Non-parallel graph of submodules of a module
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 th...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | fas |
| Published: |
Shahid Chamran University of Ahvaz
2024-02-01
|
| Series: | مدلسازی پیشرفته ریاضی |
| Subjects: | |
| Online Access: | https://jamm.scu.ac.ir/article_19109_e03dad34acdcc1f293eb90754259dddc.pdf |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | 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. |
|---|---|
| ISSN: | 2251-8088 2645-6141 |