Journal of Algebra and Its Applications

It is well known that the lattice of subspaces of a vector space over a field is modular. We investigate under which conditions this lattice is orthocomplemented with respect to the orthogonality operation. Using this operation, we define closed subspaces of a vector space and study the lattice of these subspaces. In particular, we investigate when this lattice is modular or orthocomplemented. Finally, we introduce splitting subspaces as special closed subspaces and we prove that the poset of splitting subspaces and the poset of projections are isomorphic orthomodular posets. The vector spaces under consideration are of arbitrary dimension and over arbitrary fields.

Recall that a ring $R$ is called right ${\aleph }_0$-injective if every homomorphism from a countably generated right ideal of $R$ to $R_R$ can be extended to a homomorphism from $R_R$ to $R_R$. These rings are not only a natural generalization of self-injective rings but also strongly connected with regularities of rings. Let $RG$ be the group ring of a group $G$ over a ring $R$. It is proved that $RG$ is right ${\aleph }_0$-injective if and only if (i) $R$ is right ${\aleph }_0$-injective; (ii) $G$ is finite; (iii) for each countably generated right ideal $I$ of $RG$, any right $R$-homomorphism from $I_R$ to $R_R$ can be extended to a right $R$-homomorphism from $R{G}_R$ to $R_R$.

Let $R={\oplus }_{n\in \mathbb{Z}}{R}_n$ be a strongly graded ring of type $\mathbb{Z}$ such that $R_0$ is a prime Goldie ring with its quotient ring $Q_0$. It is shown that the following three conditions are equivalent: (i) $R_0$ is a $\mathbb{Z}$-invariant generalized Dedekind ring ($G$-Dedekind ring for short), (ii) $R$ is a $G$-Dedekind ring and (iii) $R$ is a graded $G$-Dedekind ring. We describe all invertible ideals of $G$-Dedekind rings in terms of $R_0$ and $Q^g={\oplus }_{n\in \mathbb{Z}}{Q}_0{R}_n$. We provide counterexamples of $\mathbb{Z}$-invariant $G$-Dedekind rings which are not $G$-Dedekind rings.

In this note, we use the theory of test ideals and Cartier algebras to examine the interplay between the tight and integral closures in a local ring of positive characteristic. Using work of Schwede, we prove the abundance of strong test ideals, recovering some older fundamental results, and use this approach in concrete computations. In the second part of the paper, the case of Stanley–Reisner rings is fully examined.

We continue the study of Galois connections between the lattices of preradicals of two rings $A$ and $B$ induced by an adjoint pair of functors between the categories $A$-Mod and $B$-Mod. In this paper, we focus on the functor triple induced by any ring homomorphism $f:A\to B$, and particularly when it is a ring epimorphism. We give additional results when the epimorphism is flat and when it is projective.

To unify different definitions of smash Hom-products in a Hom-bialgebra $H$, we firstly introduce the notion of $𝜃$-twisted smash Hom-product $B{\#}^{𝜃}H$. Secondly, we find necessary and sufficient conditions for the twisted smash Hom-product $B{\#}^{𝜃}H$ and the twisted smash Hom-coproduct $B{\times }^{\omega }H$ to afford a Hom-bialgebra, which generalize the well-known Radford’s biproduct and the Hom-biproduct obtained in [H. Li and T. Ma, A construction of the Hom-Yetter–Drinfeld category, Colloq. Math. 137 (2014) 43–65]. Furthermore, we introduce the notion of the category of $\varpi$-Yetter-Drinfeld modules which unifies the ones of Hom-Yetter Drinfeld category appeared in [H. Li and T. Ma, A construction of the Hom-Yetter–Drinfeld category, Colloq. Math. 137 (2014) 43–65] and [A. Makhlouf and F. Panaite, Twisting operators, twisted tensor products and smash products for Hom-associative algebras, J. Math. Glasgow 513–538 (2016) 58]. Finally, we prove that the $(𝜃,\omega )$-twisted Radford’s Hom-biproduct $B_{{\times }^{\omega }}^{{\#}^{𝜃}}H$ is a Hom-bialgebra if and only if $(B,B)$ is a Hom-bialgebra in the category of ${𝜃}^{-1}{\omega }^{-1}{\alpha }_H^2$-Yetter–Drinfeld modules ${}_H^H{{\mathscr{H}}{𝒴}{𝒟}}^{{𝜃}^{-1}{\omega }^{-1}{\alpha }_H^{-2}}$, generalizing the well-known Majid’s conclusion.

Let $\sigma =\{{\sigma }_i|i\in I\}$ be a partition of the set $\mathbb{P}$ of all primes and $G$ a finite group. A group is said to be $\sigma$-primary if it is a finite ${\sigma }_i$-group for some $i$. We say that a ${\sigma }_i$-subgroup $H$ of $G$ satisfies the $\sigma$-Frobenius condition in $G$ if $N_G(H)/C_G(H)$ is a ${\sigma }_i$-group. In this paper, we determine the structure of finite groups in which every non-normal $\sigma$-primary subgroup satisfies the $\sigma$-Frobenius condition.

Let $m$ and $n$ be fixed positive integers. $R$ is called a right $(m,n)$-injective ring if every right $R$-homomorphism from an $n$-generated submodule of the right $R$-module $R^m$ to $R$ extends to one from $R^m$ to $R$; $R$ is called a right $(m,n)$-coherent ring if each $n$-generated submodule of the right $R$-module $R^m$ is a finitely presented right $R$-module. Let $M$ be a right $R$-module. We study the $(m,n)$-injectivity and $(m,n)$-coherence of the endomorphism ring $S$ of $M$. Some applications are also given.

This paper belongs to the series of articles devoted to the classification of deformed mesh algebras of Dynkin types ${𝔸}_n(n\ge 2)$, ${𝔹}_n(n\ge 2)$, ${ℂ}_n(n\ge 3)$, ${𝔻}_n(n\ge 4)$, ${𝔼}_6$, ${𝔼}_7$, ${𝔼}_8$, ${𝔽}_4$, ${𝔾}_2$, and ${𝕃}_n(n\ge 1)$. We prove here that every deformed mesh algebra of type ${𝔽}_4$ is isomorphic to the canonical mesh algebra of type ${𝔽}_4$.

In this paper, we study the conditions under which a module is a strict Mittag–Leffler module over the class ${𝒢}{ℐ}$ of Gorenstein injective modules. To this aim, we introduce the notion of ${𝒢}{ℐ}$-projective modules and prove that over noetherian rings, if a module can be expressed as the direct limit of finitely presented ${𝒢}{ℐ}$-projective modules, then it is a strict Mittag–Leffler module over ${𝒢}{ℐ}$. As applications, we prove that if $R$ is a two-sided noetherian ring, then ${𝒢}{ℐ}$ is a covering class closed under pure submodules if and only if every injective module is strict Mittag–Leffler over ${𝒢}{ℐ}$.

Let $R$ be a commutative ring with nonzero identity and $S\subseteq R$ be a multiplicatively closed subset. In this paper, we study $S$-Artinian rings and finitely $S$-cogenerated rings. A commutative ring $R$ is said to be an $S$-Artinian ring if for each descending chain of ideals ${\{I_n\}}_{n\in \mathbb{N}}$ of $R,$ there exist $s\in S$ and $k\in \mathbb{N}$ such that $s{I}_k\subseteq I_n$ for all $n\ge k.$ Also, $R$ is called a finitely $S$-cogenerated ring if for each family of ideals ${\{I_{\alpha }\}}_{\alpha \in \mathrm{\Delta }}$ of $R,$ where $\mathrm{\Delta }$ is an index set, ${\bigcap }_{\alpha \in \mathrm{\Delta }}{I}_{\alpha }=0$ implies $s({\bigcap }_{\alpha \in {\mathrm{\Delta }}^{\prime }}{I}_{\alpha })=0$ for some $s\in S$ and a finite subset ${\mathrm{\Delta }}^{\prime }\subseteq \mathrm{\Delta }.$ Moreover, we characterize some special rings such as Artinian rings and finitely cogenerated rings. Also, we extend many properties of Artinian rings and finitely cogenerated rings to $S$-Artinian rings and finitely $S$-cogenerated rings.

In this note, we present a new subgroup embedding property that can be considered as an analogue of pronormality in the scope of permutability and Sylow permutability in finite groups. We prove that finite PST-groups, or groups in which Sylow permutability is a transitive relation, can be characterized in terms of this property, in a similar way as T-groups, or groups in which normality is transitive, can be characterized in terms of pronormality.

In this paper, we state a generalization of the ring of integer-valued polynomials over upper triangular matrix rings. The set of integer-valued polynomials over some block matrix rings is studied. In fact, we consider the set of integer-valued polynomials $\mathrm{\text{Int}}{(}_s{L}_n(D))=\{f\in {}_s{L}_n(K)[x]|f{(}_s{L}_n(D)){\subseteq }_s{L}_n(D)\}$ for each $1\le s\le n-1$, where $D$ is an integral domain with quotient field $K$ and ${}_s{L}_n(D)$ is a block matrix ring between upper triangular matrix ring $T_n(D)$ and full matrix ring $M_n(D)$. In fact, we have ${}_{n-1}{L}_n(D)=T_n(D)$. It is known that the sets of integer-valued polynomials $\mathrm{\text{Int}}(T_n(D))=\{f\in T_n(K)[x]|f(T_n(D))\subseteq T_n(D)\}$ and $\mathrm{\text{Int}}(M_n(D))=\{f\in M_n(K)[x]|f(M_n(D))\subseteq M_n(D)\}$ are rings. We state some relations between the rings $\mathrm{\text{Int}}(T_s(D)),\mathrm{\text{Int}}(M_{n-s}(D))$ and the partitions of $\mathrm{\text{Int}}{(}_s{L}_n(D))$. Then, we show that the set $\mathrm{\text{Int}}{(}_s{L}_n(D))$ is a ring for each $1\le s\le n-1$. Further, it is proved that if the ring $\mathrm{\text{Int}}(M_{n-s}(D))$ is not Noetherian then the ring $\mathrm{\text{Int}}{(}_s{L}_n(D))$ is not Noetherian, too. Finally, some properties and relations are stated between the rings $\mathrm{\text{Int}}{(}_s{L}_n(D))$, $\mathrm{\text{Int}}(T_s(D))$ and $\mathrm{\text{Int}}(M_{n-s}(D))$.

In this paper, we give a recursive formula for the interesting PBW basis $E_A$ of composition subalgebras ${ℭ}_{▵}(n)$ of Ringel–Hall algebras ${ℌ}_{▵}(n)$ of cyclic quivers after [Generic extensions and canonical bases for cyclic quivers, Canad. J. Math. 59(6) (2007) 1260–1283], and another construction of canonical bases of $U_v^{+}({\widehat{𝔰𝔩}}_n)$ from the monomial bases $m^{(A)}$ following [Multiplication formulas and canonical basis for quantum affine, $𝔤{𝔩}_n$, Canad. J. Math. 70(4) (2018) 773–803]. As an application, we will determine all the canonical basis elements of $U_v^{+}({\widehat{𝔰𝔩}}_2)$ associated with modules of Loewy length $\le 3$. Finally, we will discuss the canonical bases between Ringel–Hall algebras and affine quantum Schur algebras.

We prove that the depth functions of cover ideals of balanced hypergraph have the nonincreasing property. Furthermore, we also give a bound for the index of depth stability of these ideals.

In the present note, we study configurations of codimension 2 flats in projective spaces and classify those with the smallest rate of growth of the initial sequence. Our work extends those of Bocci, Chiantini in ${\mathbb{P}}^2$ and Janssen in ${\mathbb{P}}^3$ to projective spaces of arbitrary dimension.

Let $I(G)$ be the edge ideal of a bicyclic graph $G$ with a dumbbell as the base graph. In this paper, we characterize the Castelnuovo–Mumford regularity of $I(G)$ in terms of the induced matching number of $G$. For the base case of this family of graphs, i.e. dumbbell graphs, we explicitly compute the induced matching number. Moreover, we prove that $\mathrm{\text{reg}}I{(G)}^q=2q+\mathrm{\text{reg}}I(G)-2$, for all $q\ge 1$, when $G$ is a dumbbell graph with a connecting path having no more than two vertices.

In this paper, we characterize quasi-Yetter–Drinfeld modules over a Hopf algebra $H$, which was introduced in [Y. Bazlov and A. Berenstein, Braided doubles and rational Cherednik algebras, Adv. Math. 220 (2009), 1466–1530]. We first show that the quasi-Drinfeld center of the category of $H$-modules is equivalent to the category ${}_H^H\mathcal{𝒬}\mathcal{𝒴}\mathcal{𝒟}$ of quasi-Yetter–Drinfeld modules. Next, we prove that ${}_H^H\mathcal{𝒬}\mathcal{𝒴}\mathcal{𝒟}$ is equivalent to the category of generalized Hopf bimodules. Finally, we show that ${}_H^H\mathcal{𝒬}\mathcal{𝒴}\mathcal{𝒟}$ is also equivalent to the category of quasi-coactions over some Majid’s braided group if $H$ is quasi-triangular.

We determine the Gelfand–Kirillov dimension of a weighted Leavitt path algebra $L_K(E,w)$ where $K$ is a field and $(E,w)$ a row-finite weighted graph. Further we show that a finite-dimensional weighted Leavitt path algebra over $K$ is isomorphic to a finite product of matrix rings over $K$.

We define the chordal simplicial complex by using the definition of chordal clutter introduced by Woodroofe. We show that the facet ideal of the chordal simplicial complex is Cohen–Macaulay if and only if it is unmixed. Moreover, we prove that the facet ideal of a chordal simplicial complex has infinitely many nontrivial Cohen–Macaulay modifications.