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The annihilator graph of modules over commutative rings. (English) Zbl 1488.13022

Summary: Let \(M\) be a module over a commutative ring \(R\), \(Z_*(M)\) be its set of weak zero-divisor elements, and if \(m\in M\), then let \(I_m=(Rm:_R M)= \{r\in R : rM\subseteq Rm\}\). The annihilator graph of \(M\) is the (undirected) graph \(AG(M)\) with vertices \(\tilde{Z_*}(M)=Z_*(M)\setminus\{0\}\), and two distinct vertices \(m\) and \(n\) are adjacent if and only if \((0:_R I_mI_nM)\neq (0:_R m)\cup (0:_R n)\). We show that \(AG(M)\) is connected with diameter at most two and girth at most four. Also, we study some properties of the zero-divisor graph of reduced multiplication-like \(R\)-modules.

MSC:

13A70 General commutative ring theory and combinatorics (zero-divisor graphs, annihilating-ideal graphs, etc.)
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