Journal of Algebra and Its Applications

The Chermak–Delgado lattice of a finite group $G$ is a sublattice of the subgroup lattice of $G$ that has attracted interest since its discovery. In this paper, we show that every subgroup of $G$ in the Chermak–Delgado lattice is subnormal in $G$ with subnormal depth bounded by both the depth and height function of the Chermak–Delgado lattice; we provide a nontrivial example showing that our bounds are sharp. We also show that determining whether a given subgroup $K$ is in the Chermak–Delgado lattice can be decided by examining only those subgroups of $G$ that are comparable with $K$.

Extriangulated categories were introduced by Nakaoka and Palu by extracting the similarities between exact categories and triangulated categories. Nakaoka and Palu introduced the notion of concentric twin cotorsion pairs in extriangulated categories. In this paper, let ${𝒫}=(({𝒮},{𝒯}),({𝒰},{𝒱}))$ be a concentric twin cotorsion pair in an extriangulated category and ${𝒵}={𝒰}\cap {𝒯}$, ${ℐ}={𝒮}\cap {𝒯}={𝒰}\cap {𝒱}$, we prove that ${𝒵}/{ℐ}$ has one-sided triangulated structure.

Let $R={\oplus }_{n\in {\mathbb{Z}}_0}{R}_n$ be a positively graded ring which is a sub-ring of strongly graded ring of type $\mathbb{Z}$, where $R_0$ is a Noetherian prime ring. We define a concept of ${\mathbb{Z}}_0$-invariant maximal order and show that $R$ is a maximal order if and only if $R_0$ is a ${\mathbb{Z}}_0$-invariant maximal order. If $R$ is a maximal order, then we completely describe all $v$-invertible ideals. As an application, we show that $R$ is a generalized Dedekind prime ring if and only if $R_0$ is a ${\mathbb{Z}}_0$-invariant generalized Dedekind prime ring. We give examples of ${\mathbb{Z}}_0$-invariant generalized Dedekind prime rings but neither generalized Dedekind prime rings nor maximal orders.

A numerical semigroup is irreducible if it cannot be obtained as an intersection of two numerical semigroups containing it properly. If we only consider numerical semigroups with the same Frobenius number, that concept is generalized to atomic numerical semigroup. Based on a previous one developed to obtain all irreducible numerical semigroups with a fixed Frobenius number, we present an algorithm for computing all atomic numerical semigroups with a fixed Frobenius number.

If $𝜃$ is a subgroup property, a group $G$ is said to satisfy the double chain condition on $𝜃$-subgroups if it admits no infinite double sequences

Neural rings and ideals, as introduced by Curto, Itskov, Veliz-Cuba, and Youngs, are useful algebraic tools for organizing the combinatorial information of neural codes. However, they are neither graded nor local, and consequently, many algebraic approaches to understanding such rings fail. In this paper, we introduce an operation, called “polarization,” that allows us to relate neural ideals with squarefree monomial ideals, which are very well studied and known for their nice behavior in commutative algebra. We use this notion of polarization to construct free resolutions of neural ideals. We also relate the structure of polarizations of neural ideals to combinatorial properties of codes.

Let $R$ be a ring (not necessarily commutative) and $({ℒ},{𝒜})$ a bi-complete duality pair. We investigate the notions of (flat-typed) $({ℒ},{𝒜})$-Gorenstein rings, which unify Iwanaga–Gorenstein rings, Ding–Chen rings, AC-Gorenstein rings and Gorenstein $n$-coherent rings. Using an abelian model category approach, we show that ${\mathrm{\text{K}}}_{\mathrm{\text{ac}}}(\mathrm{\text{Proj}})$ and ${\mathrm{\text{K}}}_{\mathrm{\text{ac}}}(\mathrm{\text{Inj}})$, the homotopy categories of all exact complexes of projective and injective $R$-modules, are triangulated equivalent whenever $R$ is a flat-typed $({ℒ},{𝒜})$-Gorenstein ring.

In this paper, first we study dual representations and tensor representations of Hom-pre-Lie algebras. Then we develop the cohomology theory of regular Hom-pre-Lie algebras in terms of the cohomology theory of regular Hom-Lie algebras. As applications, we study linear deformations of regular Hom-pre-Lie algebras, which are characterized by the second cohomology groups of regular Hom-pre-Lie algebras with the coefficients in the regular representations. The notion of a Nijenhuis operator on a regular Hom-pre-Lie algebra is introduced which can generate a trivial linear deformation of a regular Hom-pre-Lie algebra. Finally, we introduce the notion of a Hessian structure on a regular Hom-pre-Lie algebra, which is a symmetric nondegenerate 2-cocycle with the coefficient in the trivial representation. We also introduce the notion of an ${𝒪}$-operator on a regular Hom-pre-Lie algebra, by which we give an equivalent characterization of a Hessian structure.

Additive decompositions over finite fields were extensively studied by Brawely and Carlitz. In this paper, we study the additive decomposition of polynomials over unique factorization domains.

Let ${𝔽}_q$ be the finite field of order $q$, where $q$ is a power of odd prime $p$. Assume that $\gamma$, $\lambda$ are nonzero elements of the finite field ${𝔽}_q$ such that ${\gamma }^{p^s}=\lambda$. In this paper, we determine the $b$-distance of $\lambda$-constacyclic codes with generator polynomials ${(x^{\eta }-\gamma )}^i$ of length $\eta p^s$, where $b\le \eta$ and $0\le i\le p^s$. As an application, all maximum distance separable (MDS) $b$-symbol constacyclic codes of length $\eta p^s$ over ${𝔽}_q$ are established. Among other results, we construct several classes of new MDS symbol-pair codes with minimum symbol-pair distance six or seven by using repeated-root cyclic codes of length $4p$ and $\ell p$, respectively, where $\ell$ is an odd prime.

In this paper, we study the relative negative $K$-groups $K_{-n}(f)$ of a map $f:X\to S$ of schemes. We prove a relative version of Weibel’s conjecture; i.e. if $f:X\to S$ is a smooth affine map of Noetherian schemes with $dimS=d$, then $K_{-n}(f)=0$ for $n>d+1$ and the natural map $K_{-n}(f)\to K_{-n}(f\times {𝔸}^r)$ is an isomorphism for all $r>0$ and $n>d.$ We also prove a vanishing result for relative negative $K$-groups of a subintegral map.

This paper explores various notions of projective, injective and flat dimensions, and the different versions of each dimension are compared to each other. Homological characterizations of these dimensions are provided in terms of bounded resolutions and vanishing of appropriate derived functors over a non-positive DG-ring. Some homological identities related to these homological invariants, such as Auslander–Buchsbaum formula, Bass formula and derived depth formula, are proved over a commutative Noetherian DG-ring.

We show that every polynomial in $\mathbb{Z}[x]$ defines an endomorphism of the $d$-ary rooted tree induced by its action on the ring ${\mathbb{Z}}_d$ of $d$-adic integers. The sections of this endomorphism also turn out to be induced by polynomials in $\mathbb{Z}[x]$ of the same degree. In the case of permutational polynomials acting on ${\mathbb{Z}}_d$ by bijections, the induced endomorphisms are automorphisms of the tree. In the case of ${\mathbb{Z}}_2$, such polynomials were completely characterized by Rivest in [Permutation polynomials modulo $2^w$, Finite Fields Appl. 7(2) (2001) 287–292]. As our main application, we utilize the result of Rivest to derive the condition on the coefficients of a permutational polynomial $f(x)\in \mathbb{Z}[x]$ that is necessary and sufficient for $f$ to induce a level transitive automorphism of the binary tree, which is equivalent to the ergodicity of the action of $f(x)$ on ${\mathbb{Z}}_2$ with respect to the normalized Haar measure.

In this paper, we introduce the concept of the associate class graph of zero-divisors of a commutative ring $R$, denoted by ${\mathrm{\Gamma }}_A(R)$. Some properties of ${\mathrm{\Gamma }}_A(R)$, including the diameter, the connectivity and the girth are investigated. Utilizing this graph, we present a new class of counterexamples of Beck’s conjecture on the chromatic number of the zero-divisor graph of a commutative ring.

The notion of a PRINC domain was introduced by Salce and Zanardo [Products of elementary and idempotent matrices over integral domains, Linear Algebra Appl. 452 (2014) 130–152], motivated by the investigation of the products of idempotent matrices with entries in a commutative domain. An integral domain $R$ is a PRINC domain if every two-generated invertible ideal of $R$ is principal. PRINC domains are closely related to the notion of a unique comaximal factorization domain, introduced by McAdam and Swan [Unique comaximal factorization, J. Algebra 276 (2004) 180–192]. In this paper, we prove that there exist large classes of PRINC domains which are not comaximal factorization domains, using diverse kinds of constructions. We also produce PRINC domains that are neither comaximal factorization domains nor projective-free.

We study rings $R$ which are sums of a subring $R_1$ and an additive subgroup $R_2$. We prove that if $R_1$ is a prime radical ring and $R_2$ satisfies a polynomial identity, then $R_1$ is nilpotent modulo the prime radical of $R$. Additionally, we show that if $R_1$ is a $\mathrm{\text{PI}}$ ring, then the prime radical of $R_1$ is nilpotent modulo the prime radical of $R$. We also obtain a new condition equivalent to Koethe’s conjecture.

Let $k$ be an algebraically closed field of characteristic 0. Further, $R=\frac{k[X_1,\dots ,X_n]}{〈f+1〉}$, where $f$ is a polynomial in $k[X_1,\dots ,X_n]$ such that $f\in 〈\frac{\partial f}{\partial X_1},\dots ,\frac{\partial f}{\partial X_n}〉$. We show that the module of derivations of $R$, namely $DerR$ is generated by $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ elements. We also compute the generators explicitly. It is well known that, in such cases, $DerR$ is stably free of rank $n-1$.

As a special case, when $n=3$ and $f$ is quasi-homogeneous, we give an explicit minimal generating set for $DerR$, consisting of two derivations.

In this paper, we define and study weak monoidal Hom-Hopf algebras, which generalize both weak Hopf algebras and monoidal Hom-Hopf algebras. Let $H$ be a weak monoidal Hom-Hopf algebra with bijective antipode and let ${\mathrm{\text{Aut}}}_{wmHH}(H)$ be the set of all automorphisms of $H$, we introduce a category ${}_H{{𝒲}{ℳ}{\mathscr{H}}{𝒴}{𝒟}}^H(\alpha ,\beta )$ with $\alpha ,\beta \in {\mathrm{\text{Aut}}}_{wmHH}(H)$ and construct a braided $T$-category ${𝒲}{ℳ}{\mathscr{H}}{𝒴}{𝒟}(H)$ having all the categories ${}_H{{𝒲}{ℳ}{\mathscr{H}}{𝒴}{𝒟}}^H(\alpha ,\beta )$ as components.

From Morita theoretic viewpoint, computing Morita invariants is important. We prove that the intersection of the center and the $n$th (right) socle $Z{S}^n(A):=Z(A)\cap {\mathrm{\text{Soc}}}^n(A)$ of a finite-dimensional algebra $A$ is a Morita invariant; this is a generalization of important Morita invariants — the center $Z(A)$ and the Reynolds ideal $Z{S}^1(A)$.

As an example, we also studied $Z{S}^n(FG)$ for the group algebra FG of a finite $p$-group $G$ over a field $F$ of positive characteristic $p$. Such an algebra has a basis along the socle filtration, known as the Jennings basis. We prove certain elements of the Jennings basis are central and hence form a linearly independent subset of $Z{S}^n(FG)$. In fact, such elements form a basis of $Z{S}^n(FG)$ for every integer $1\le n\le p$ if $G$ is powerful. As a corollary we have ${\mathrm{\text{Soc}}}^p(FG)\subseteq Z(FG)$ if $G$ is powerful.

Let $F$ be a finite field, $R=T_n(F)$ the ring of all $n\times n$ upper triangular matrices over $F$, $Z(R)$ the set of all zero-divisors of $R$, i.e. $Z(R)$ consists of all $n\times n$ upper triangular singular matrices over $F$. The total graph of $R=T_n(F)$, denoted by ${𝒯}(\mathrm{\Gamma }(R))$, is a graph with all elements of $R$ as vertices, and two distinct vertices $x,y\in R$ are adjacent if and only if $x+y\in Z(R)$. In this paper, we determine all automorphisms of the total graph ${𝒯}(\mathrm{\Gamma }(R))$ of $R=T_n(F)$.