Journal of Algebra and Its Applications

In this paper, we introduce and study the class $S$-${ℱ}$-ML of $S$-Mittag-Leffler modules with respect to all flat modules. We show that a ring $R$ is $S$-coherent if and only if every ideal is in $S$-${ℱ}$-ML, if and only if $S$-${ℱ}$-ML is closed under submodules. As an application, we obtain the $S$-version of Chase Theorem: a ring $R$ is $S$-coherent if and only if any direct product of copies of $R$ is $S$-flat, if and only if any direct product of flat $R$-modules is $S$-flat. Consequently, we provide an answer to the open question proposed by Bennis and El Hajoui [On $S$-coherence, J. Korean Math. Soc. 55(6) (2018) 1499–1512].

We characterize the invertible matrices over a class of semirings such that the set of additively invertible elements is equal to the set of nilpotent elements. We achieve this by studying the liftings of the orthogonal sums of elements that are “almost idempotent” to those that are idempotent. Finally, we show an application of the obtained results to calculate the diameter of the commuting graph of the group of invertible matrices over the semirings in question.

Let $S=k[x_1,\dots ,x_n]$ be a polynomial ring in $n$ variables over a field $k$. When $I$, $J$ and $K$ are monomial ideals of $S$ generated by powers of the variables $x_i$, it is proved that $\mathrm{\text{reg}}(IJK)\le \mathrm{\text{reg}}(I)+\mathrm{\text{reg}}(J)+\mathrm{\text{reg}}(K)$. If $n\le 4$, the same result for the product of a finite number of ideals as above is proved.

In this paper, we study holomorphic $\mathbb{Z}$-graded vertex superalgebras. We prove that all such vertex superalgebras of central charge $8$ and $16$ are purely even. For the case of central charge $24$ we prove that the weight-one Lie superalgebra is either zero, of superdimension $24$, or else is one of an explicit list of 1332 semisimple Lie superalgebras.

This paper gives explicit formulas for the reduction number and the Castelnuovo–Mumford regularity of projective monomial curves.

A subgroup $H$ of a group $G$ is said to be pronormal in $G$ if $H$ and $H^g$ are conjugate in $〈H,H^g〉$ for each $g\in G$. Some problems in Finite Group Theory, Combinatorics and Permutation Group Theory were solved in terms of pronormality, therefore, the question of pronormality of a given subgroup in a given group is of interest. Subgroups of odd index in finite groups satisfy a native necessary condition of pronormality. In this paper, we continue investigations on pronormality of subgroups of odd index and consider the pronormality question for subgroups of odd index in some direct products of finite groups. In particular, in this paper, we prove that the subgroups of odd index are pronormal in the direct product $G$ of finite simple symplectic groups over fields of odd characteristics if and only if the subgroups of odd index are pronormal in each direct factor of $G$. Moreover, deciding the pronormality of a given subgroup of odd index in the direct product of simple symplectic groups over fields of odd characteristics is reducible to deciding the pronormality of some subgroup $H$ of odd index in a subgroup of ${\prod }_{i=1}^t{\mathbb{Z}}_3\wr {\mathrm{\text{Sym}}}_{n_i}$, where each ${\mathrm{\text{Sym}}}_{n_i}$ acts naturally on $\{1,\dots ,n_i\}$, such that $H$ projects onto ${\prod }_{i=1}^t{\mathrm{\text{Sym}}}_{n_i}$. Thus, in this paper, we obtain a criterion of pronormality of a subgroup $H$ of odd index in a subgroup of ${\prod }_{i=1}^t{\mathbb{Z}}_{p_i}\wr {\mathrm{\text{Sym}}}_{n_i}$, where each $p_i$ is a prime and each ${\mathrm{\text{Sym}}}_{n_i}$ acts naturally on $\{1,\dots ,n_i\}$, such that $H$ projects onto ${\prod }_{i=1}^t{\mathrm{\text{Sym}}}_{n_i}$.

Let $n\ge 2$, let ${{ℛ}}_n$ be the graph consisting of one vertex and $n$ loops and let ${{ℛ}}_{n^{-}}$ be its Cuntz splice. Let $L_n=L({{ℛ}}_n)$ and $L_{n^{-}}=L({{ℛ}}_{n^{-}})$ be the Leavitt path algebras over a unital ring $\ell$. Let $C_m$ be the cyclic group on $2\le m\le \infty$ elements. Equip $L_n$ and $L_{n^{-}}$ with their natural $C_m$-gradings. We show that under mild conditions on $\ell$, which are satisfied, for example, when $\ell$ is a field or a principal ideal domain, there are no unital $C_m$-graded ring homomorphisms $L_n\to L_{n^{-}}$ nor in the opposite direction.

Since topoi were introduced, there have been efforts putting mathematics into the context of topoi. Amongst known topoi, the topoi of sheaves or presheaves over a small category are of special interest. We have here as the base topos that of sheaves over a monoid $M$ as a one object category. By means of closure operators we then obtain categories of sheaves related to the right ideals of $M$. These categories have already been studied but we give these categories a more thorough treatment and reveal some additional properties. Namely, for a weak topology determined by a right ideal $I$ of $M$, we show that the category of sheaves associated to this topology is a subtopos of $MSet$ (the presheaves over $M$) and determine the Lawvere–Tierney topology yielding the same subtopos, which is the Lawvere–Tierney topology associated to the idempotent hull of the (not necessarily idempotent) closure operator associated to $I$. We will then find conditions under which the subcategory of separated objects turns out to be a topos, and in the last section, we find conditions under which the category of sheaves becomes a De Morgan topos.

Let $R$ be a commutative ring and $S$ a given multiplicative closed subset of $R$. In this paper, we introduce the new concept of $S$-torsion exact sequences (respectively, $S$-torsion commutative diagrams) as a generalization of exact sequences (respectively, commutative diagrams). As an application, they can be used to characterize two classes of modules that are generalizations of projective modules.

The class of ordinary linear constant coefficient differential equations is naturally embedded into a wider class by associating differential equations to (convergent) Laurent series. It is thought that these more general differential equations can be used as an alternative description of some special functions.

In this paper, we prove an isomorphism theorem for the case of refinement monoids with a group $\mathrm{\Gamma }$ acting on it. Based on this, we show a version of the well-known Jordan–Hölder theorem in this framework. The central result of this paper states that — as in the case of modules — a monoid $T$ has a $\mathrm{\Gamma }$-composition series if and only if it is both $\mathrm{\Gamma }$-Noetherian and $\mathrm{\Gamma }$-Artinian. As in module theory, these two concepts can be defined via ascending and descending chains, respectively.

Let $A_{𝔮}=𝕜〈x,y〉/(x^2,xy+𝔮yx,y^2)$ be the quantum exterior algebras over a field $𝕜$ with char $𝕜\ne 2$, ${\mathrm{\Lambda }}_q$ be the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-Galois coverings of $A_q$. In this paper, a basis of each degree of Hochschild cohomology of ${\mathrm{\Lambda }}_q$, the cup product and the Gerstenhaber bracket on Hochschild cohomology of ${\mathrm{\Lambda }}_q$ are given.

Motivated by the idea of perspectivity of rings and modules, we introduce perspective elements of a ring. We notably prove that perspective elements form a proper subset of special clean elements, and that perspectivity of elements is a left-right symmetric property. An equational characterization of perspective elements is also given and examples are provided.

Let $A\subseteq B$ be an extension of integral domains and $\mathrm{\Gamma }$ a commutative, additive, cancellative torsion-free monoid. Let $B[\mathrm{\Gamma }]$ be the semigroup ring of $\mathrm{\Gamma }$ over $B$ and set ${\mathrm{\Gamma }}^{*}=\mathrm{\Gamma }\setminus \{0\}$. Suppose that $\mathrm{\Gamma }\cap -\mathrm{\Gamma }=\{0\}$. Then $R=A+B[{\mathrm{\Gamma }}^{*}]$ is a subring of $B[\mathrm{\Gamma }]$. In this paper, we study primal elements in $A+B[{\mathrm{\Gamma }}^{*}]$ domains. As an application, we characterize when the construction $A+B[{\mathrm{\Gamma }}^{*}]$ is a (pre-)Schreier domain.

Let $K$ be a field and $D$ be a finite-dimensional central division algebra over $K$. We prove a variant of the Nullstellensatz for $2$-sided ideals in the ring of polynomial maps $D^n\to D$. In the case where $D=K$ is commutative, our main result reduces to the $K$-Nullstellensatz of Laksov and Adkins–Gianni–Tognoli. In the case, where $K=ℝ$ is the field of real numbers and $D$ is the algebra of Hamilton quaternions, it reduces to the quaternionic Nullstellensatz recently proved by Alon and Paran.

Quasi-symmetric designs are the special type of $2$-designs with two block intersection numbers $x$ and $y$. In this paper, we study the automorphism groups of quasi-symmetric designs with block intersection numbers $0$ and $2$. We prove that for a quasi-symmetric design ${𝒟}$ with $x=0$ and $y=2$ satisfying that $\lambda$ does not divide $r$, if $G\le Aut({𝒟})$ is flag-transitive, then $G$ must be point-primitive. We further obtain that $G$ is either of affine type or almost simple type by the O’Nan Scott Theorem. Moreover, when the socle of $G$ is sporadic, ${𝒟}$ is a unique $2$-$(22,6,5)$ design and $G=M_{22}$ or $M_{22}:2$.

The Peirce decomposition of a Jordan algebra with respect to an idempotent is well known. This decomposition was taken one step further and generalized recently by Hall, Rehren and Shpectorov, with their introduction of axial algebras, and in particular primitive axial algebras of Jordan type (PAJs for short). It turns out that these notions are closely related to three-transposition groups and vertex operator algebras. De Medts, Peacock, Shpectorov and M. Van Couwenberghe generalized axial algebras to decomposition algebras which, in particular, are not necessarily commutative. This paper deals with decomposition algebras which are non-commutative versions of PAJs.

Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. In this paper, we show that the idempotent completion of an extriangulated category admits a natural extriangulated structure. As an application, we prove that the idempotent completion of a recollement of extriangulated categories is still a recollement.

For the centrally extended Heisenberg double of SL₂, a classification of its prime and primitive ideals is obtained. For each primitive ideal, an explicit set of generators is given.

The commuting graph ${𝒞}(\mathrm{\Omega },G)$ of a finite group $G$ has vertex set as $\mathrm{\Omega }\subseteq G$, and any two distinct vertices $x,y$ are adjacent if $x$ and $y$ commute with each other. In this paper, we first study the perfect codes of ${𝒞}(\mathrm{\Omega },G)$. We then find the universal adjacency spectra of the join of two regular graphs, join of two regular graphs in which one graph is a union of two regular graphs, and generalized join of regular graphs in terms of adjacency spectra of the constituent graphs and an auxiliary matrix. As a consequence, we obtain the adjacency, Laplacian, signless Laplacian, and Seidel spectra of the above graph operations. As an application of the results obtained, we calculate the adjacency, Laplacian, signless Laplacian, and Seidel spectra of ${𝒞}(G,G)$ for $G\in \{{\mathrm{\text{D}}}_n,{\mathrm{\text{Dic}}}_n,{\mathrm{\text{SD}}}_n\}$, where ${\mathrm{\text{D}}}_n$ is the dihedral group, ${\mathrm{\text{Dic}}}_n$ is the dicyclic group and ${\mathrm{\text{SD}}}_n$ is the semidihedral group. Moreover, we provide the exact value of the spectral radius of the adjacency, Laplacian, signless Laplacian, and Seidel matrix of ${𝒞}(G,G)$ for $G\in \{{\mathrm{\text{D}}}_n,{\mathrm{\text{Dic}}}_n,{\mathrm{\text{SD}}}_n\}$. Some of the theorems published in [F. Ali and Y. Li, The connectivity and the spectral radius of commuting graphs on certain finite groups, Linear Multilinear Algebra 69 (2019) 1–14; T. Cheng, M. Dehmer, F. Emmert-Streib, Y. Li and W. Liu, Properties of commuting graphs over semidihedral groups, Symmetry 13(1) (2021) 103] can be deduced as corollaries from the theorems obtained in this paper.

For a character $\chi$ of a finite group $G$, the number ${\chi }^c(1)=\frac{[G:\mathrm{\text{ker}}\chi ]}{\chi (1)}$ is called the co-degree of $\chi$. In this paper, we classify the finite groups in which at most two irreducible characters have equal co-degrees.