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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.

For a commutative ring R with identity, the ideal-based zero-divisor graph, denoted by ?I (R), is the graph whose vertices are {x ? R\I|xy ? I for some y ? R\I}, and two distinct vertices x and y are adjacent if and only if xy?I. In this paper, we investigate an annihilator ideal-based zero-divisor graph, denoted by ?Ann(M)(R), by replacing the ideal I with the annihilator ideal Ann(M) for an R-module M. We also study the relationship between the diameter of ?Ann(M) (R) and the minimal prime ideals of Ann(M). In addition, we determine when ?Ann(M)(R) is complete. In particular, we prove that for a reduced R-module M, ?Ann(M) (R) is a complete graph if and only if R ? Z2?Z2 and M ? M1?M2 for M1 and M2 nonzero Z2-modules.

In this paper, we consider the behavior of polynomial rings over generalized quasi-Baer rings and show that the generalized quasi-Baer condition on a ring R is preserved by many polynomial extensions.