On the torsion graph and von Neumann regular rings
Keyword(s):
Let R be a commutative ring with identity and M be a unitary R-module. A torsion graph of M, denoted by ?(M), is a graph whose vertices are the non-zero torsion elements of M, and two distinct vertices x and y are adjacent if and only if [x : M][y : M]M = 0. In this paper, we investigate the relationship between the diameters of ?(M) and ?(R), and give some properties of minimal prime submodules of a multiplication R-module M over a von Neumann regular ring. In particular, we show that for a multiplication R-module M over a B?zout ring R the diameter of ?(M) and ?(R) is equal, where M , T(M). Also, we prove that, for a faithful multiplication R-module M with |M|?4,?(M) is a complete graph if and only if ?(R) is a complete graph.
1974 ◽
Vol 17
(2)
◽
pp. 283-284
◽
Keyword(s):
2006 ◽
Vol 2006
◽
pp. 1-6
Keyword(s):
1971 ◽
Vol 4
(1)
◽
pp. 57-62
◽
Keyword(s):
Keyword(s):
1969 ◽
Vol 12
(4)
◽
pp. 417-426
◽
Keyword(s):
1974 ◽
Vol 19
(1)
◽
pp. 89-91
◽
Keyword(s):
2009 ◽
Vol 08
(05)
◽
pp. 601-615
Keyword(s):
2011 ◽
Vol 39
(9)
◽
pp. 3242-3252
◽