Non-parallel graph of submodules of a module

Document Type : Original Paper

Authors

ِDepartment of mathematics shahid chamran university of ahvaz , ahvaz iran

Abstract

A non-parallel submodules graph of M, denoted by G ∦ (M), is an undirected simple graph whose vertices are in one-to-one correspondence with all non-zero proper submodules of M and two distinct vertices are adjacent if and only if they are not parallel to each other. In this paper, we investigate the interplay between some of the module-theoretic properties of M and the graph-theoretic properties of G ∦ (M) . It is shown that if G ∦ (M) is connected, then diam(G ∦ (M)) ≤ 3 and if G ∦ (M) is not connected, then G ∦ (M) is a null graph. It is proved that G ∦ (M) is null if and only if M contains a unique simple submodule. In particular, M is a strongly semisimple R -module if and only if G ∦ (M) is a complete graph, and from this it follows that if G ∦ (M) is complete, then every R -module with finite Goldie dimension is Artinian and Noetherian. In addition, G ∦ (M) is a finite star graph if and only if M∼= Z pq, for some distinct prime numbers p and q.

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References

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History

  • Receive Date: 16 November 2023
  • Revise Date: 12 March 2024
  • Accept Date: 20 May 2024

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