Preface to the Special Issue “Advanced Research in Pure and Applied Algebra”

This is a continuation of the work initiated with a previous Special Issue entitles “Advanced Research in Pure and Applied Algebra” published in the MDPI journal Mathematics. Among the 48 submissions received for this Special Issue, the editors selected ten articles that successfully passed the peer-review process, and were then published in the journal in the period from March 2024 to November 2025. The selected contributions delve into the intricate structures of rings, algebras, and their representations, exploring deep interconnections with quantum mechanics, group theory, and topological algebra.

The works presented in this volume reflect a broad spectrum of contemporary algebraic research. A significant theme is the exploration of various types of derivations and their impact on the structure of rings. For instance, Alsowait et al. in (Contributions 2), in their two contributions, investigate how generalized reverse derivations and generalized (α, β)-derivations influence the commutativity of quotient rings modulo a prime ideal, providing essential conditions under which these structures become commutative. Complementing this, Hummdi et al. in (Contributions 3) present a comprehensive study on homoderivations in semiprime rings, establishing several conditions that force these maps to act as commuting maps on Lie ideals, thus generalizing several classical results in this area. Another prominent thread is the study of non-commutative and deformed algebraic structures. Zhu and Teng offer a sophisticated treatment of the cohomology and crossed modules of modified Rota–Baxter pre-Lie algebras, linking their findings to infinitesimal deformations and extension theory. In a similar way, Zhong and Tang bridge the gap between one- and two-parameter quantum groups by elucidating the precise connections between their derivation algebras, providing a valuable tool for parallel development in both theories. Furthermore, Sun et al. undertake a detailed analysis of the ribbon elements in the quantum double of a generalized Taft–Hopf algebra, offering a complete classification based on the parity of the parameters involved. The special issue also features novel algebraic constructions. Saad et al. in (Contributions 5) introduce the intriguing concept of i-commutative rings, a new class defined by idempotent-driven commutativity conditions, and explore their fundamental properties with illustrative matrix examples. Cheng contributes to tropical algebra by establishing equivalent standard forms for a class of tropical matrices and describing the structure of their centralizer groups, revealing new insights into their algebraic symmetry. The connection between algebra and quantum physics is powerfully demonstrated by Nieto-Chaupis, who derives the canonical commutation relation of quantum mechanics from the Witt algebra (Virasoro algebra with null central charge). This groundbreaking work suggests a profound and previously unexplored link between fundamental quantum observables and this infinite-dimensional Lie algebra. Finally, Al-Omari and Al-Shomrani advance graded module theory by introducing and studying G-weak graded rings and modules, proving a Maschke-type equivalence of categories for strongly graded rings in this generalized setting.

The Guest Editor wishes to express their deepest gratitude to all the authors for their excellent contributions, which have significantly enriched this Special Issue. We are also profoundly indebted to the numerous anonymous reviewers for their rigorous peer review, insightful comments, and invaluable suggestions that greatly enhanced the quality of the manuscripts. Furthermore, we acknowledge the excellent collaboration with the publisher, the constant assistance provided by the MDPI associate editors in bringing this project to the end, and the great support of the Managing Editor of this Special Issue, Ms. Jialin Su.