Journal of Algebra and Its Applications

A braided Frobenius algebra is a Frobenius algebra with a Yang–Baxter operator that commutes with the operations, that are related to diagrams of compact surfaces with boundary expressed as ribbon graphs. A heap is a ternary operation exemplified by a group with the operation $(x,y,z)\mapsto x{y}^{-1}z$, that is ternary self-distributive. Hopf algebras can be endowed with the algebra version of the heap operation. Using this, we construct braided Frobenius algebras from a class of certain Hopf algebras that admit integrals and cointegrals. For these Hopf algebras we show that the heap operation induces a Yang–Baxter operator on the tensor product, which satisfies the required compatibility conditions. Diagrammatic methods are employed for proving commutativity between Yang–Baxter operators and Frobenius operations.

Given a bivariate system of polynomial equations with fixed support sets $A,B$ it is natural to ask which multiplicities its solutions can have. We prove that there exists a system with a solution of multiplicity $i$ for all $i$ in the range $\{0,1,\dots ,|A|-|conv(A)\ominus B|-1\}$, where $A\ominus B$ is the set of all integral vectors that shift B to a subset of $A$. As an application, we classify all pairs $(A,B)$ such that the system supported at $(A,B)$ does not have a solution of multiplicity higher than $2$.

It is now well known that the conditions used by Auslander to define the Gorenstein projective modules on Noetherian rings are independent. Recently, Ringel and Zhang adopted a new approach in investigating Auslander’s conditions. Instead of looking for examples, they investigated rings on which certain implications between Auslander’s conditions hold. In this paper, we investigate the relative counterpart of Auslander’s conditions. So, we extend Ringel and Zhang’s work and introduce other concepts. Namely, for a semidualizing module $C$, we introduce weakly $C$-Gorenstein and partially $C$-Gorenstein rings as rings representing relations between the relative counterpart of Auslander’s conditions. Moreover, we introduce a relative notion of the well-known Frobenius category. We show how useful are $C$-Frobenius categories in characterizing weakly $C$-Gorenstein and partially $C$-Gorenstein rings.

Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal in a polynomial ring over a field $𝕂$. In this paper, we obtain upper bounds for the Castelnuovo–Mumford regularity of symbolic powers of certain classes of edge ideals. We also prove that for several classes of graphs, the regularity of symbolic powers of their edge ideals coincides with that of their ordinary powers.

In this paper, we construct some model structures corresponding Gorenstein $({ℒ},{𝒜})$-modules and relative Gorenstein flat modules associated to duality pairs, Frobenius pairs and cotorsion pairs. By investigating homological properties of Gorenstein $({ℒ},{𝒜})$-modules and some known complete hereditary cotorsion pairs, we describe several types of complexes and obtain some characterizations of Iwanaga–Gorenstein rings. Based on some facts given in this paper, we find new duality pairs and show that ${𝒢}{ℐ}$ is covering as well as enveloping and ${𝒢}{ℱ}$ is preenveloping under certain conditions, where ${𝒢}{ℐ}$ denotes the class of Gorenstein $({ℒ},{𝒜})$-injective modules and ${𝒢}{ℱ}$ denotes the class of Gorenstein $({ℒ},{𝒜})$-flat modules. We give some recollements via projective cotorsion pair $({𝒢}{𝒫},{{𝒢}{𝒫}}^{\perp })$ cogenerated by a set, where ${𝒢}{𝒫}$ denotes the class of Gorenstein $({ℒ},{𝒜})$-projective modules. Also, many recollements are immediately displayed through setting specific complete duality pairs.

The investigation of orthodox $P$-restriction semigroups was initiated by Jones in 2014 as generalizations of orthodox ${}^{*}$-semigroups. The aim of this paper is to further study orthodox $P$-restriction semigroups based on the known results of Jones. After establishing a construction theorem for orthodox $P$-restriction semigroups, we introduce proper $P$-restriction semigroups (which are necessarily orthodox) and prove that every (finite) orthodox $P$-restriction semigroup has a (finite) proper cover. Our results enrich and extend existing results for restriction semigroups and orthodox ${}^{*}$-semigroups.

Let ${𝒢}$ denote the variety generated by infinite-dimensional Grassmann algebras, i.e. the collection of all unitary associative algebras satisfying the identity $[[z_1,z_2],z_3]=0$, where $[z_i,z_j]=z_i{z}_j-z_j{z}_i$. Consider the free algebra $F_3$ in ${𝒢}$ generated by $X_3=\{x_1,x_2,x_3\}$. We call a polynomial $p\in F_3$ symmetric if it is preserved under the action of the symmetric group $S_3$ on generators, i.e. $p(x_1,x_2,x_3)=p(x_{\xi 1},x_{\xi 2},x_{\xi 3})$ for each permutation $\xi \in S_3$. The set of symmetric polynomials forms the subalgebra $F_3^{S_3}$ of invariants of the group $S_3$ in $F_3$. The commutator ideal $F_3^{\prime }$ of the algebra $F_3$ has a natural left $K[X_3]$-module structure, and ${(F_3^{\prime })}^{S_3}$ is a left $K{[X_3]}^{S_3}$-module. We give a finite free generating set for the $K{[X_3]}^{S_3}$-module ${(F_3^{\prime })}^{S_3}$.

Let $(X,{𝒯})$ be a topological space. By the Skula topology (or the $b$-topology) on $X$, we mean the topology $b({𝒯})$ on $X$ with basis the collection of all ${𝒯}$-locally closed sets of $X$, the resulting space $(X,b({𝒯}))$ will be denoted by $b(X)$. We show that the following results hold:

• (1)$b(X)$ is an Alexandroff space if and only if the $T_0$-reflection ${\mathrm{\text{T}}}_0(X)$ of $X$ is a $T_D$-space.
• (2)$b(X)$ is a Noetherian space if and only if ${\mathrm{\text{T}}}_0(X)$ is finite.
• (3)If we denote by $X^{*}$ the Alexandroff extension of $X$, then $b(X^{*})={(b(X))}^{*}$ if and only if $(X,{𝒯})$ is a Noetherian quasisober space.

We also give an alternative proof of a result due to Simmons concerning the iterated Skula spaces, namely, $b(b(b(X)))=b(b(X))$.

A space is said to be clopen if its open sets are also closed. In [R. E. Hoffmann, Irreducible filters and sober spaces, Manuscripta Math. 22 (1977) 365–380], Hoffmann introduced a refinement clopen topology $\mathrm{\text{Clop}}({𝒯})$ of ${𝒯}$: The indiscrete components of $\mathrm{\text{Clop}}(X)$ are of the form $C_x=\overline{\{x\}}\cap {𝒪}(x)$, where $x\in X$ and ${𝒪}(x)$ is the intersection of all open sets of $X$ containing $x$ (equivalently, $C_x=\{y\in X:\overline{\{x\}}=\overline{\{y\}}\}$). We show that $\mathrm{\text{Clop}}(X)=b(b(X))$

Let $R$ and $S$ be commutative rings with identity, $J$ be an ideal of $S$, and let $f:R\to S$ be a ring homomorphism. The amalgamation of $R$ with $S$ along $J$ with respect to $f$ denoted by $R{\bowtie }^{f}J$ was introduced by D’Anna et al. in 2010. In this paper, we investigate some properties of the comaximal graph of $R$ which are transferred to the comaximal graph of $R{\bowtie }^{f}J$, and also we study some algebraic properties of the ring $R{\bowtie }^{f}J$ by way of graph theory. The comaximal graph of $R$, $\mathrm{\Gamma }(R)$, was introduced by Sharma and Bhatwadekar in 1995. The vertices of $\mathrm{\Gamma }(R)$ are all elements of $R$ and two distinct vertices $a$ and $b$ are adjacent if and only if $Ra+Rb=R$. Let ${\mathrm{\Gamma }}_2(R)$ be the subgraph of $\mathrm{\Gamma }(R)$ generated by non-unit elements, and let $J(R)$ be the Jacobson radical of $R$. It is shown that the diameter of the graph ${\mathrm{\Gamma }}_2(R)\setminus J(R)$ is equal to the diameter of the graph ${\mathrm{\Gamma }}_2(R{\bowtie }^{f}J)\setminus J(R{\bowtie }^{f}J)$, and the girth of the graph ${\mathrm{\Gamma }}_2(R)\setminus J(R)$ is equal to the girth of the graph ${\mathrm{\Gamma }}_2(R{\bowtie }^{f}J)\setminus J(R{\bowtie }^{f}J)$, provided some special conditions.

In this paper, we determine the distance Laplacian spectra of graphs obtained by various graph operations. We obtain the distance Laplacian spectrum of the join of two graphs $G_1$ and $G_2$ in terms of adjacency spectra of $G_1$ and $G_2$. Then we obtain the distance Laplacian spectrum of the join of two graphs in which one of the graphs is the union of two regular graphs. Finally, we obtain the distance Laplacian spectrum of the generalized join of graphs $G_i$, where $1\le i\le n$, in terms of their adjacency spectra. As applications of the results obtained, we have determined the distance Laplacian spectra of some well-known classes of graphs, namely the zero divisor graph of ${\mathbb{Z}}_n$, the commuting and the non-commuting graph of certain finite groups like ${\text{D}}_n$ and ${\mathrm{\text{Dic}}}_n$, and the power graph of various finite groups like ${\mathbb{Z}}_n$, ${\text{D}}_n$ and ${\mathrm{\text{Dic}}}_n$. We show that the zero divisor graph and the power graph of ${\mathbb{Z}}_n$ are distance Laplacian integral for some specific $n$. Moreover, we show that the commuting and the non-commuting graph of ${\text{D}}_n$ and ${\mathrm{\text{Dic}}}_n$ are distance Laplacian integral for all $n\ge 2$.

Local dimension is an ordinal valued invariant that is in some sense a measure of how far a ring is from being local and denoted $\mathrm{\text{ldim}}(R)$. The purpose of this paper is to study the local dimension of ring extensions such as homomorphic image, trivial ring extension and the amalgamation of rings.

Let $R$ be a ring and let $n$ be an arbitrary but fixed positive integer. We characterize those rings $R$ whose elements $a$ satisfy at least one of the relations that $a^n+a$ or $a^n-a$ is a nilpotent whenever $n\in \mathbb{N}\setminus \{1\}$. This extends results from the same branch obtained by Danchev [A characterization of weakly J(n)-rings, J. Math. Appl. 41 (2018) 53–61], Koşan et al. [Rings with $x^n-x$ nilpotent, J. Algebra Appl. 19 (2020)] and Abyzov and Tapkin [On rings with $x^n-x$ nilpotent, J. Algebra Appl. 21 (2022)], respectively.

In this study, we study principal admissible representations for the affine Lie superalgebra $𝔬𝔰𝔭{(1|2n)}^{ˆ}$. Using the character formula of irreducible admissible representations of $𝔬𝔰𝔭{(1|2n)}^{ˆ}$, we calculate a character formula of $W$-modules which are obtained from the quantized Drinfeld–Sokolov reduction and principal admissible representations. As a by-product, we obtain the minimal series modules of the Neveu–Schwarz algebra through the $W$-modules arising from the principal admissible modules over $𝔬𝔰𝔭{(1|2)}^{ˆ}$.

In this paper, we show that integral fusion categories with rational structure constants admit a natural group of symmetries given by the Galois group of their character tables. Based on these symmetries, we generalize a well-known result of Burnside from representation theory of finite groups. More precisely, we show that any row corresponding to a non-invertible object in the character table of a weakly integral fusion category contains a zero entry.

Let $G$ be a finite group. A subgroup $H$ of $G$ is called to be $S$-permutable in $G$ if $H$ permutes with all Sylow subgroups of $G$. A subgroup $H$ of $G$ is said to be $SS$-supplemented in $G$ if there exists a subgroup $K$ of $G$ such that $G=HK$ and $H\cap K$ is $S$-permutable in $K$. In this paper, we investigate $p$-nilpotency of a finite group. As applications, we give some sufficient and necessary conditions for a finite group belongs to a saturated formation.

We obtain a common generalization of the results by Wong and Birkenmeier-Kim-Park, respectively, which say that a reduced ring with unity is strongly (respectively, weakly) regular if and only if all of its prime homomorphic images are division rings (respectively, simple domains). Our arguments are different from those in the known proofs and are quite simple. They also give a characterization of weakly regular reduced rings without unity. This characterization implies in particular that the class of weakly regular reduced rings forms a radical class. However, even if a weakly regular reduced ring has no unity, its prime homomorphic images must be simple domains with unity. In the second part of the paper, we study reduced rings whose prime homomorphic images are simple domains (not necessarily with unity).

Noncatastrophic encoders are an important class of polynomial generator matrices of convolutional codes. When these polynomials have coefficients in a finite field, these encoders have been characterized as polynomial left prime matrices. In this paper, we study the notion of noncatastrophicity in the context of convolutional codes when the polynomial matrices have entries in the finite ring ${\mathbb{Z}}_{p^r}$. In particular, we study the notion of zero left prime in order to fully characterize noncatastrophic encoders over the finite ring ${\mathbb{Z}}_{p^r}$. The second part of the paper is devoted to investigate free and column distance of convolutional codes that are free finitely generated ${\mathbb{Z}}_{p^r}$-modules. We introduce the notion of $b$-degree and provide new bounds on the free distances and column distance. We show that this class of convolutional codes is optimal with respect to the column distance and to the free distance if and only if its projection on ${\mathbb{Z}}_p$ is.

In this paper, given a morhism $\phi$ with its pseudo core inverse ${\phi }^{Ⓓ}$ and a morphism $\eta$ such that $1_X+{\phi }^{Ⓓ}\eta$ is invertible, a necessary and sufficient condition and two sufficient conditions are presented under which the additive property, namely ${(\phi +\eta )}^{Ⓓ}={(1_X+{\phi }^{Ⓓ}\eta )}^{-1}{\phi }^{Ⓓ}$ holds. Several interesting results about additive properties of core inverses of bounded linear operators presented in Huang et al. are generalized to the case of pseudo core inverse of morphism. Also, many results regarding additive properties of core-EP inverses of complex matrices studied by Ma and Stanimirović are extended to the cases of morphism.

Let $k$ be a commutative artinian ring and ${𝒜}$ a small Ext-finite Krull–Schmidt $n$-abelian $k$-category with enough projectives and injectives. We introduce two full subcategories ${{𝒜}}_r$ and ${{𝒜}}_l$ of ${𝒜}$ in terms of the representable functors from the stable category of ${𝒜}$ to category of finitely generated $k$-modules. Moreover, we define two additive functors ${\tau }_n$ and ${\tau }_n^{-}$, which are mutually quasi-inverse equivalences between the stable categories of this two full subcategories. We give an equivalent characterization on the existence of $n$-Auslander–Reiten sequences using determined morphisms.

Nottingham algebras are a class of just-infinite-dimensional, modular, $\mathbb{N}$-graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. Homogeneous components of a Nottingham algebra have dimension one or two, and in the latter case they are called diamonds. The first diamond occurs in degree $1$, and the second occurs in degree $q$, a power of the characteristic. Many examples of Nottingham algebras are known, in which each diamond past the first can be assigned a type, either belonging to the underlying field or equal to $\infty$.

A prospective classification of Nottingham algebras requires describing all possible diamond patterns. In this paper, we establish some crucial contributions towards that goal. One is showing that all diamonds, past the first, of an arbitrary Nottingham algebra $L$ can be assigned a type, in such a way that the degrees and types of the diamonds completely describe $L$. At the same time we prove that the difference in degrees of any two consecutive diamonds in any Nottingham algebra equals $q-1$. As a side-product of our investigation, we classify the Nottingham algebras where all diamonds have type $\infty$.

In this paper, we study the derivation module of the ring of invariants of $k[X,Y]$ under the linear action of dihedral groups $D_{n,q}<GL(2,k)$ mentioned in a paper by Riemenschneider [Die Invarianten der endlichen Untergruppen von $GL(2,ℂ)$, Math. Zeitsch. 153 (1977) 37–50]. We obtained an explicit generating set for the derivation module of $R=k{[X,Y]}^{D_{n,q}}$. We show that $\mu ({\mathrm{\text{Der}}}_k(R))\le O(D_{n,q})$.

In this paper, we construct a self-dual complete resolution of a module defined by a pair of embedded complete intersection ideals in a local ring. Our construction is based on a gluing construction of Herzog and Martsinkovsky and exploits the structure of Koszul homology in the embedded complete intersection case. As a consequence of our construction, we produce an isomorphism between certain stable homology and cohomology modules.