A torsion graph determined by equivalence classes of torsion elements and associated prime ideals
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In this paper, we define the torsion graph determined by equivalence classes of torsion elements and denote it by $A_{E}(M)$. The vertex set of $A_{E}(M)$ is the set of equivalence classes $\{[x]\mid x\in T(M)^{*}\}$, where two torsion elements $x, y\in T(M)^{*} $ are equivalent if $ann(x)= ann(y)$. Also, two distinct classes $[x]$ and $[y]$ are adjacent in $A_{E}(M)$, 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 $A_{E}(M)$ has degree two if and only if it is an associated prime of $M$.
Abstract
In this paper, we define the torsion graph determined by equivalence classes of torsion elements and denote it by $A_{E}(M)$. The vertex set of $A_{E}(M)$ is the set of equivalence classes $\{[x]\mid x\in T(M)^{*}\}$, where two torsion elements $x, y\in T(M)^{*} $ are equivalent if $ann(x)= ann(y)$. Also, two distinct classes $[x]$ and $[y]$ are adjacent in $A_{E}(M)$, 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 $A_{E}(M)$ has degree two if and only if it is an associated prime of $M$.
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