2016/04/06 by D. D. Anderson, Driss Bennis, Anderson, D. D. +5 · 1 citation
Mathematics · #13A02 #13A05 #13A15 #13B99 #13E05 #13F05 #13F30 (Primary) #16S99 #17A99 (Secondary) #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1604.01486
openalex publication_date 2016/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The notion of trivial extension of a ring by a module has been extensively studied and used in ring theory as well as in various other areas of research like cohomology theory, representation theory, category theory and homological algebra. In this paper we extend this classical ring construction by associating a ring to a ring R and a family M=(Mi)i=1n of n R-modules for a given integer n≥ 1. We call this new ring construction an n-trivial extension of R by M. In particular, the classical trivial extension will be just the 1-trivial extension. Thus we generalize several known results on the classical trivial extension to the setting of n-trivial extensions and we give some new ones. Various ring-theoretic constructions and properties of n-trivial extensions are studied and a detailed investigation of the graded aspect of n-trivial extensions is also given. We end the paper with an investigation of various divisibily properties of n-trivial extensions. In this context several open questions arise.