ which divides neither a term in the support of $g_i$ nor in the support of $f_j$ for $j\ne i$. The possible numbers of such sets of polynomials are shown to be governed by the Gröbner fan of $I$. The dimension of the cotangent space of $R$ at a $K$-linear maximal ideal is a lower bound for the embedding dimension, and if we find coherently separating polynomials corresponding to this bound, we know that we have determined the embedding dimension of $R$ and found an optimal re-embedding.

In this paper, we introduce the notions of $S_{\varphi }$-polynomial and $S$-minimal value set polynomial where $\varphi$ is a polynomial over a finite field ${𝔽}_q$ and $S$ is a finite subset of an algebraic closure of ${𝔽}_q$. We study some properties of these polynomials and we prove that the polynomials used by Garcia, Stichtenoth and Thomas in their work on good recursive tame towers are $S_{\varphi }$-minimal value set polynomials for $\varphi =x^m$, whose $S$-value sets can be explicitly computed in terms of the monomial $x^m$.

Let $R$ be a domain. It is proved that if $\mathrm{\text{wG-gldim}}(R)<\infty$, then the class of finitely generated $G$-flat modules and the class of finitely generated $G$-projective modules coincide. It is also proved that an integrally closed domain $R$ is a Prüfer domain if and only if $\mathrm{\text{FP}}-{\mathrm{\text{id}}}_R(R)\le 1$, if and only if $\mathrm{\text{wG-gldim}}(R)\le 1$.