Extending structures for Rota-Baxter family Hom-associative algebrasWang, Junwen; Zhang, Yuanyuan; Chu, Yanjun
doi: 10.1080/00927872.2025.2500691pmid: N/A
Abstract In this paper, we first define extending datums and unified products of Rota-Baxter family Hom-associative algebras, and theoretically solve the extending structure problem. Moreover, we consider flag datums as an application, and give an example of the extending structure problem. We then introduce matched pairs of Rota-Baxter family Hom-associative algebras, and theoretically solve the factorization problem. Finally, we define deformation maps on a Rota-Baxter family Hom extending structure, and theoretically solve the classifying complements problem.
Socle of a C*-algebraRouzbehani, Mohammad; Amini, Massoud; Asadi, Mohammad B.
doi: 10.1080/00927872.2025.2503303pmid: N/A
Abstract We introduce the notion of socle for a C*-algebra and study C*-algebras with essential socle (property (ES)) and those with residual (ES). We show that in the class of C*-algebras with residual (ES), a C*-algebra is strongly purely infinite iff it is purely infinite iff it is weakly purely infinite, giving a partial answer to a question of Kirchberg and Rørdam. For a separable or postliminal C*-algebra A with Goldie dimension n, we show that the basic structure spaces of A and its socle both have cardinal n.
The influence of s-semipermutable subgroups on the p F -hypercenter of a finite groupSun, Jian; Yu, Haoran; Xu, Xiaowei
doi: 10.1080/00927872.2025.2505063pmid: N/A
Abstract Let p be a prime dividing the order of a finite group G, e a nonnegative integer, S ∈ Syl p ( G ) , NEG , and P a normal subgroup of S containing S ∩ N with | P | ≥ p e + 1 . Suppose that | S ∩ N | ≤ p e , and for every normal subgroup H of P with order p e , H ∩ N is s-semipermutable in G. Let U = ( Φ ( P ) ∩ N ) O p ′ ( N ) . In this paper, we prove that UEG , and N / U is p-hypercyclically embedded in G / U . The above result generalizes some main results of Lei et al. [Weakly s-semipermutable subgroups and the p F -hypercenter of finite groups, Commun. Algebra (2022)].
Auslander conditions and cotorsion pairsLiu, Zhiyang; Wu, Dejun; Wang, Yongduo
doi: 10.1080/00927872.2025.2505068pmid: N/A
Abstract Let Λ be an artin algebra. Assume that A C ( D A C ) is the class of left Λ -modules satisfying the (dual) Auslander condition. It is shown that Λ Λ satisfies the Auslander condition if and only if A C is a resolving subcategory of Λ - Mod if and only if D A C is a coresolving subcategory of Λ - Mod if and only if each injective left Λ -module satisfies the dual Auslander condition if and only if each Gorenstein injective left Λ -module satisfies the dual Auslander condition. As an application, it is proven that if Λ satisfies the Auslander condition, then Λ is a Gorenstein ring if and only if ( ⊥ ( D A C < ∞ ) , D A C < ∞ ) is a cotilting-like cotorsion pair.
Averaging operators on groups and Hopf algebrasZhang, Huhu; Gao, Xing
doi: 10.1080/00927872.2025.2505071pmid: N/A
Abstract Rota-Baxter operators on groups were studied quite recently. Motivated mainly by the fact that weight zero Rota-Baxter operators and averaging operators are Koszul dual to each other, we propose the concepts of averaging group and averaging Hopf algebra, and study relationships among them and the existing averaging Lie algebras. We also show that an averaging group induces a disemigroup and a rack, respectively. As the free object is one of the most significant objects in a category, we also construct explicitly the free averaging group on a set.
Multiplicative lattices with absorbing factorizationReinhart, Andreas; Ulucak, Gülşen
doi: 10.1080/00927872.2025.2505079pmid: N/A
Abstract In [24], Yassine et al. introduced the notion of 1-absorbing prime ideals in commutative rings with nonzero identity. In this article, we examine the concept of 1-absorbing prime elements in C-lattices. We investigate the C-lattices in which every element is a finite product of 1-absorbing prime elements (we denote them as OAFLs for short). Moreover, we study C-lattices having 2-absorbing factorization (we denote them as TAFLs for short).
Automorphism groups of C-Cayley graphs on almost simple groupsTian, Yao; Li, Xiaogang
doi: 10.1080/00927872.2025.2506704pmid: N/A
Abstract A Cayley graph Cay ( G , S ) is called a C-Cayley graph if S ⊊ G ∖ { e } is a generating set which is closed under conjugation and taking inverse. Let G be an almost simple group which is an abelian extension over its socle T and S ⊊ G ∖ { e } a generating set which is closed under conjugation and taking inverse. If there exist cosets T z 1 , … , T z m not contained in S such that G = 〈 ∪ i = 1 m ( T z i ∩ S ) 〉 , then it was shown in this paper that the automorphism group of the C-Cayley graph Cay ( G , S ) is isomorphic to ( R ( G ) ⋊ Aut ( G , S ) ) ⋊ 〈 τ G 〉 , where τ G is the map on G sending every element to its inverse. As a consequence, the automorphism group of the C-Cayley graph Cay ( G , S ) is completely determined in this paper when |G:T| is either 1 or a prime. This generalizes previous work on the automorphism groups of C-Cayley graphs Cay ( G , S ) when G is either a symmetric group or an alternative group.
Relative Gorenstein projective and injective modules on ring extensionsGuo, Shufeng; Zou, Zhiyong; Huang, Yanxia; Huang, Yifei
doi: 10.1080/00927872.2025.2506708pmid: N/A
Abstract A ring extension is a ring homomorphism that preserves identities. Motivated by the theory of Gorensteion homological algebra, we introduce the definitions of relative Gorensteion projective and relative Gorensteion injective modules over arbitrary ring extensions and study their basic properties. Let f : S → R be a ring extension. We prove that the category of all relative Gorenstein projective modules is a relative resolving subcategory of R ‐Mod , which is closed under direct sums and direct summands. Additionally, every R-module with finite relative Gorenstein projective dimension admits a relative Gorenstein projective precover. Furthermore, we provide a series of homological descriptions of relative Gorenstein projective dimensions. Dually, all the results concerning relative Gorenstein projective modules have a corresponding relative Gorenstein injective version. Additionally, we introduce the concept of quasi-Frobenius extensions, and demonstrate that quasi-Frobenius extensions encompass both semisimple extensions and Frobenius extensions.