Graph-Induced by Modules via Tensor Product  

作　　者：Mohammad Jarrar Mohammad Jarrar(Department of Applied Mathematics, Palestine Technical University-Kadoorie, Tulkarem, Palestine)

机构地区：[1]Department of Applied Mathematics, Palestine Technical University-Kadoorie, Tulkarem, Palestine

出　　处：《Applied Mathematics》2024年第12期840-847,共8页应用数学(英文)

摘　　要：This paper investigates the connections between ring theory, module theory, and graph theory through the graph G(R)of a ring R. We establish that vertices of G(R)correspond to modules, with edges defined by the vanishing of their tensor product. Key results include the graph’s connectivity, a diameter of at most 3, and a girth of at most 7 when cycles are present. We show that the set of modules S(R)is empty if and only if R is a field, and that for semisimple rings, the diameter is at most 2. The paper also discusses module isomorphisms over subrings and localization, as well as the inclusion of G(T)within G(R)for a quotient ring T, highlighting that the reverse inclusion is not guaranteed. Finally, we provide an example illustrating that a non-finitely generated module M does not imply M⊗M=0. These findings deepen our understanding of the interplay among rings, modules, and graphs.This paper investigates the connections between ring theory, module theory, and graph theory through the graph G(R)of a ring R. We establish that vertices of G(R)correspond to modules, with edges defined by the vanishing of their tensor product. Key results include the graph’s connectivity, a diameter of at most 3, and a girth of at most 7 when cycles are present. We show that the set of modules S(R)is empty if and only if R is a field, and that for semisimple rings, the diameter is at most 2. The paper also discusses module isomorphisms over subrings and localization, as well as the inclusion of G(T)within G(R)for a quotient ring T, highlighting that the reverse inclusion is not guaranteed. Finally, we provide an example illustrating that a non-finitely generated module M does not imply M⊗M=0. These findings deepen our understanding of the interplay among rings, modules, and graphs.

关 键 词：Graph Theory Commutative Ring Tensor Product CONNECTED DIAMETER Semisimple Ring 

分 类 号：O15[理学—数学]