Intersection graph of idealizations

• Mahmoodi, Akram ^[1] ; Vahidi, Alireza ^[1] ; Manaviyat, Raufeh ^[1] ; Alipour, Roghaieh ^[1]
  1. [1] Payame Noor University

     Payame Noor University

     Irán

     
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 75, Fasc. 3, 2024, págs. 693-702
• Idioma: inglés
• DOI: 10.1007/s13348-023-00407-7
• Texto completo no disponible (Saber más ...)
• Resumen

  • Let R be a commutative ring with identity. The intersection graph of ideals of a ring R is an undirected simple graph denoted by \Gamma (R) whose vertices are in a one-to-one correspondence with non-zero proper ideals and two distinct vertices are joined by an edge if and only if the corresponding ideals of R have a non-zero intersection. Let M be a unitary R-module and let R\ltimes M be the idealization of M in R. In this paper, we investigate the interplay between the algebraic properties of R\ltimes M and the graph-theoretic properties of \Gamma (R\ltimes M) . Under some conditions on the ring R and the module M, we determine the exact form of ideals of R\ltimes M and characterize all rings R and modules M for which \Gamma (R\ltimes M) is a star graph. Also, we give a necessary and sufficient condition under which R\ltimes M is uniform and then we conclude that when \Gamma (R\ltimes M) is a complete graph. Among other results, some graph-theoretic properties of \Gamma (R\ltimes M) such as domination number, connectedness and grith are obtained.

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