A TORSION GRAPH DETERMINED BY EQUIVALENCE CLASSES OF TORSION ELEMENTS AND ASSOCIATED PRIME IDEALS

In this paper, we define the torsion graph determined by equivalence classes of torsion elements and denote it by AE(M). The vertex set of AE(M) is the set of equivalence classes {[x] | x ∈ T(M)*}, where two torsion elements x, y ∈ T(M)* are equivalent if ann(x) = ann(y). Also, two distinct classes [x] and [y] are adjacent in AE(M), provided that ann(x)ann(y)M = 0. We shall prove that for every torsion finitely generated module M over a Dedekind domain R, a vertex of AE(M) has degree two if and only if it is an associated prime of M.